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If X is smooth, then: 1) L1 HO(X) ~ Z, 2) If>: L1H1(X).=t H1(XjZ) is an isomorphism, 3) L1 H2(X) ~ NS(X),
4) with respect to 3), the natural transformation
is the first Chem class, and 5) L1Hk(X) = 0 for k > 2. As a consequence of 3) above we have the naturally defined Lefschetz operators L: LB Hk(X) -+ LB+l Hk+2(X) given by multiplication by the class of a fixed, very ample line bundle in L1 H2(X). By 4) above, this map transforms under If> to the standard Lefschetz opertor, given by multiplication by C1 (L). Theorem 7.1 together with the inner and other products, gives the existence of many non-trivial groups L* H*(X). For example, L* H* (IP'n) -+ H* (IP'nj Z) is surjective. This is true also for abelian varieties. Moreover, the :Fe and Hodge filtrations agree for products of elliptic curves. §8. Chern classes. Let X be a variety and denote by Vect~(X) the equivalence classes of rank-q algebraic vector bundles which are generated by their global cross-sections. This space can be identified with 11'0 of the space ~
Mor (X, gq (IP'n»
n
When gq (11m) is the Grassmannian of co dimension -q planes in IP'n. Using results discussed in IlL!, one can define Chern classes for such bundles in morphic cohomology.
206
Theorem 8.1 ([32]). functors
H. BLAINE LAWSON, JR.
For any q
> o there
is a natural transformation of
q
Vec4(X)
~ EBL'H 2,(X) ,=0
with the property that q
Vect~(X)
~ EBH 2'(XjZ) 8=0
is the standard total Chern class.
§9. An existence theorem. Using 8.1 and results of Grothendieck one can prove the following. Theorem 9.1 ([32]). Let X be a smooth projective variety. Then every class in H2* (X j Q) which is Poincare dual to the homology class of a (rational) algebraic cycle is represented by a rational linear combination of effective algebraic cocycles.
In other words at the level of rational cohomology there are at least as many algebraic co cycles as there are algebraic cycles. In the next section we shall discuss an even stronger theorem, namely Poincare duality at the level of L *H* . §10. A Kronecker pairing with L.H*. It is shown in [32] that for any projective variety X there is a pairing
whenever 2p ::; k ::; 2s,
which when p = 0 carries over, under the natural transformation ~, to the standard Kronecker pairing Hk(Xj Z) ® Hk(Xj Z) -+ Z. In the next section we examine an even more striking pairing betwen these theories.
Chapter VI - Duality It is an striking fact the two theories L.H. and L* H· whose definitions are so completely different (one in terms of cycles and other in terms of morphsims) actually admit a Poincare duality map which carries over under the natural transformations cJ> to the standard Poincare duality map. For smooth varieties this map is an isomorphism!
SPACES OF ALGEBRAIC CYCLES
207
§1. Definition. The duality map is generated in an deceptively simple fashion. Suppose X and Y are projective varieties. Then for each s, a 5 s 5 dimc(Y), there is a natural inclusion
(1.1) as the submonoid of codimension-s cycles on X x Y which are equidimensional over X. (See V.2.1). This engenders a map
ZS(XjY)
(1.2)
---4
ZS(X x Y)
of group completions. Suppose now that Y
= c,N,
i.e., consider the two cases Y
= IP'N and Y =
IP'N-l and pass to a quotient. Then (1.2) yields a map
(1.3) where n = dimc(X), and the homotopy equivalence on the right comes from the Algebraic Suspension Theorem: Zp(X) 5:!!! ZP+l(X x C). (See 11.1 and IV. 5). Taking 1I"2s-k in (1.3) gives a Duality homomorphism
which is defined in [33], where the following is proved. Theorem 1.1 ([33]). For any projective variety X of dimension n, the natural transformations to singular theory give a commutative diagram
UHk(X) ~ L n- sH2n-k(X)
~l Hk(Xj Z) ~
l~ H 2n -k(Xj Z)
where'D is the standard Poincare duality map (given by cap product with the fundamental class of X.) §2. The duality isomorphism: L* H* ~ L n - .. H 2n above lead to the following conjecture.
••
The considerations
Conjecture 2.1 (Friedlander-Lawson). For X and Y smooth and projective, the map {1.2} is a homotopy equivalence. E. Friedlander and the author verified this in several cases, including the case 8 = 1. Ofer Gabber then suggested that a general proof could be obtained from a good version of the Chow Moving Lemma for Families. Such a Moving Lemma has now been proved by Friedlander and the author [89]. The result has some independent interest. It holds over arbitrary infinite fields, and applies to
208
H. BLAINE LAWSON, JR.
classical questions concerning the Chow ring. More importantly here, it leads to the following result. Theorem 2.2 ([33]). Conjecture 2.1 is true. In particular, for any smooth projective variety X of dimension n, the duality map
is an isomorphism for all k
~
2s.
An analogous duality result holds for quasi-projective varieties. Details of this appear in [88]. This result has a number of non-obvious consequences. Note for example the isomorphism L S H2s(X) ~ L n - s H 2(n-s) (X) = An-s which relates families of affine varieties over X to cycles modulo algebraic equivalence inside X. Note also that this gives a complete computation of morphic cohomology for a number of spaces, including all generalized flag manifolds (pr'ojective spaces, Grassmannians, etc., c.f. IV.6.l.). In particular, for such spaces we have isomorphisms
for all k,8 with 28
~
k, and the transformations
ZS(X;C') --+ Map(X,ZB(C')) are homotopy equivalences for all n
~
s.
Another consequence of duality is that it gives rise to Gysin "wrong way" maps of L· H· and L.H. for general morphisms between smooth varieties. Such maps were constructed in [28]. Here however the maps have additional naturality properties which have importance for applications of the theory. REFERENCES
[1]
Almgren, F.J. Jr.,Homotopy groups of the integral cycle groups, Topology 1 (1962), 257-299. [2] Atiyah, M.F.,K-theory and reality, Quart. J. Math. Oxford (2), 17 (1966), 367-386. [3] Atiyah M F. and Jones J. D., Topological aspects of Yang-Mills theory, Comm. Math. Phy. 61 (1978), 97-118. [4] Atiyah M.F. and MacDonald I., Commutative Algebra, Addison-Wesley, London, 1969. [5] Bloch, S., Algebraic cycles and higher K-theory, Adv. Math. 61 (1986), 267-304.
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[6]
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Barlet, D., Espace analytique reduit des cycles analytiques complexes compacts d'un espace analytique complex de dimension finie. pages 1-158 Fonctions de plusieurs variables complexes II. (Seminaire F. Norguet 74/75) Lectures Notes in Math Vol. 482, Springer, Berlin, 1975.
[7] Boyer, C.P., Lawson, Jr H.B., Lima-Filho, P., Mann, B., and Michel-
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sohn, M.-L .., Algebraic cycless and infinite loop spaces, Invent. Math., 113 (1993), 373-388. Bloch, S and Ogus, A.,Gerten's conjecture and the homology of schemes, Ann. Scient. Ecole Norm. Sup. (4e) 7 (1974), 181-202. Bott, R., The space of loops on a Lie group, Michigan Math. J. 5 (1958), 35-61. _ _ , The stable homotopy of the classical groups, Ann. of Math. 70 (1959), 179-203. Boyer, C.P., Hurtubise, J.C., Mann, B.M., and Milgram, R.J., The Topology of Instanton Moduli Spaces I: The Atiyah-Jones Conjecture, Ann. of Math., 137 (1993), 561-609. Cohen, F.R., Cohen, R.L., Mann, B.M. and Milgram, R.J., The Topology of Rational Functions and Divisors of Surfaces, Acta Math. 166(3) (1991), 163-221. Chow, W.-L., On the equivalence classes of cycles in an algebraic variety, Ann. of Math. 64 (1956),450-479. Chow, W.-L. and van der Waerden B.L., Zur algebraischen geometrie, IX: mer sugerordnete Formen und algebraische Systeme von algebraischen Mannigfaltigkeiten, Math. Ann. 113 (1937), 692-704. Deligne, P. , Theorie de Hodge II and III, Publ. Math. IHES 40 (1971), 5-58 and 44 (1975) 5-77. de Rham G., VarieUs DifJerentiables, Hermann, Paris, 1960. Dold, A. and Thom, R., Une generalisation de la notion d'espace fibre. Applications aux produits symetriques infinis, C.R. Acad. Sci. Paris 242 (1956), 1680-1682. _ _ , Quasifaserungen und unendliche symmetrische produkte, Ann. of Math. (2) 67 (1956), 230-281. Elizondo, J., The Euler-Chow Series for Toric Varieties, PhD. thesis, SUNY Stony Brook, August, 1992. _ _ , The Euler Series of Restricted Chow Varieties, Composito Math. 94 (1994), 279-310. Friedlander, E., Homology using Chow varieties, Bull. Amer. Math. Soc. 20 (1989), 49-53. _ _ , Algebraic cycles, Chow varieties and Lawson homology, Compositio Math. 77 (1991), 55-93. _ _ , Filtrations on algebraic cycles and homology, to appear in Annales d'Ecole Norm. Sup.
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[24]
_ _, Some computations of algebraic cycle homology, K-Theory, 8 no. 3, (1994), 271-286. [25] Federer, H., Some theorems on integral currents, Trans. Amer. Math. Soc. 117 (1965), 43-67. [26] _ _, Geometric measure theory, Springer-Verlag, New York, 1969. Federer, H. and Fleming, W., Normal and integral currents currents, Ann. of Math. (2)72 (1960), 458-520. [28] Friedlander, E. and Gabber, 0., Cycle spaces and intersection theory, Topological Methods in Modern Math., Publish or Perish Press, Austin, Texas, 1993, 325-370. [29] Friedlander, E. and Mazur, B., Filtrations on the homology of algebraic varieties, Memoir of the Amer. Math. Soc., no. 529 (1994). [30] _ _, Correspondence homomorphisms for singular varieties, to appear in Ann. Inst. Fourier, Grenoble. [31] Friedlander, E. and Lawson, Jr., H.B., A theory of algebraic cocycles, Bull. Amer. Math. Soc. 26 (1992),264-267. [32] _ _ , A theory of algebraic cocycles, Ann. of Math. 136 (1992),361-428. [33] _ _, Duality relating spaces of algebraic co cycles and cycles, Preprint, 1994. [34] Fulton, W., Intersection theory, Springer, New York, 1984. [35] Gabber, 0., Letter to Friedlander, Sept., 1992. [361 Gajer P., The intersection Dold-Thorn Theorem, Ph.D. Thesis, S.U.N.Y. Stony Brook, 1993. [37] Grothendieck, A., La theorie des classes de Chern, Bull. Soc. Math. France 86 (1958),137-154. [381 _ _, Standard conjectures on algebraic cycles, Algebraic Geometry (Bombay Colloquium), Oxford Univ. Press 1969, 193-199. [39] _ _ , Hodge's general conjecture is false for trivial reasons, Topology 8 (1969), 299-303. [40] Hain, R., Mixed Hodge Structures on homotopy groups, Bull. Amer. Math. Soc. 14 (1986), 111-114. [41] _ _, The de Rham homotopy theory of complex algebraic varieties I and II, K-theory I (1987), 171-324, and 481-494. [42] Harvey, R., Holomorphic chains and their boundaries, Several Complex Variables, Proc. Sympos. Pure Math. Vol. 30, Amer. Math. Soc. 1977, 309-382. [431 Hartshorne, R., Algebraic Geometry, Graduate Texts in Math. Springer, Berlin, 1977. [44] Hironaka, H., 7riangulation of algebraic sets, in Algebraic Geometry, Proc. Sympos. Pure Math. 29 (1975), 165-185. [45] Hoyt, W., On the Chow bunches of different projective embeddings of a [27]
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211
projective variety, Amer. J. Math. 88 (1966),273-278. Lawson, H. B. Jr, The topological structure of the space of algebraic varieties,Bull. Amer. Math. Soc. 17, (1987), 326-330. _ _ , Algebraic cycles and homotopy theory, Ann.ofMath. 129 (1989),253291. Lam, T.-K., Spaces of Real Algebraic Cycles and Homotopy Theory, Ph.D. thesis, SUNY,Stony Brook, 1990.
Lewis, J., A Survey of the Hodge Conjecture, Les Publications CRM, Univ. de Montreal, Montreal ,Quebec, 1991. Lima-Filho, P.C., Homotopy groups of cycle spaces, Ph.D. thesis, SUNY, Stony Brook, 1989. _ _ , Lawson homology for quasiprojective varieties, Compositio Math 84 (1992), 1-23. _ _ , Completions and fibrations for topological monoids, Trans. Amer. Math. Soc., 340 (1993), 127-147. _ _, On the generalized cycle map, J. Diff. Geom. 38 (1993), 105-130. _ _ , On the topological group structure of algebraic cycles, Duke Math. J. 75, no. 2 (1994),467-491. Lawson, H.B. Jr, Lima-FiIho, P.C. and M.-L. Michelsohn, M.-L., Algebraic cycles and equivariant cohomology theories, to appear. _ _, The G-suspension theorem for affine algebraic cycles, preprint, 1995. Lawson, H.B. Jr. and Michelsohn, M.-L., Algebraic cycles, Bott periodicity, and the Chern characteristic map, The Math. Heritage of Hermann Weyl, Amer. Math. Soc., Providence, 1988, pp. 241-264. _ _, Algebraic cycles and group actions in Differential Geometry, Longman Press, 1991, 261-278. Lewis, L.G., May, P. and Steinberger, M., Equivariant stable homotopy theory, Lecture in Math., Vol. 1213, Springer, Berlin, 1985. Lawson, H.B. Jr and Yau, S.-T., Holomorphic symmetries, Ann. Sci. Ecole Norm. Sup. (4) 20 (1987), 557-577. MacDonald, I.G., The Poincare polynomial of a symmetric product, Proc. Cambridge. Phil. Soc., 58 (1962) 563-568. McDuff, D. and Segal, G., Homology fibrations and the "group completion" theorem, Invent. Math. 31 (1976), 279-284. May, J.P., Eoo Ring Spaces and Eoo Ring Spectra, Lecture Notes in Math. Vol. 577, Springer, Berlin, 1977. Morrow, J.and Kodaira, K., Complex Manifolds, Holt-Reinhart-Winston, New York, 1971. Mann, B.M. and Milgram, R.J., Some Spaces of Holomorphic Maps to Complex Grassmann Manifolds, J. Diff. Geom. 33 (1991),301-324.
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Moore, J., Semi-simplicial complexes and Postnikov systems, Sympos. Intern. Topologia Algebraica, Univ. Nac. Aut6noma de Mexico and UNESCO, Mexico City, 1958, pp, 232-247. [67] Nori, M., Algebraic cycles and Hodge-theoretic connectivity, Invent. Math. 111 (1993), 349-373 . [68] Roberts, J., Chow's moving Lemma, Algebraic geometry (Oslo 1970, F. Oort Ed.), Wolters-Noordhoff Publ., Groningen, 1972,89-96. [69] Quillen, D., Higher algebraic K-theory I, Lecture Notes in Math. Vol. 341, Springer, (1973), 85-147. [70] Samuel, P., Methodes d'algebre abstrait en geometrie algebrique, Springer, Heidelberg, 1955. [71] Segal, G., The multiplicative group of classical cohomology, Quart. J. Math. Oxford Ser. 26 (1975), 289-293. [72] _ _, The Topology of Rational Functions, Acta Math.143 (1979),39-72. [73] Snaith V.P., The total Chern and Stiefel- Whitney Classes are not infinite loop maps, lllinois J. Math. 21 (1977),300-304. [74] Shafarevich, LR., Basic Algebraic Geometry, Springer, New York, 1974. [75] Steiner, R., Decompositions of groups of units in ordinary cohomology, Quart, J. Math. Oxford 90 (1979),483-494. [76] Suslin A. and Voevodsky, V., Singular homology of abstract algebraic varieties, Harvard Preprint, 1993. [77] Taubes, C.H., The Stable Topology of Self-Dual Moduli Spaces, J. Diff. Geom. 29 (1989), 163-230. [78] Totaro, B., The maps from the Chow variety of cycles of degree 2 to the space of all cycles, MSRI Preprint, 1990. [79] _ _ , The total Chern class is not a map of multiplicative cohomology theories, Univ. of Chicago Preprint, 1991. [80] Whitehead, Elements of Homotopy Theory, Springer, New York, 1974. [81] Gelfand, LM., Krapanov, M.M. and Zelevinsky, A.V., Discriminants, Resultants and Multidimensional Determinants, Birkhauser Press, Boston, 1994. [82] Kraines, D. and Lada, T., A counterexample to the transfer conjecture, In P.Hoffman and V. Snaith (Eds.) Algebraic Topology, Waterloo, L.N.M. no. 741, Springer-Verlag, New York, 1979, pages 588-624. [83] Kozlowski, A., The Evana-Kahn formula for the total Stiefel- Whitney class, Proc. A. M. S. 91 (1984),309-313. [84] _ _, Transfer in the groups of multiplicative units of the classical cohomology rings and Stiefel- Whitney classes, Proc. Res. Inst. Math. Sci. 25 (1989),59-74. [85] Shuota, M. and Yokoi, M., Triangulations of subanalytic sets and locally suanalytic manifolds, Trans A.M.S. 286 (1984),727-750.
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213
Steiner, R., Infinite loop structures on the algebraic k-theory of spaces, Math. Proc. Camb. Philos. Soc. 1 no. 90, (1981),85-111. Flannery, C., Spaces of algebraic cycles and correspondence homomorphisms, to appear in Advances in Math. Friedlander, E., Algebraic cycles on normal quasi-projective varieties, Preprint, 1994. Friedlander, E. and Lawson H.B., Jr., Moving algebraic cycles of bounded degree, Preprint, 1994. Friedlander, E. and Voevodsky, V., Bivariant cycle cohomology, Preprint, 1994. Gajer, P., Intersection Lawson homology, M.S.R.1. Preprint no. 00-94, 1994. Lima-Filho, P., On the equivariant homotopy of free abelian groups on G-spaces and G-spectra, Preprint, 1994. Lawson, H.B. Jr, Lima-Filho, P. and Michelsohn, M.-L., On equivariant algebraic suspension, Preprint, 1994. Michelsohn, M.-L., Steenrod cycles, Preprint, 1994. STATE UNIVERSITY OF NEW YORK AT STONY BROOK STONY BROOK, NEW YORK
SURVEYS IN DIFFERENTIAL GEOMETRY, 1995 Vol. 2 ©1995, International Press
Problems on rational points and rational curves on algebraic varieties Yu. I.
MANIN
§l. Introduction 0.1. Basic problems. In this report, we review some recent results, conjectures, and techniques related to the following questions. Question 1. Let V be a (quasi)projective algebraic variety defined over a number field k. How large is the set of rational points V(k)? Question 2. Let V be a compact Kahler manifold. How large is the set of rational curves in V, or the space of analytic maps pl -+ V? More precisely, in the arithmetic setting we choose a height function hL : V (k) -+ R, and want to understand the behavior of
Nv(H) := card {x E V(k) ! hL(x) :5 H}
(0.1)
as H -+ 00. In the geometric setting, we replace the (logarithmic) height by the degree of the curve with respect to the Kahler class, coinciding with its volume with respect to the Kahler metric (Wirtinger's theorem). IT the degree is bou\lded by H, the space of rational curves is a finite-dimensional complex space, and we migh~ be interested in the number of its irreducible components, their dimensions, their characteristic numbers, etc. 0.2. A heuristic reasoning. In order to see what geometric properties of V influence the behavior of the two sets, let us start with the following naive reasoning. Let V = V (nj d1 , ••. , dr) be a smooth complete intersection in pn given by the equations Fi(xo, ... ,xn ) = 0, i = 1, ... ,r, where Fi is a form of degree di . 0.2.1. Arithmetic setting. Assuming that Fi have integral coefficients we take Q as the ground field. Every rational point is represented by a primitive (n+ I)-ule of integer-valued coordinates x = (xo, ... , x n ) E znp~~ . A standard (exponential) height function is h(x) = maxi(!Xil). There are about Hn+l primitive (n + I)-pies of height :5 H. A form Fi takes about Hd; values on this set. Assume that the probability of taking the zero value is about H- d ;, and that the conditions Fi = 0 are statistically independent. Then we get a conjectural growth order (?) for the number of points of the height::; H in V(Q).
(0.2)
215
RATIONAL POINTS AND CURVES
0.2.2. Geometric setting. Now we will allow Fi to have complex coefficients, and endow Vee) with the metric induced by the Fubini-Study metric on pn. We normalize it in such a way that a line in pn has degree (volume) 1. Consider a projective line p1 = Proj e[to, t1l. Any map cp : p1 ~ pn can be written as
(to: tt}
1--+
(fo(to, tt) : ... : fn(to, t1»
where Ii are forms of some degree k ~ 0 not vanishing identically and relatively prime. Denote by Mk(pn) the space of all (n + I)-pIes of forms of degree k (except (0, ... ,0» up to a common scalar factor. Obviously, ~ p(n+1)(k+1)-1.
Mk(pn)
The space Mk(pn) C Mk(pn) is Zariski open and dense. Similarly, denote by Mk (V) the space of maps p1 ~ V of degree k. Its closure Mk(V) C Mk(pn) is defined by a system of polynomial equations on the coefficients of /i's derived from
Fi (fo (to, td,··· , fn(to, td)
= OJ i = 1, ••• , r.
(0.3)
Clearly, (0.3) furnishes kdi + 1 homogeneous equations of degree di corresponding to the monomials tgt~dj+1-a. It follows that r
r
dimMk(V) ~ (n+l)(k+l)-I- ~)kdi+l)
= k(n+l- Ldi)+dim V;
i=l
(0.4)
i=l r
deg Mk(V) ~
II d~di+1.
(0.5)
i=l
0.3. Discussion. a). Since the geometric degree of a curve corresponds to the logarithmic height of a point (with respect to the same ample class), the r.h.s. of (0.2) and (0.4) predict the same qualitative behavior of the number of points, resp. of the dimension of the space of maps, depending on the sign of n + 1- E~l~' Now, this last number is essentially the anticanonical class of V: r
-Kv ~ Ov(n+ 1-
Ld
i)
(0.6)
i=1
in the Picard group of V. Boldly extrapolating from the complete intersection case, we may expect many rational curves and points when -Kv is ample (V is a Fano manifold), and few when K v is ample. The intermediate case K v = 0 must be more subtle. For example, if we disregard the difference between Mk(V) and Mk(V) and assume that (0.4) is an exact equality, we expect a dim (V)-dimensional family
216
YU.I.MANIN
of parametrized rational curves on V of any degree k. If in addition dim V = 3 = dim Aut pl, we expect only a finite number nk of rational (unparametrized) curves of degree k belonging to V for all k ~ 1. For quintics in p5, this was conjectured by Clemens (cf. below). b). These expectations are fulfilled when dim V = 1 that is, when V is a smooth compact curve. More precisely, when -Kv is ample, genus of V is zero, V may be a non-trivial form of plover a non-closed field k which has no k points. However, after a quadratic extension of k, V will become pl, and the point count with re$pect to an anticanonical height gives an asymptotic formula agreeing with (0.2). Moreover, the count of maps pl ~ pl is unconstrained. When Kv = 0, one gets Nv(H) '" c(logHy/2 in view of the Mordell-Weil theorem for elliptic curves, so that (0.2) is still valid if one interprets the r.h.s. as "O(H e ) for anye > 0". Moreover, there are no maps pI ~ V of degree k~1.
Finally, when Kv > 0 one gets Nv(H) parametrized rational curve is constant.
= 0(1)
(Faltings' theorem), and any
c). Starting with dimension two, the situation becomes much more complex and problematic. Let us start with geometry. For smooth m-dimensional Fano varieties, Mori proved that through every point passes a rational curve of (-Kv)-degree ~ m + 1). Moreover, any two points can be connected by a chain of rational curves. But a quantitative picture of the space Map (Pl, V) remains unknown. For varieties (Kv ample) of general type, we expect only a finite dimensional family of unparametrized rational curves. However, this was,proved only for varieties with ample cotangent sheaf which is a considerably stronger assumption. Finally, for manifolds with Kv = 0 (and Kahler holonomy group SU), physicists recently suggested a fascinating conjectural framework for the curve count which we will review in the second part of this report. Passing to the arithmetic case, let us notice first that (0.2) can be proved by the circle method over Q, when n + 1 is large in comparison with E di and the necessary local conditions are satisfied (see below). On the other hand, already for n = 3, r = 1, d = 3, (0.2) may fail for the following reason: it predicts the linear growth for Nv(H), but V may contain a projective line defined over Q (there are 27 lines over Q) in which case counting points only on this line we already get Nv(H) ~ cH2. Therefore, if anything like (0.2) may be expected in general, we must at least stabilize the situation by allowing ground field extensions and deleting some proper subvarieties tending to accumulate points. Moreover, in the case K v = 0 we may have to delete infinitely many subvarieties to achieve the predicted O(H~) estimate. We elaborate this program in Section 1 below. Its goal, roughly speaking, lies in establishing a (conjectural) direct relation between the distribution of rational points on V and the geometry of rational curves on V. In addition, there exists a well known analogy between rational curves and rational points. In Arakelov geometry, rational points on V become "horizontal arithmetical curves" on a Z model of V, endowed with an Hermitean metric
RATIONAL POINTS AND CURVES
217
at arithmetical infinity. In the framework of this analogy, the height becomes literally an arithmetical intersection index. We want to draw attention to an unexplored aspect of this analogy: what in arithmetics corresponds to the local deformation theory of embedded curves? Here is a relevant fragment of the geometric deformation theory. In the following V denotes a quasiprojective variety defined over an algebraically closed field k, and Map (pi, V) is the locally closed finite quasiprojective scheme parametrizing morphisms pi -+ V. For simplicity, in the next Proposition we consider only the unobstructed case. 0.4. Proposition. Let cp be a morphism pi -+ V, [cp] E Map (pi, V) the corresponding closed point, and Tv the tangent sheaf to V. If HI (pI, cp* (Tv)) = 0, then [cp] is a smooth point, and the local dimension of Map(pl, V) at [cp] equals dimHO(Pi,cp*(Tv)). For a proof of a more general statement, see Mori [19]. Assume now that cp is an immersion, and V is smooth in a neighbourhood of cp(pl). Then we have the following sequence of locally free sheaves on pI:
(0.7) where N[~l is the normal sheaf. Hence N[~l ~ ffi:~fO(mi)' s = dim(V). Recall also that TPi ~ 0(2). We can now prove that (0.4) becomes exact equality locally on Map (PI, V) if cp(V) is nicely immersed infinitesimally: 0.4.1. Corollary. Assume in addition that mi Then [cp] is smooth, and
~
dim[~l Map(pl, V) = degcp*(-Kv)
-1 for all i
= 1, ... , s-1.
+ dim V
(0.8)
which coincides with the r.h.s. of (0.,.0 in the complete intersection case. Proof. The smoothness of [cp] follows from Prop. 0.4. Put now
= {ilmi == -I}, a = card (A), B == {ilmi ~ O}, b = card (B).
A
We have a+b = s -Ii degcp*(-Kv) == 2+ LA mi + LBmj = 2-a+ LBffli (take the determinant of (0.7», and, again from (0.7), dim[~l Map (Pi, V)
=
dimHo('7ju) + dimH)(N[~l) = 3 + LB(mi + 1) = 3 + b + LBmi
= 3 + b + deg cp* (- K v) - 2 + a
= dim V +
deg cp*(-Kv).
In particular, when dim V = 3 and - K v = 0, every immersed curve with normal sheaf O( -I)ffiO( -1) must be isolated because the local dimension ofthe lllap space equals dim V = 3 and this is accounted for by reparametrizations. The simplest example when this may occur generically is that of a smooth quintic threefold V. In fact, H. Clemens conjectured that a generic smooth
218
YU.I.MANIN
quintic contains only finitely many smooth rational curves of arbitrary degree k, and that all of them have normal sheaf O( -1) ffi O( -1). Sh. Katz proved partial results in this direction: see [13], [14]. 0.5. Problem. Establish an analog of the geometric deformation theory for embedded arithmetical curves. Specifically, we have 0.6. Problem. Find conditions on arithmetical normal sheaf (or higher order infinitesimal neighborhoods) of an arithmetical curve which are necessary for the generic point of this curve to lie on a rational curve. (We want to find an exact expression of the feeling that an arithmetical curve is deformable only if its generic point lies on a rational curve). 0.7. Rational curves in other contexts. Besides algebraic geometry and number theory, the study of rational curves was recently motivated by quantum field theory and symplectic geometry. We will finish this Introduction with a brief discussion of some relevant ideas. 0.7.1. Physics. Physicists start with a space of maps Map (82 , V) where the target space V is endowed with a Riemannian metric 9 and an action functional 8: Map (8 2 , V) ~ R. V can be thought of as a space-time with a possibly non-trivial gravity field and topology. Any r.p: 8 2 ~ V defines a world-sheet of an one-dimensional object, a "string", which replaces the classical image of point-particle. Alternatively, one can think about 8 2 as a two-dimensional space-time in its own right. Then (V, g) in a neighborhood of r.p(8) represents classical fields on 8. Action of a virtual world-sheet r.p: 8 2 ~ V is usually given by a Lagrangian density which must be integrated over 8 2 • Here we will look only at the simplest action functional 8(r.p) = f vol (r.p*(g». (0.9)
ls2
In other words, 8(r.p) is just the surface of the world sheet. Non-trivial stationary points of this action are just minimal surfaces. The path integral quantization of this theory in the stationary phase approximation involves a summation over these minimal surfaces Imagine now that (V, g) is not just a Riemannian manifold, but a complex Kahler one. It is well known that in this case minimal surfaces in V (actually, minimal submanifolds of any dimension) are precisely complex subvarieties (Wirtinger's theorem). A physical context in which V acquires a natural Kahler structure arises in string compactification models where V appears as a Planck size compact chunk of space-time adding missing six real dimensions to the classical fourdimensional space-time. 0.7.2. Symplectic geometry. The basic mathematical structure of the classical mechanics is a triple (V 2 n,w,H) where v 2 n is a smooth manifold, w is a closed non-degenerate 2 form on V 2 n, and H is a function on V called Hamiltonian. Given such a triple, we want to understand the geometry of the flow defined by the vector field X on V such that dH = ix(w). In particular,
RATIONAL POINTS AND CURVES
219
we want to know how a domain of initial positions B C V may change with time. Any Hamiltonian flow preserves the symplectic volume v(B) = w n . On the other hand, certain unstable flows like geodesic flows on hyperbolic manifolds severely distort B: a small ball eventually becomes spread allover V forming a fractal-like structure. Nevertheless, (exp(tX)B,w) remains symplectomorphic to B because Lie x(w) = dix(w) + ixdL.J = O. V.I.Arnold in the sixties suggested that exp(tX)B should satisfy some additional constraints displaying then unknown "symplectic rigidity" properties. M.Gromov's work confirmed these expectations. He proved in particular that the unit ball
IB
2n
(Bl
= {xl L
n
x~ < I}, w =
L dxi AdXi+n)
i=1
is not symplectomorphic to any open subset of n
(V1-e:
= {xl Ixl < 1- e}, w = 'LdXi" dXi+n). i=1
Gromov's argument involves rational curves in the following ingenious way. Notice first that in the example above we envision the two symplectic spaces Bl and V1 -e: not in terms of w but rather in terms of the standard Euclidean metric ds 2 = E(dxi)2. But if we are considering pairs (g,w) consisting of a quadratic and an alternate form, say, on a linear space E, there is a natural subclass of such pairs corresponding to Hermitean forms, which can be characterized by the existence of a complex structure J : E ~ E, J2 = -1 such that w(Jx, y) = g(x, y), g(Jx, y) = -w(x, y). Applying this to tangent spaces of a symplectic manifold (V, w) and shifting attention from (w, g) to (w, J) we come to the following notion due to Gromov. An almost complex structure J on V is tamed by w, if g(x, x) := w(Jx,x) > 0 for any tangent vector x, that is, if 9 + iw defines a Hermitean metric on the tangent bundle to V. Now, even though J may be non-integrable, its restriction on surfaces is integrable, so that it makes perfect sense to speak about holomorphic maps pI ~ (V, J). M.Gromov derives his results from a thorough study of such rational curves, establishing existence of curves of small volume. (In a similar vein, rational curves of small degrees play the crucial role in the Mori theory.) E.Witten used Gromov's construction as a deformation device allowing one to correctly count the number of rational curves on Calabi-Yau manifolds; cf. also [15]. This paper is structured as follows. §1 is devoted to the analytic methods to count rational points on projective varieties, whereas §2 reviews the algebrogeometric approach. In §3 we turn to the curve count, explaining the simplest example of Calabi-Yau mirrors. Finally, §4 is devoted to the explanation of toric mirror constructions. For the most part, proofs are omitted.
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220
SECTION I COUNTING RATIONAL POINTS §1. Analytic methods 1.1. Heights on projective varieties. Let k be an algebraic number field. Denote by M" the set of all places of k; for v E k, let kv be the completion of kat v. Define the local norm 1.lv : k~ -t R* by the following condition: if J.t is a Haar measure on k;;, then J.t(aU) = lalvJ.t(U) for each measurable subset U. Let x E pn(k) be a point in a projective space endowed with a homogeneous coordinate system. H coordinates of x are (xo, ... , x n ), Xi E k, put hex)
II
=
mF(lxil v).
(1.1)
vEMk
The product formula shows that this is well defined. More generally, let V be a projective variety defined over k, and L = (L, s) a pair consisting of a very ample invertible sheaf L and a finite set of sections s = {so, ... sn} C rev, L) generating L. For a point x E V (k) and an arbitrary choi~e of a local section u of L non-vanishing at x we put (1.2)
(1.3) In particular, consider the anticanonical height hw-1 on pn(k) defined by the (n + l)-th tensor power of (0(1); {xo, ... ,Xn}). Then hW-l(X) = h(x)n+1 where hex) is given by (1.1). When s in the definition of L is replaced by another generating set of sections, hL is multiplied by exp(O(l)). The resulting set of height functions consists of Weil's heights. There is a different choice of additional structure allowing one to define height functions directly for not necessarily ample sheaves: the Arakelov heights are obtained by choosing an appropriate set of v-adic metrics 1~.lIv on all L ® kv and putting, for L = (L, {1I.lIv}), hL{x) =
II
Ilu(x)lI;l.
vEM.
These heights are also multiplicative with respect to the obvious tensor product, and up to exp 0(1) are independent on the choice of local metrics and coincide with the respective Weil heights. For a subset U C V(k), put
Nu(Lj H) = card {x
E UlhL{x) ~ H}.
(1.4)
For ample L, this number is always finite. We want to understand its behavior as H -t 00. In this section, we review main situations when an asymptotic
221
RATIONAL POINTS AND CURVES
formula for (1.4) is known. In all cases which I am aware of, such a formula is of the type Nu(L;H) = cH.Bu(L)(logH)tu(L)(I + 0(1)) (1.5) for some constants c > 0, .Bu(L) ~ 0, tu(L) ~ O. The archetypal result is the following theorem due to Schanuel: 1.2. Theorem. Put d = [k: Q]. Then -1
Npn(k)(Ld
;H)
= c(n,k)H + h
c(n, k)
= (k(n + 1)
(
{O(H 1 / 2 10gH) O(H 1-1/d(n+1») 2rl +r2 7rr2 ) n+1
1)1/2
for d = n otherwise;
= 1,
(1.6)
R
-(n + I)r 1 +r 2 -1 tv
.
(1.7)
Here h denotes the class number of k, and (k its Dedekind zeta, r1 (resp. r2) is the number of its real (resp. complex) places, D the absolute value of the discriminant, R the regulator, and tv the number of roots of unity in k. The main feature of (1.6) is that N pn(k)(w- 1 ;H) grows asymptotically linearly in H, whatever the dimension n and the ground field k are. This becomes possible only because we have chosen local norms 1.1 .. as Haar multipliers. Therefore the height function (1.1) is non-invariant with respect to ground field extensions: if we replace k by k' :::> k, hex) becomes h'(x) = h(x)[k':kJ so that pn(k) does not contribute to the main term of the asymptotic formula for Npn(k') (w- 1 ; H) : essentially, we count only "new points". Schanuel proved (1.7) by reducing the problem to that of counting lattice points in a large domain. The volume of the domain furnishes the leading term, and if the boundary is not too bad, we get an asymptotic formula. We will now sketch an alternate approach via zeta functions. 1.3. Zetas. Consider the following abstract setting. Let U be a finite or countable set, and hL : U -+ ~ a counting function (this means that Nu(L; H) defined by (1.4) is finite for all H). Assume moreover that Nu(L; H) = O(He) for some c > O. Put (1.8) xEU
The better we understand the analytical properties of Zu(L; s), the more precise information about Nu(L; H) we can obtain. We will distinguish here four levels of precision. Level 0: Convergence abscisse. Put
.B = .Bu(L)
=
inf {a
I Zu(L; s)
converges for Re(s)
> a}.
(1.9)
This is well defined and invariant if one replaces h by exp(O(I))h. In particular, if hL is a Weil or Arakelov ample height, .B depends only on the isomorphism class of the relevant ample sheaf L. It gives the following information about Nu(L; H):
f3 (L) _ u
-
{-oo
if U is finite; lim sup log Nu{L;H) > 0 otherwise. 10gH
-
(1.10)
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222
In other words, if f3
~
0, we have for all
N (L· H) _ u, -
>0:
g
{O(Hf3+~),
Levell: a Tauberian situation. Assume that {3 t = tu(L) ~ 0 we have Zu(Lj s)
(1.11).
n(Hf3-~).
= (8 -
= {3u(L)
~ 0, and for some
(1.12)
{3)-tG(s),
where G(f3) '" 0, and G(s) is holomorphic in a neighborhood of Re(s) ~ {3. In this case
Nu(LjH) =
~~1 ~ (log
H)t-l(l
+ 0(1)).
(1.13)
In particular, assume that U = U1 X ... X Urn, hL(Ul, ... , Urn) = hLl (Ul) ... {3ui(Li),ti tu,(Li) whenever they are defined, and {i I {3 f3i}. Using the zeta-description of these numbers, one readily sees that
hLm(urn)· Put f3i /3 = maxi (f3i), J
= =
=
f3u(L)
=
= /3,
tu(L)
=L
(1.14)
ti.
iEJ
Formula of the type (1.13) is valid for (U, h L ) if the Tauberian condition is assumed only for Ui, hLi with i E J.
Level 2: analytic continuation to a larger halfplane. Instead ofaxiomatizing the situation, I will only remind the contour deformation technique. Let us start with the formula valid for f3' > f3: Nu(Lj H)
=
l
f3 '+ioo
(3'-ioo
HB -Zu(Lj s)ds.
(1.15)
8
In favorable case, one can integrate instead along a vertical line Re(s) = 7 < {3 adding the contribution of poles Zu(Lj s) for 7 < Re(s) < f3'. This contribution constitutes the leading term of the asymptoticsj it will be of the type cHf3 P(log H) where P is a polynomial if Zu(Lj s) has a pole at s f3 as its only singularity in 7 < Re( s) < f3'. The integral over Re( s) = 7 will grow slower, possibly as O(Hf3-~), if Zu has no more poles in Re(s) > 7, and can be appropriately majorized. To accomplish the necessary estimates, one has sometimes to first replace Nu(Lj s) by an appropriate average, and the r.h.s. of (1.15) by something like f3 '+ioo H' Z ( ) . I (3'-ioo 8(8H) U Lj S whIch converges b etter.
=
Level 3: explicit formulas. IT one has a well-behaved meromorphic continuation of Zu(Lj s) to the whole complex plane, one can sometimes push f3' to -00 in (1.15) and obtain a precise formula for Nu(Lj H) as a series over all poles of Zu(Lj s).
RATIONAL POINTS AND CURVES
223
1.4. A generalization of Schanuel's theorem. The behavior of the height zeta-function (1.8) is well understood only for two classes of projective manifolds: a) Abelian varietiesj b) homogeneous Fano manifolds. If U = V(k), V is an Abelian variety, and L is an ample symmetric sheaf on V, one can use Neron-Tate's height hL to count points. Denote by W the image of V(k) in V(k) ® R, and let t be the order of V(k}tors. Then hL(X) = exp(q(x mod V(k}tors) where q is a positive quadratic form on V(k) ®R so that our zeta is a theta-function:
L exp( -q(y)s).
Zu(Lj s) = t
(1.16)
yEW
Hence, if r := rk V(k)
> 0,
we have (3
NV(k) (Lj H)
> 0, and
= clogr / 2 H(1 + 0(1)).
(1.17)
Notice that the convergence abscisse Re(s) = 0 is also the natural boundary for Zu(Ljs). For abelian varieties, Kv = 0 so that (1.17) matches our naive expectation (0.2). Let us turn now to homogeneous Fano varieties. 1.4.1. Theorem. Every homogeneous Fano variety V is isomorphic to a generalized flag space P \ G where G is a semi-simple linear algebraic group, and P is a k-rational conjugacy class of parabolic subgroups. If V(k) i- 0, we can take P to be a parabolic subgroup defined over k. For a proof, see Demazure [10]. Flag spaces P \ G admit a distinguished class of heights which can be defined in terms of Arakelov metrics invariant with respect to maximal compact subgroups of the adelic group of G. For such heights, the zeta function of V = P \ G becomes essentially one of the Langlands-Eisenstein series. Their deep theory developed by Langlands allows one to use the technique of contour integration of the Level 3 above, and prove the following theorem, generalizing 1.2: 1.4.2. Theorem. If V is a homogeneous Fano variety with V(k) i- 0, then for a distinguished anticanonical height we have
Nv(-Kv;H) == Hp(logH)(1
+ H-
E)
(1.18)
where e > 0, and p is a polynomial of degree rk Pic(V) - 1. For a proof, see [12]. In particular, (3v(-Kv) = 1. This theorem can be extended to the distinguished heights corresponding to other invertible L. It must be stressed however that, even for projective line, there are natural situations when the relevant heights are not distinguished. This happens on accumulating Fano subvarieties, when a height is induced from the ambient space: see the next section. In the homogeneous case,the asymptotic is of the same form. A very interesting question of charactering the coefficient of the leading term directly in terms of the anticanonical height was recently attacked by E. Peyre. The simplest variety for which the analytic properties of Zu beyond the convergence abscisse are unknown is the affine Del Pezzo surface of degree 5 over
224
YU.I.MANIN
Q which can be obtained by blowing up four rational points on p2 and then deleting all 10 exceptional curves. One reason for this may be a wrong choice of the function itself. The mirror conjecture on the curve count on, say, threedimensional quintics, furnishes analytic continuation for a geometric version of the height zeta where the contribution of the curve x is (logh(x))3 l~~(Z)' . rather than our simple-minded h(X)-B. It would be quite important to guess a version of Zu(Lj s) with good analytic properties. 1.5. Circle method. We will now briefly explain a classical approach to counting points which is efficient for Fano hypersurfaces and complete intersections (mostly over Q) with many variables. Let X be a finite set, F: X -+ Z a function, and e(a) = e 2 11"ia. Put 8(a)
= 8(X.F) (a) = L
(1.19)
e(aF(x)).
zeX
Then
I F(x) = O} =
card {x E X
11
8(a)da.
(1.20)
A useful version of this formula refers to the case of a vector function F (F1 , ••• , Fr) : X -+ zr. Then a = (al, ... , a r ) varies in a unit cube, aF(x) E aiFi(x) , 8(a) is again defined by (1.19), and
card {x E X
I F(x) = O} =
11 1
..•
1
8(a)OOI . .. da r .
= =
(1.21)
The circle method, when it works, gives a justification to the following heuristic principle: 1.5.1.
Circle principle. •
finite set of rational points a'
Under favorable circumstances, there exists a a(i)
a(i)
= {:hr, ... , =r-} and small cubes I( i) ql qr
centered at
these points ("major arcs") such that
11 1
1 •.•
o
dal ... dar
=
L! i
0
8(a)dal ... dar
+ {a small remainder term}.
[(I)
To get some feeling of why it might be true, and what it implies, let us look at the case r = 1. First of all, the values of 8(a) at rational points are related to the distribution of values of F(x) modulo integers: 8(0)
=
1 card (X)j8(2) = card {x
L
S(~) = q
p
e27riap/q
I F(x)
card {x
even} - card {x
I F(x) == p
I F(x)
odd};
mod q}.
mod q
= [1, ... , N] with large N, F(x) = x 2 , then S(~) is approximately ~ x {a Gauss sum} decreasing as :Jq for large q « N.
If X
RATIONAL POINTS AND CURVES
225
Hence we may expect that Sea) is relatively small (in comparison with the number N of its summands) outside of a neighborhood of the set of rational points with denominators bounded in terms of N. In the classical additive problems with large number of summands k, the remainder term can be effectively damped as k -t 00, because (1.22) For example, in Waring's problem of degree n with k summands, (X, F)
= ([0, ... , [MI/n]], x? + ... + Xk -
M)
so that k
card {(Xi)
I Lxf = M} =
1 I
[Ml/n) e-27riaM(
0
i=l
L
e 27rOZn )kda.
z=o
Below we review some results of W. Schmidt [24] who applied the circle method to the intersections of hypersurfaces in a projetive space over Q. In fact, he worked with the corresponding affine cone, but this only changes the coefficient in the asymptotic formula. 1.5.2. The setting. Consider a finite system of r-forms in s variables of degrees ~ 2:F = {FI , •• . , Fr }, with integral coefficients. Let V be the variety {Fi = O} in the affine space. Let rd be the number of forms of degree d, and r = Ei rio W. Schmidt proved an asymptottic formula of the type (0.2) in the cases where "the number of variables is large, and the forms are not too degenerate." Both conditions are used as a refined substitute for the classical damping effect (1.22). Let us state them more precisely. A. Many variables. The basic bound is written in terms of the number
v(r2, ... ,rk)
= max {s I for some F and some prime p, F(Qp) = 0}.
In other words, s > v(r2" .. , rk), implies p-adic solvability for all p and all F with a given vector degree. B. Degeneracy. The degeneracy is measured in terms of the tensor rank, well known in the computational complexity theory. Specifically, for one form F put h(F) = min {h
I
there exist non-constant forms AI,BI, ... ,Ah,Bh E Q[XI, ... ,Xs ] such that F = AIBI + ... + AhBh }.
For a system of forms of the same degree F = {Fi }, put
Finally, for a general system of forms put hd 1.5.3. Theorelll. Assume that
= h(degree d part of F).
YU.I.MANIN
226
a). hd ~ 24dd!rdkv(r2, ... ,rk)' b). dim VCR) ~ s - L~=2 rio Then the number of integral points of V in
{Ixil
~ H} is
where the constant J.L > 0 is a product of local densities. 'l\uning to the base of the cone V, we again see the linear growth rate with respect to an anti canonical height, at least when this base is only mildly singular so that the anticanonical sheaf exists and is given by the same formula as for the smooth complete intersections.
§2. Algebro-geometric methods 2.1. Accumulating subvarieties. The analytic methods described in §1 work efficiently only for those Fano varieties which are either homogeneous or complete intersections with many variables (or, more invariantly, oflarge index). Moreover, their success seems to be connected with the fact that the rational points are uniformly distributed with respect to a natural Tamagawa measure. Algebra-geometric data suggest that generally we may not expect such a uniformity, and that rational points tend to concentrate upon proper subvarieties. Below we will discuss several ways to make this idea precise. Let U be a quasiprojective variety over a number field k. a. Zariski topology. Denote by V the closure of U(k) in Zariski topology. H a compactification of U is a curve of genus > 1, then V is a proper subvariety of U. This fancy way to state Faltings' theorem leads to the generalized Mordell conjecture: we expect that V is a proper subvariety of U whenever U is birationally equivalent to a variety of general type. Roughly speaking, this means that the description of U(k) can be divided into two subproblems: to understand the distribution of rational points on varieties with K ~ 0, and to understand the distribution of such subvarieties in varieties of general type. This pattern is characteristic for all definitions of accumulation.
=
b. Hausdorff topology. Let k Q. B. Mazur recently suggested that U(Q) may be Hausdorff dense in the space of R-points of its Zariski closure V. If this is universally true, it implies that Z cannot be a Q-Diophantine subset subset of Q so that not all Q-enumerable subsets are Q-Diophantine. (Recall that E C Qn is Q-Diophantine if it is a projection of U(Q) C Qn+m for some affine U defined over Q).In particular, Matiyasevich's strategy of proving (he algorithmic undecidability of Diophantine equations over Z would not work for Q. c. Measure theory. Again for simplicity working over Q consider the limit
227
RATIONAL POINTS AND CURVES
of the averaged delta-distributions over rational points Xi E U(Q) ordered, say, by increasing height. IT such a limit exists, the support of J.L provides a notion of accumulating subset which may be finer than the topological closure. d. Point count according to the polynomial growth rate. The following notion was suggested in [5]: choose a height function hL on (a projective closure of) U and call a Zariski closed subset V C U accumulating with respect to hL if (3u(L)
= (3v(L) > (3u\V(L),
where the growth order (3 is defined by (1.9) or equivalently (1.10). One easily sees that there exists a unique minimal accumulating subset Vi; putting UI = U \ Vi and applying the same reasoning to UI etc, one gets a sequence of Zariski open subsets (2.1) such that Ui \ Ui+! is the minimal hL-accumulating subset in Ui' A description of (2.1) and the corresponding growth order sequence (2.2)
is the natural first goal in understanding U(k), which can be best attacked by algebro-geometric means. We will now report on the results of [18], [17] concerning mostly Fano varieties, in particular surfaces and threefolds. 2.2. Invariant a and reductions. Let V be a projective manifold (we can also allow mild singularities). Denote by N:/f (resp. N~mple) the closure of the cone generated by effective (resp. ample) classes in NS(V) ® R where NS is the Neron-Severi group. For an invertible sheaf L, put a(L)=inf{p/q
I
P,qEZ,q>O,p[L]+qKvEN:If }·
H V is Fano and L is ample then a(L) > O. The following two results allow us to reduce in certain cases the calculation of (3u (L) to that of {3u ( - K v), if a( L) is considered as a computable geometric invariant. 2.2.1. Theorem on the upper bound. aj. For every e > 0, there e:cists a dense Zanski open subset U(e) C V such that/or aI/V C U(e) we have fJu(L) 5 a(L)fJu(-Kv) +e.
(2.3)
b). Q in addition a(L) is rational (and positive), there e.xtsts a dense open subset U C V such that for all U' C U we have (3u,(L) :5 o:(L){3u(-Kv).
(2.4)
Proof. a). Take p/q very close to a(L) such that p[L] + qKv is effective. Then p/q = a(L) + 11 with small 11 > O. Denote by U(p, q) the complement
YU.I.MANIN
228
to the support of base points and fixed components of IpL + qKvl. For all x E U(p, q)(k) , we have hpL+qK(X) ;:: c' > 0 i.e. hL(x) > ch~k(X), sO that f3U(p,q)(L)
b). IT 0:
~
!!.f3U(p,q) (-Kv) q
= (o:(L) + 11)f3U(p,q) (-Kv).
= p/q, we can put U = U(p, q).
Remark. This Theorem shows that it is important to know whether o:(L) is rational for all ample L on Fano manifolds. This is true for surfaces in view of the Mori polyhedrality theorem and the convex duality of and N~mple' For threefolds, V. V. Batyrev showed that it is a (rather non-trivial) consequence of Mori's technique. In higher dimensions, this is an open problem.
N:"
2.2.2. Theorem on the lower bound. manifold V, assume that o:(L)[L]
Given an ample L on a Fano
+ Kv E aN~mple n aN:!f'
(2.5)
Then o:(L) is rational. Assume in addition that o:(L)[L] + Kv := I belongs to exactly one face of aN~mple of codimension one. Then the contraction morphism associated to this face has a fiber F which is a non-singular Fano variety of dimension;:: 1, and we have for any U ::> V, (2.6)
Condition (2.5) is a strong one. However, if it is not satisfied for L, one can sometimes ameliorate the situation by an appropriate birational modification ofV. Whenever both inequalities (2.4) and (2.6) hold, we can get the best possible result f3u(L) o:(L) in the case where f3u(-K) 1 for appropriate open subsets of subsets of V and F. We have already noticed in §1 that analytic methods when applicable give exactly this result. We will show below that this also seems to be a tendency for surfaces and threefolds, but only after deleting the accumulating subvarieties. The following results heavily depend upon classification theorems. Geometric classification is done over a closed ground field; we generally dispose of subtler problems by passing to a finite extension of the ground field.
=
=
2.3. Del Pezzo surfaces. Fano manifolds of dimension two are called the del Pezzo surfaces. They split into ten deformation families. Two of them are homogeneous (P 2 and pi x pi) so that point count on them reduces to the Schanuel's theorem. Family {Va}, 1 ~ a ~ 8, consists of surfaces that can be obtained by blowing up a points on p2 in a sufficiently general position. We call a surface Va split (over k), if these a points can be chosen k-rational. Every surface Va contains a finite number of exceptional curves ("lines"); they are all k-rational if Va is split. Denote by Ua the complement to these lines, and put Aa = Va \ U a . The following Theorem is proved in [18]: 2.3.1. Theorem. Let Va be split. Then the following hold.
RATIONAL POINTS AND CURVES
229
a). fiA,,(-Kv) = 2. b). We have the/allowing estimates/or{3uA(-Kv) :={3a· For k Q: {31 = {34 = 1; {35 ~ 5/4; {36 ~ 5/3. For general k: {31 = ... = {33 = 1; {34 ~ 6/5; (35 ~ 3/2. The results for a = 5 and a = 6 have especially direct Diophantine interpretation, since V5 is an intersection of two quadrics in p4, and V6 is a cubic in p3. We see that if all lines on these surfaces are rational they are accumulating, and, for k = Q, the remainder term Nu,,(-K,H) is O(H 5/4+t:) (resp. O(H5/3+ E )). A proof of Theorem 2.3.1 given in [18] consists of two parts. The cases a ~ 4 are treated directly, by representing Va as a blow-up of p2, comparing height on Va with height on p2, and using explicit number-theoretical properties of the height. The remaining cases are treated via an inductive reasoning which shows that {3a+l ~ :=:{3a'
=
= ...
2.4. Fano threefolds. This case was treated in [17] where the following linear lower bound was established: 2.4.1. Theorem. For any Fano threefold V over a number field k and any Zariski open dense subset U c V, there exists a finite extension k' of k such that if k" contains k', then NU®kll (K, H) > cH for some c > 0 and large H. In particular, /3U®kll ~ 1. The proof is based upon a description of all 104 deformation families of Fano threefolds obtained by Fano, Iskovskih, Shokurov, Mori, and Mukai. Studying this description, one can derive the following: 2.4.2. Main Lemma. Every Fano threefold over a closure of the ground field becomes isomorphic to a member of at least one of the following families: a}. A generalized flag space P \ G. b}. A Fano threefold covered by rational curves C with (Kv.C) ~ 2. c}. A blow-up of varieties of the previous two groups. Group a) is treated via Eisenstein series. For the group b), it suffices to count points on a single rational curve invoking the Schanuel theorem. Finally, a blowup diminishes the anticanonical height in the complement of the exceptional set and increases the number of such points of bounded height. 2.5. Length of arithmetical stratification. We conjecture that for Fano manifolds, the length of the sequence (2.1) of the complements to accumulating subsets is always finite. However, it can be arbitrarily long. 2.5.1. Proposition. For every n ~ 1, there exists a Fano manifold W of dimension 2n over Q and an ample invertible sheaf L on it such that the sequence (2.1) for (W, L) is of length ~ 27n + 1. Proof. For n = 1, take for W a split del Pezzo surface Vs. Representing it as a blow-up of six rational points on p2, denote by A the inverse image of OP2 (1), and by ll, ... ,127 the exceptional classes, of which h, ... , l6 are represented by inverse images of blown up points. Choose a large positive integer N and small Positive integers Cl, ..• , E."6. Take for L a class approximately proportional to -Kv: L = 3NA-(N -cdh - .. ·-(N -c6)l6. Choose the parameters (N,ci) in such a way that (l.,L) i (lj,L) for all i '" jj 1:::; i,j:::; 27j (1.,L) < ~N. Theorem 2.3.1 then shows that the 27 lines will be consecutive accumulating
230
YU.I.MANIN
subvarieties, with the growth orders (L~';)' and the complement to them will have f3 < 3~' so that the total length is at least 28. For n ~ 2, take n pairs (Vi, L,) of this type. Arrange parameters (Ni , ... ,E~) in such a way that the spectra of the growth orders for various (Vi, L,) do not intersect. Then put W = Vl X ... X Vn,L = pri(Ll) ® .•. ®pr~(Ln). From (1.14) one easily sees that the spectrum of the growth orders will have length at least 27n + 1 (one can even get 28n - 1).
EL
2.5.2. Conjecture. If V is a manifold with K v = 0 on which there exist rational curves of arbitrarily high degree defined over a fixed number field, then the arithmetical stratification with respect to any ample sheaf L is infinite, and the consecutive growth orders tend to zero. The first non-trivial case of this conjecture is furnished by certain quartic surfaces, and more general K3-surfaces. In this case, the accumulating subvarieties must consist of unions of rational curves of consecutive L-degrees. However, the problem of understanding rational curves on K3-surfaces is difficult, in particular because it is "unstable": even the rank of the Picard group depends on the moduli. It is expected that some stabilization occurs starting with tree-dimensional Calabi-Yau manifolds. We will devote the next Section to the highly speculative and fascinating picture whose contours were discovered by physicists.
SECTION II COUNTING RATIONAL CURVES §3. Calabi-Yau manifolds and mirror conjecture 3.1. Classification of manifolds with Kv = O. In this Section, we discuss some conjectural identities involving, on the one hand, characteristic series for the numbers of rational curves of all degrees on certain manifolds V with K v = 0, and on the other hand, hypergeometric functions expressing periods of "mirror dual" manifolds W in appropriate local coordinates. From the physical viewpoint, such identities mean that certain correlation functions of a string propagating on V coincide with other correlation functions of a string propagating on Wj the passage from V to W involves also a Lagrangian change ("A- and B- models" of Witten [25]).
Recent physical literature contains a wealth of generalizations of these identities involving curves of arbitrary genus on varieties with K v ~ O. However, no single case of these conjectures has been rigorously proved. Therefore we have decided to concentrate upon the simplest case, that of Calabi-Yau threefolds. In the framework of Kahler geometry, they can be introduced by means of the following classification theorem. Let us call a Kahler manifold V to be irreducible if no finite unramified cover of V can be represented as a non-trivial direct product.
231
RATIONAL POINTS AND CURVES
3.1.1. Theorem. For any compact Kiihler manifold V with Kv = 0, there exist a finite unramified cover V' and its decomposition into irreducible factors
V' ~
II Ti II Sj x II Ck X
i
j
k
such that following hold: a}. T; are Kiihler tori. b}. Sj are complex symplectic manifolds, (i.e., they admit everywhere nondegenerate closed holomorphic B form), but not tori. c}. Ck are neither tori nor symplectic. Irreducible Kahler manifolds of the type Ck can be called Calabi- Yau manifolds; in the physical literature this name is sometimes applied to any manifold with K v = O. The smallest dimension of a complex torus is 1, of a symplectic manifold 2 (any symplectic surface is a K3 surface); strictly Calabi-Yau manifolds occur first in dimension three. Classification of Calabi-Yau threefolds is a wide open problem; one does not know even whether they belong to a finite number of deformation families. Most of known examples are constructed as anticanonical hypersurfaces of Fano varieties W, or more generally, as "anticanonical complete intersections": V = niDi, Ei Di E 1 - K wi. Every Kahler manifold belongs to the realm of three geometries: Riemannian, symplectic, and complex (or algebraic). Theorem 3.1.1 is basically a Riemannian statement (de Rham theorem on the holonomy groups). The curve count, seemingly a pure complex problem, at present can be properly approached only from the symplectic direction revealing its "quasi-topological" nature. In this report we will concentrate upon algebro-geometric aspects of this vast and complex picture. 3.2. The structure of the mirror conjecture. Consider a Calabi-Yau threefold V and a complete local deformation family W z , Z E Z of CalabiYau threefolds. We will say that V and Wz are mirror related if a certain characteristic function F counting maps
O. (This is possible because Ak has locally finite measure in a neighbourhood of each of its points, but may not have locally finite measure in a neighbourhood of points in the closure Ak and this may intersect At, l:l k.) (2) It is also proved in [32] that 9 u (z) is a.e. constant on each of the sets in the finite collection referred to in the above theorem, and that sing'U has a (unique) tangent plane in the Hausdorff distance sense at 1[m-almost all points z E sing u, and u itself has a unique tangent map at 1-£m-almost all points of singu. (See the discussion of [32] for terminology.) There is an important refinement of Theorem 1 in case (1.13)
dim singu $ m
for all energy minimizing maps into N. In this case the conclusion of Theorem 1 holds with m in place of n - 3: Theorem 2. If u, N are as in Theorem 1, m $ n - 3 is a non-negative integer, and (1.13) holds, then for each closed ball Ben, B n sing'U is the union of a finite pairwise disjoint collection of locally m-rectifiable locally compact subsets. Remarks. (1) As for Theorem 1, again 9 u (z) is constant a.e. on each of the sets in the finite collection referred to in the statement, sing u has a tangent space in the Hausdorff distance sense, and also 'U has a unique tangent map, at 1-£m-almost all points of sing'U. In [26], [28] there are also results about singular sets (albeit for special classes of energy minimizing maps and stationary minimal surfaces), which, unlike the results here, were proved using "blowup methods". In particular we have Theorem 3. If N = 8 2 with its standard metric, or N is 8 2 with a metric which is sufficiently close to the standard metric of 8 2 in the C 3 sense, then singu can be written as the disjoint union of a properly embedded (n - 3)dimensional C 1 ,I'-manifold and a closed set 8 with dim 8 ~ n - 4. If n = 4, then 8 is discrete and the C 1 ,1' curves making up the rest of the singular set have locally finite length in compact subsets of n. For further discussion and proofs, we refer to [27]. There is an analogue of Theorem 2 which applies to an arbitrary subplanifold M in a mulitiplicity one class M of stationary minimal submanifolds:
LEON SIMON
254
Here and subsequently we let
(1.13')
m = max{dim singM : ME M};
this maximum exists and is an integer E {O, ... ,n - I}, as shown in the discussion following 2.7 below. Theorem 4. Suppose M is a multiplicity one class of stationary minimal surfaces as in 1.11, supposem is as in 1.1:1, and ME M. Then for each x E singM there is a neighbourhood Uz of x such that sing M nuz is a finite union of locally m-rectifiable locally compact subsets.
1.14 Remark. Analogous to the remarks after Theorems 1, 2 we have in addition that SM(Z) is constant a.e. on each of the sets in the finite collection referred to in the statement of the theorem, sing M has a tangent space in the Hausdorff distance sense, and also M has a unique tangent map, at 1£m-almost all points of sing M. We give the detailed proof of Theorem 4 and Remark 1.14 in §7 below; as we pointed out in the introduction, the proof involves only very minor technical modifications of the proof of Theorem 2 given in [32]. In view of the examples in 1.12, we thus have in particular the following: Theorem 5. (i) H M is the regular set of an n-dimensional mod 2 mass minimizing current in R n+1c (n, k ~ 2 arbitrary), then the singular set singM is locally a finite union of locally (n - 2)-rectifiable, locally compact subsets. (ii) H M is the regular set of an arbitrary n-dimensional mass minimizing current in R n+l, then sing M can locally be expressed as the finite union of locally (n - 7)-rectifiable, locally compact subsets. (Except for the local compactness result, part (i) of the above theorem is also proved in [26] by using "blowup" methods, which are quite different than the techniques used in the proof of Theorem 4.) In addition to the above results, there are also more special results, proved using blowup techniques in [26], analogous to the results for energy minimizing maps described in Theorem 3. For example, we have the following: Theorem 6. Suppose the m of (1.13) is equal to (n - 1). H M E M, C(O) = C~O) x R E C n Tan zo M with C~O) a I-dimensional cone consisting of an odd . number of rays emanating from 0, and SCCD) (0) = mincET Sc (0), then there is p > 0 such that sing M n Bp(xo) is a properly embedded (n - I)-dimensional C1,a manifold. Theorem 7. If V is an n-dimensional stationary integral varifold in some open set U C Rn+k, and Xo E U with 1 < Sv(xo) < 2, then sing V n Bp(xo) is the union of an embedded (n - I)-dimensional c1,a manifold and a closed set of dimension :5 n - 2. If n = 2 we have the more precise conclusion that there is p > 0 such that either sing V n Bp(xo) is a properly embedded c1,a Jordan arc with endpoints in 8Bp(xo) or else is a finite union of properly embedded locally c1,a Jordan arcs of finite length, each with one endpoint at Xo and one endpoint in 8B p(xo).
RECTIFIABILITY OF THE SINGULAR SETS
255
For some special (but important) cla.oTp, E S n T2 -,. \T2 -1o-1, j = 1, ... ,Qk, k ~ 2, with
Cl , C2 depend only on n, m, and the Zk,j
S n T2 -1o \T2 -1o-1 C U~~~ax(k-2,2) U7.!1
k ~ 2.
B,P/22-l (Zl,j),
For the proof of this lemma, we refer to [32]. 3.8 Remarks. (1) It is important for later application that C does not depend on d, nor indeed on S. Of course one has to keep in mind that if the set S is very badly behaved (like a Koch curve for example), then the sets Tp can all reduce to the empty set for sufficiently small p, in which case the lemma has correspondingly limited content. (2) As part of the proof given in [32], it is shown that To is contained in the graph of a Lipschitz function defined over {OJ x Rm and with Lipschitz constant 5 Cd, so automatically ll m L To has total measure 5 C. 4 Area Estimates for Submanifolds in M. Here we continue to assume that M E M. Points in Rn+k will be denoted (x,y) E Rl+k X Rm, and we continue to use the notation r = Ixl and w = lxi-IX E SlH-l for x E Rl+k\ {OJ. We are often going to use the variables (r,y) = (Ixl,y) corresponding to a given point (x,y) E Rl+k X Rm, and it will be convenient to introduce the additional notation Bt
= {(r, y)
for given Yo Also,
> 0, r2+IY12 < p2}, E Rm and p > o. : r
B{: (y)
and we let vr(x, y)
where PT.J.
VT! Vy
= PT.J.
(-,,,)
Bt(yo)
= {(r, y)
: r
> 0, ly-yol2 < p2}
= M n Bp(Y),
be defined on M by
(-.v)
M(lxl-1x,0),
Vyi
= PT.J.
(-.,,)
M(elH+j),
i = 1, ...
, m,
M denotes orthogonal projection of Rn+A: onto the normal space
T(-;,y)M. Notice that we thus have m
ZI~
= l-IV'MrI2,
v;
m
m
=I>;i = L IPT(~.,,)M(elH+j)12 =L(I-IV'My I2). j
j=1
j=1
j=1
In particular, if e is any vector in {O} x R m, then IPT.J. M(e)125IeI2vy2. (-,,,)
The main inequality of this section is given in the following theorem:
264
LEON SIMON
4.1 Theorem. H ( E (0, ~), (3 > 0 then there are G = G({3,k,n) > 0, 1/ = 1/({3, k, n, () > 0 and a = a({3, k, n) E (0,1) such that the following holds: If p-n\B~ (0)\ ~ (3, 0 E M, w~lp-n\B~ (0)1-9M(0) < 11 and p-n-2 1B:'(0) r2(11:+ v;) < 11, then there is C E I with singC C {OJ X Rm, satisfying p-n-2
f
dist«x,y),C)2
< (,
B:'(O)
where M(r, y)
= M n Sr,y.
In proving Theorem 1 we shall need three lemmas, each of which is of some independent interest. The first of these gives some important general facts about C E Ii we use the notation of 2.11, and define
TJO) = {r:: (4.2)
r: is a compact (i- I)-dimensional embedded minimal submanifold of Sl+k-l with (().w,y) : ). > 0, Y E R m , w E r:} E 7/3}.
If r: is a compact (i-I)-dimensional embedded minimal submanifold of Sl+k-l , and if 'r/J is a cj section of the normal bundle of Cover r: (we write 'r/J E Cj (r:; Col», then we continue to let Gr; ('r/J) denote the "spherical graph" defined in §2 and Ad'r/J) the corresponding area functional as in 2.15. Notice that if I'r/Jlci is small enough (depending on r:), and if j ~ 1, then Gr;{'r/J) will be an embedded GLsubmanifold of Sl+k-l. Under suitable circumstances, we can also express appropriate parts of M E M as a spherical graph taken off a cone C E C. specifically, if n C C is open and if u is a cj section of the normal bundle of Cover n (we write u E Cj(n; Col» with L:;=o rj-1lDjul $ 'Y, with 'Y sufficiently small depending only on C (and not depending on the domain n), then we can define the spherical graph Gc{u) (analogous to 2.14) by Gc(u) = {(I + Ixl- 2Iu(x,y)1 2)-1/2«X,y) + u(x,y»)}; Gc(u) is then an embedded Ci-submanifold of RR+k. We can also define the area functional Ac(u) (analogous to 2.15) over C for such u E GI(n; Col) by Ac(u) = IGc(u)l.
Then we have the following:
RECTIFIABILITY OF THE SINGULAR SETS
265
TjO)
rj0),
4.3 Lemma. For each {3 > 0, is compact in the sense that if ~j E then there is a subsequence converging in the Hausdorff distance sense to an element ~ E Also, there is (1 (l({3,n,k) E (0, such that, if~lJ ~2 E
rj0).
iJ
=
rJ°) and E2 can be expressed as a spherical graph G!;l.,p ofa C3 function .,p taken
with 11/Jlc3(!;I) < (lJ then IE11 = 1~21. Furthermore there are constants E (0,1J and a = a({3, n, k) E (0,1) such that if ~1 E rj0) and jf ~2 (not necessarily in rj0») can be expressed as a spherical graph G!;l1/J of a C3 section 1/J of the normal bundle of E1 with 11/Jlc3(!;1) < (2, then Off~l
(2
= (2({3, n, k)
IIE 1 1-IE211 2- Q ::;
r
j!;1
IQ!;11/J12,
where Q!;1 denotes the minimal surface operator on E1 (i.e., Q!;1 (1/J) is the Euler-Lagrange operator of the area functional A(1/J) == IG!;l (1/J) I of spherical graphs over Ed. Remark. Thus we have a uniform Lojasiewicz inequality for a whole C3 neighbourhood of and also, by the first part of the above lemma, the area
rj0),
rj0) , and there are only finitely many values of the area corresponding to E E rj0) . Proof of Lemma 4.3. The compactness of TjO) is a direct consequence of
is constant on the connected components of of
the estimates of 2.12 and the compactness 1. 11 (b) for M. Next suppose there is no such (1. Then there must be sequences E j , Ej in converging in the
rJ°)
Hausdorff distance sense to a common limit E E
rJ°) but with
(1) According to the Lojasiewicz inequality of 2.14 we have a = O'(E) > 0 such that
= o(~) E (0,1) and
0'
(2)
IIG!;1 (1/J)I_I~1111-Q/2 ::; CIIQ!;l (1/J)II£2(!;I)'
11/Jlcs(!;t)
< 0'.
Therefore for all sufficiently large j we can apply this with graph!;1 (1/J) = E j , lS j in order to deduce that lEd = IE21, thus contradicting (1). Now if the inequality of the lemma fails, then there are sequences Ej E and 1/Jj E C 3 sections of the normal bundle of Cj over Ej, with Cj the cone determined by Ej , with Ej converging to a given E E 7)0) and with l1/Jjlcs but such that
rJ°)
(3)
where OJ .J.. 0 as j -+ 00. Thus IEjl = lEI for all sufficiently large j by the first part of the proof above, and (3) contradicts (2), because IIQ!;j(,pj)II£2(!;j) is geometrically the £2-norm of the mean curvature vector of G!;j (,pj) integrated over Ej and (since Ej is approaching E in the Ct-norm) this is proportional to the £2 norm of the mean curvature vector of lSj = G!;j(,pj) when lSj is expressed as a spherical graph taken off E.
266
LEON SIMON
4.4 Lemma. Let a E (0,1] and {3 > o. There is TJ = TJ(n, k,{3) E (0,1) sum that if B¢!/s(O)\{(x,y) : Ixl ~ a/16} = Gcu with C = Co x R m E /p, u a C 3 (CnB 7u / s (0)\{(x,y) : Ixl ~ a/16};C.L) function and 3
L
sup aJ-IIDjul ~ TJ, cnB T. /8 (0)\{(Z,II): tzl:5u/16} j=O then
and
B:./
IV",II(rl - l IM(r,y)l)1
sup 4 \{(",II): ":5u/S}
~ Ca- 1- n f
JB!;'\{(Z,II): Izl:5u/16}
r2(,,~ + ,,;).
Here V",II means the gradient with respect to the variables (r,y) E B;;l C = C({3, n, k), andu(r, y) denotes the function on ~ defined byu(r, y)(w) = u(rw, y), and ~ = Co n Sl+k-l .
Proof. As discussed in §2, the Euler-Lagrange operator v E C2(~; C.L) is characterized by the integral identity
QEV
for
so in particular
Also (see, e.g., the discussion of [26]) the Euler-Lagrange operator Qc of the area functional over C has the form
where
t::. ",11 v= ;:z=r 1 ~ + L,,3=1 "'~ 8"ltj~) 8,. (ri-l~) 8,. 8 II' , and
Notice that if Qcu = 0 in some region (3)
We also recall that the linear operator
nc
C, then by definition,
RECTIFIABILITY OF THE SINGULAR SETS
267
is a linear elliptic operator of the form
luv = fj.r,1/v
+ r- 2LE,uV •
where LE,u is a linear elliptic self-adjoint operator on functions v E C 2 (I:i Cl.). In particular (using the notation introduced prior to 4.3) if M = Gcu with u E C2(Oi Cl.) for some 0 C C, then since M t = M - telH+j is a minimal surface for each t, and M t = graphc Ut, where Ut(x, y) = u((x, y) + tel+k+j), then we have QCUt == 0 on a domain Ot = 0 - tel+k+j, and hence v = u1/; == ftu((x, y) + tel+k+j)lt=o is a solution of
luv
=0
for each j = 1, ... ,m. Also since M t = (1 + t)M is a minimal surface for each t with It I < 1, and M t = graphut, where Ut(x,y) = (1 + t)-lu((1 + t)(x,y», then we have similarly that v = RUR - U == ftUtlt=o is also a solution of this equation. But RUR - U = ((x, y) . D)u - u == r (u/r)r + E;:1 yiu1/;' so we have the equations ".
(4)
lu(u1/;)
= 0,
lu(r2(u/r)r)
= -lu(Eyiu ,,;) = i==1
-2fj.1/u. •
Notice that the operator lu W has the form
+ r- 2fj.EW + r-1a . VEw + r- 2b. w with lal, Ibl ::; C(n, k,.B) on B 7cr / S (0)\{(x, y) : Ixl < u /16}. Then the standard fj.r,1/w
Cl,a Schauder theory for such linear operators ([12]) gives 1
(5)
L
sup lui D i u1/12 ::; Cu- n Bf"/D \{(x,1/): Ixl 0, a = a(n, k, f3) E (0,1) such that if p-nlM n Bp(O)1 ~ {3, 0 E M, and w;;-lp-nIM n Bp(O)I- 9M(0) < 1/, then the inequality
p-n-2 [
JMnB sp /4(0)\{(Z,II): Izl 0 Lemma 4.3 we have that there is a such that (1) implies
=
Ir 1 -'IM(r,y)I_IEI1 1 -
=
a/2
:$ CIIQI;u(r,y»lIi2(E)
=
for each (r, y) E Bt/s(O) with r ~ p/16, where C C(n, k, (3). Then the required inequality holds by virtue of Lemma 4.4; notice that the hypothesis S
LpilDjul :$ 71
sup
B~/8(0)\{(z,y): Iz l:5p/16} j=O
required in Lemma 4.4 is satisfied (with C( in place of 71) due to 2.12 and the inequality (1) above. . We shall need the following corollary of the above lemma later.
(> 0, (3 > 1 there is 710 = 71o«(,{3,n, k) > 0 such that the following holds. Suppose C E 7 with singC = to} x Rm, M E M with p-nlM n Bp(O)1 :$ (3, w;lp-nIM n Bp(O)I- 9M(O) < 710,0 E M, and also
4.6 Corollary. For any given
p-n-2
f
JB~(O)\{(z,y): Izl })
[
M
.
(O,II)\{(Z,II): Ixl; •• (Zj,k) we must have p/2H2 ~ O'z; .•. Thus, in any case, if OJ,k does not intersect B u,; .• (Zj,k) we can apply (9) with 0' = p/2 j - 1, Y = Zj,k (so Y = 0 in case i = k = 1), to deduce
1
r
Ivr,y(r 1- l IM(r, y)l) I r l - 1 drdy
OJ.,,
(16)
SC
~ r In; .• J
r2(v; + v~) dwr l - 1 drdy.
st-l
On the other hand if OJ,k does intersect B;J,; .• (Zj,k), then
i
~ 2 (by (6» and
~ 2-;-2 p, so OJ,k C BttT;.• (Zj,k) C Ui~OUi (Yi). Hence by summing in (16) and using (14), (15), we conclude that
O'j,k
r
JB:/2(O)\(UjBt"op; (II;»
(17)
r
IVr,lI(r 1 - l IM(r, y)l) I r l - 1drdy
SC
r
JB~(O)
r2(v; + v~) dxdy.
Notice also that using the monotonicity 1.7' and the definition (5) of have that for each j CT-,n- 2
II,
1
B~.. v; (0,11;)
1 0'-,n-2 r 2 , , ., , of the (j onto L should be in such uniformly general position in BD'«(I»,
(:n
Proof of Lemma 5.B. By definition
so in particular
(w - Zj)1. - (w - zo)1. = (zo - Zj)1., and by the hypothesis we then have that on M m
"Yp2vl ~ C
(2)
L«w - Zj)1. - (w - ZO)1.)2. j=O
On the other hand using (1) with j = 0 we also have on
B:t (zo) that
(3) Combining (2) and (3) we then have m
riv:£ + p2vl ~ CL I(w - Zj)1.12 j=O
as claimed, Notice that the reverse inequality m
C- 1
L
I(w - Z;)1.12 ~ (rLvrL )2 + p2 vl
;=0
B:t
follows directly from (1) on (zo). Next notice (Cf, (1) above) that at any point WEB:! «(0)
(4)
(w - (;)1. == (w _ w')1. + (w' _ ('.)1. + «('. _ (;)1. 31. 3 1. = rLvrL + (Pdw - (;)) +
«(i - (;) ,
Taking differences in (4) we see that «(; - (0)1.
= -(w -
Since l(j - (; I = dist( (j, Zo then see that On U
(;)1.
+ (w -
+ L),
a2vI :::; C
L j=O
+ «(i - (;)1. - (I> -
m
ICw -
(0)1.,
by using the given hypothesis on the (; we
m
(5)
(0)1.
(;).1.12
+C L j=O
dist 2 C(j, Zo
+ L),·
278
LEON SIMON
Going back to (4) again we thus also conclude that on m
B:! «(0)
m
rill~L ~ C ~ I(w - (;)1.12 + C ~ dist 2 ('j, Zo + L), ;=0
j=O
which proves the required upper inequality for riv~L + u2IDLVI 2. The reverse inequality follows directly from (4) and the triangle inequality. The final inequality of the lemma is simply a matter of combining two of the previous inequalities, so this completes the proof of the lemma. In the proof of Theorem 5.7 we shall want to apply the main area estimate established in Theorem 4.1 of §4, and this requires that we check the hypothesis that M is L 2-sufficiently close to some e E T with sing e = {OJ x R m in the appropriate ball. 5.9 Lemma. For any given, > 0 there is if 5.1, 5.4, 5.5, 5.6 hold with f ~ fO, then
where the notation is as in 5.B, and dist«x, y), Zo + L) ~ p/16}j e1.),
U
fO
= fo(n, k, (J, () > 0 such that
E C3«zo
+ e) n B 2p/ 3(ZO)\{(x,y)
3
~pi-lIDjulo3 ~,
sup
(.zo+C)nB 2p /s(.zO)\{(Z,II): di8t«Z,II),.zo+L)~p/16} j=O
for some fO(n, k, (J)
e
E TO/3 with sing e
= L, 6c(O) = (}o.
Furthermore there is fO = ~ fO, then for all z E S+
> 0 such that if 5.1, 5.4, 5.5, 5.6 hold with f
dist 2 (z, Zo + L)
e 5: Cp-n {
J{(Z'II)EB~/4(.zo): rL~p/4}
(rlll;L + (p + Iz - zol)211i + I«x, y) - z)1.1 2)
5: Cf(p + Iz - zol)2 Remark. It is not assumed that Iz - zol is small herej Zo, z are unrelated points in S+. Proof of Lemma 5.9. Evidently we can assume without loss of generality that Lin 5.6 is {OJ xRm. To prove the first inequality, notice that by Lemma 5.8 above we have m
r~lI;o
+ p211~ 5: C L: I«x, y) - Z;)1.12, ;=0
RECTIFIABILITY OF THE SINGULAR SETS
279
where ro = Ix .... e.zo I, rovro = PTJ. M(X ..... e.zo' 0), Zo = (e.zo ,1].1'0)' Integrating (-.,,) this inequality over the ball Bp(zo) and noting that 5.5 implies
(1) we then have the first inequality as claimed. In view of the first inequality, the first part of Lemma 4.5 guarantees that the second and third inequalities of the lemma hold for some C E T with sing C = {O} X Rm and
(2) and 3
(3)
Lpi-1lDiulos
sup
:$
C.
(.z0+C)nB 2p / S (.z0)\{(z.u): Iz-t· o l$p/16} i=O
We agree that f and Care chosen smaller than the minimum distance between distinct elements of {ec(O) : C E Tp}. Then (2) gives ec(O) = 90 • We next claim that (for Csmall enough in (3», for any E RitA:,
e
where C = C(n, k, (3) is fixed (independent of e, u), provided fO = fo(n, k, (3) > o is small enough. Indeed otherwise by (2) and (3), after rescaling and trans-
lating so that p = 1 and Zo = 0, we would have a sequence Mi E M, with o E sing M Ci E Top with sing C i = C~O) X R m, and points E Sl+k-l such that
ej
j,
3
lei nB1 (0)1:$ (3,
sup
Lp iD ujlo8 ~ 0 as j ~ i
i
00,
CjnB2 / S (0)\{(z.u): IzI9/16} i=O
and
(5)
ej --+ eE Sl+A:-l,
Notice we also have
(6)
li~ inf ec; (0) 3~OO
>1
by virtue of 2.1. Using 1.11(b) we can assume that C j ~ C locally in the lIausdorff distance sense in Rn+k, Mj --+ C in B 2 / 3 (0)\{(x,y) : Ixl ~ 1/16} and that (e,O).L 0 on C. But this, together with stationarity of C, implies that C is invariant under translations in the direction of (e,O), which means
=
LEON SIMON
280
sing C contains the line through 0 in the direction of (~, 0), contradicting the fact that sing C = {O} x R m. (Notice that C is not a linear subspace because 9c(0) > 1 by (6) and upper-semicontinuity 2.3.) Thus (4) is established. On the other hand we have, using the notation Zo == (~zo, 17.10)' z = (~z, 17.1), (~.zo - ~.z, 0)1.
= «x, y) -
z)1. - «x, y) - zo)1. - (0, y -17.1)1.,
and hence
Integrating this identity over M and using (4) with
~
==
~.zo
-
~.z
yield
I~.z - ~.zo12
~ Cp-n [ (r~v~o + (p + Iz - zol)2v~ + I«x, y) J1y-".0 l O. Then there is do = do(n, k, (3) > 0 such that the following holds. H 5.1-5.5 hold, and 810 ~ d ~ do, then for any p E (0, lIe] we
have the estimate
where a
= a(n, k, (3) E (0,1) and 9 :
9(n, k, (3) E (0,
312]'
Proof. The proof is based on the L2 estimates of the previous section. As mentioned above, we assume
(1)
L O,l
= {O} x Rm.
T:
Take p E (0, ~]. If = 0, then we have nothing further to prove, so assume that T: =F 0, and take an arbitrary point Wo E T: n S+. By definition of Tt(C Tt), there is a point Wo E Bp(wo) n S+ such that
(2)
B 2P (wo)ns+c{w: dist(w,wo+{O} xRm) 0 such that
9M(WO) ~ O'-nIB~(wo)1 ~ 9M(WO)
(1.2)
+ 10,
=
0' E (0,0'0].
Also, by monotonicity 1.9' we have the identity
(1.3) for each z, r such that Bp(z) CUM. Since BO'(z) C B(l+E)O'(WO) for any z E B::(wo), from 1.2 we deduce that
z E B::(wo), 0' ~ 0'0/2, provided that 9M(Z) ;;:: 9M(WO) and that 0'0 O'o(M, wo, e) > 0 is sufficiently small. Let 8+
take
Wl
= {z E B O'o/2(WO)
E 8+ n B EO'o/4(WO),
E
0'1
: 9M(Z) ;;:: (Jo},
E (0,100'0/4] and define
(1.4) where (7.5)
'7Wl,O'l
(x, y)
== all«x, y) - wt}. Then the above inequality gives
=
LEON SIMON
292
where
(7.6)
0 E S+(Wl,O'l)
={Z E Bl(O) : 9 M(z)
~ Oo} = BdO) n7]w l,/T1S+,
Notice that S+(Wl, 0'1) corresponds exactly to the S+ of §§5, 6 with Xi in place of M. Also, recall that by Lemma 2.16, we can, and we shall, assume that 0'0 == O'o(M, wo, f) is chosen small enough so that S+ has the €-approximation property of 2.16 and hence S+(Wl,O't} does also. Thus (Cf. 5.4)
(7.7)
S+(Wl, 0'1) n B/T(Z) C the (fO')-neighbourhood of Lz,tT,
for each Z E S+(Wl,O'l) and each 0' E (0,1], where Lz,tT is an m-dimensional affine space containing z. We fix these affine spaces in the sequel. Without loss of generality we assume L O,l
(7.8)
= {O} x Rm.
We emphasize that 7.5 and 7.7 hold automatically if 0'0 = O'O(f,U,WO) is chosen sufficiently small. We henceforth assume 0'0(10, u, WO) has been so chosen, and we continue to take M as in 7.4. Notice also that by 7.4 (choosing a new 10 if necessary) 5.1, 5.3, 5.5 all hold with S+(Wt,O'l) in place of S+ and with 00 = 9M(WO)' Thus we can apply the results of §5, §6 with M In place of M, and with S+ = S+(W1,0'1), 00 = 9 M(wo). Before we begin, we need to establish the following lemma, which is a simple inequality for real numbers:
7.9 Lemma. If 0 < a
0 and 0.2-a C
~ f3(b - a),
then
= C(f3, 0:) > O.
b/ a > 2 we have trivially that a-1+ 0 /2 _ b-1+ 0 /2 > _ Ca-1+ 0 /2 > _ C a -0/2 ,
so the required inequality holds in this case. In case b/a
~
2 we have
a-1+0/ 2 - b-1+0 / 2 = (1 - 0:/2)C-2+ 0 /2(b - a) for some c E (a, b) 1 - 0:/2 -0/2 b - a > a -2since a ~ b/2 4 a -0
> 13(1 - 0:/2) -
4
-0/2
a
since a 2- 0 ~ f3(b - a),
so again the required inequality is satisfied, and the lemma is proved.
Proof of Theorem 4. Let T:, J.t+ (corresponding to given a with 10 < a/8, and with M as in 7.4 in place of M) be as in §6. a ~ ao(n,k,f3) and € < fJ/B will be chosen later. Now, with M as in 7.4, by virtue of 7.3,7.5, we can apply all the results of §6 to M, and hence (1)
RECTIFIABILITY OF THE SINGULAR SETS
293
with 1/1 the deviation function of §6 with M in place of M, where () 0, and 0 = o(n, k, (3) E (0,1). In view of Lemma 7.9 we can use (1) to get
(ITt,. '")
where 10
(ITt '")
-1+./' ?:
-1+./' -
(2)
= fT+ 1/1. i
= (}(n, k, (3) >
CI;·/',
Then starting with p = ~ we can iterate the inequality (2)
in order to obtain j
= 1,2, ... ,
and hence
!
(3)
T"': ., /4
where 2'}' = 0/(2 - 0)
I:(U +
< C3·-1-2'1' 12'1' 0 ,
.1. '#' -
> 0.
Since (j
+ 1)1+'1' -
1)1+'1' - p+'1')!+
;=0
1/1
j = 1,2, ... ,
p+'1'
~ CI~'1'
~
Cp, this implies
I:r
1-'1'
~ CI~'1'.
;=1
T 8 ; /4
Using summation by parts we obtain that
so that
(4) where d is defined on Tt by 4
(5) Now for z E
2-A: d(x,y)= { 0
if (x,y) E Ti-~ \T2t:..~-1' k ~ 2, if (x,y) E Tri.
S+ n Tt and (x, y) E Tt we claim that
(6)
where R,,(x,y) = I(x,y) - zl· Here we include z E Tt, in which case d(z) = 0 so (6) says d(x, y) :5 4R,,(x, y), 'v' (x, y) E To prove this we can of course
Tt
LEON SIMON
294
Ti-.
k
assume d(x, y) > 0, so take any w = (x, y) E \Ti-.-l for some ~ 2, and consider cases as follows: Case (a): Z E T2~' for some q ~ k + 2. (If Z E To+, then this case will be applicable 'V q ~ k + 2.) Then by Remark 3.4(2)(d) of §3 we have Iw - zi ~ 2- k - 2 = 2- k /4 = d(w)/4. Case (b): Z E Ti-. \Ti-'-l with q 5 k + 1. In this case, if we assume that w ~ B.!!i!J.(z), then (keeping in mind that Z E S+ and d(z) = 2- q in case 2 z E T2~' \T2~'-1)' we have Iw - zl ~ 2- q - 1 ~ 2- k- 2 = d(w)/4. Thus (6) is always satisfied as claimed. Now inequality (4) states that
I((x, ~n~2z).L12 dJ.L+(z) dxdy 5
[ 1iog dl1+"Y [
(7)
iT;
18+
CI~'"I,
z
so that by interchanging the order of integration we deduce that
[
(8)
i'1+ T!
!log dl1+"Y I((x, y) - z).L1 2 dxdy < 1"Y R n +2 o· z
for all z E S+ with the exception of a set of J.L+ -measure 5 CPo. (We must keep in mind here that there will in general be lots of points z E S+ which are not in the support of J.L+. and these have J.L+ -measure zero, so in particular (8) need not hold for them.) In view of (6), (8) implies
1
(9)
T;\B¥(z)
- z).L1 2 dxdy -< 1"Y I logRz 11+'"1 I((x, y) R n +2 o. z
for all z E S+ with the exception of a set of J.L+ -measure 5 C IJ . Next note that according to Lemma 3.7 we have a countable set S = {Zj,k : j = 1, ... ,Qk, ~ 2} c S+ n such that
k
Tt
(10)
Zj,k E Ti-. \Ti-.-l, so d(Zj,k)
(11)
=
00
J.L
C1 6m / 2
= 2- k ,
j
= 1, ... ,Qk,
k ~ 2,
Q.
L
2- mk L:[Zk,j]
k=2
j=1
+ C2 1l m
L Tt,
c, = C,(m), i = 1, 2,
and
(12)
S n Ti-. \T:J-.-l C U~~'!'ax(k-2,2) U~l
B61/22-Io
(Zl,j)
Now let £0 C S be the collection of all Zj,k E S such that
'V k ~ 2.
295
RECTIFIABILITY OF THE SINGULAR SETS
and let £1 C Td" be the collection of all z E Td" such that
(14)
i
1\B¥(z)
- z).L1 2 dxdy > IJ IlogRz 11+'Y I«x, y) R n +2 o· z
Since 1'+(£0 U £d :$ Clri by (9), by (11) we have that
(15)
L
d(w)m
+ 1lm (£I)
:$ CIJ,
C = C(n, k, &).
wEt:o
Now take any Z E Tt n S+ \To+. From (12) it follows that for some Zj,1c E Sj if this Zj,1c ¢ £0 then by (9)
Z
E
Bd(Zi,.)/4(Zj,lc)
(16)
Regardless of whether Zj,k E £0 or not, by (10) and Remark 3.4(2)(d) (with k + 2, k + 1 in place of l, k) we have that z E B d (IIJ,.)/4(Zj,lc) C Rn\Ti=-.-:I so that
(17) Thus by (16), (17), for any
Z
E S+ n Tt\Td",
(18)
On the other hand if z E Td"\£l, then by definition d(z) -= 0, and (9) implies
(19)
- z).L1 hfTi\B¥(z) IlogRz 11+'Y I«x, y) Rn+2 II
2
dxdy
< l'Y, -
0
(18) and (19) are the main estimates, Using them we now want to check that We have all the hypotheses needed to apply the rectifiability lemma of §2 (in case p = 1 and S = S+), For this purpose, we first assume (20)
no ball Bp(z) with
Z
E B 5 / S (0)
n S+ and p E [k, 11 has a o-gap.
By the definition of o-gap in §3, from (20) it follows that (20)'
S+ has no .to-gaps in the ball Bl(O).
LEON SIMON
296
Also by Definition 3.1 and the definition of d(z) we have (20)"
d(z)
~ 3~
provided E is sufficiently small (depending on E, n, k, {3), which we subsequently assume. Using (20),7.7, 7.S, and the fact that E < 6/S we obtain
(21) In this case (19) implies
1
(22)
Q\B¥(z)
- z)1.12 dxdy < I! I logRz 11+"Y I«x, y) R n +2 0' z
H,
for any z E Tt\ct, where Q = Bs/s(O) n {(x,y) ~ Ixl ~ and (IS) implies that for any z E (8+ n Bl/2(0)\Tt)\(Uz; .• eEoBd(z; .• )/4(Zj,k)) there is always a point E 8+ n Tt\Tt such that
z
(23)
1
Q\B~(i)
Z
_Z)1.12 dxdy IlogR-I1+"YI«x,y) z R n +2 i
E B d (i)/4(Z),
< I! -
0'
d(z) ~ ~d(z).
Now take an arbitrary point z E (8+ n Bl/2(0)\Tt)\(UZ; .• EEoBd(z; .• )f4(Zj,k)) and let z be as in (23). 8+ = 8+(Wt,0'1) has no 6-gaps in Bp(z) for p ~ d(z), and hence for all p E [~, II we can select ZI, ... ,Zm E 8+ n Bp(z) such that 5.6 holds with z in place of Zo and with 'Y depending only on n, k, {3. Let 7] E (0,6 3 ] be given and let L be as in 5.6 (with z in place of zo). For E small enough (depending d(;' on 7], n, k, {3) and for p E [~, ~l we have all the hypotheses needed to apply Lemma 5.9 with 2p in place of p and with 7] in place of C. Hence there is C(p) E Top with singC(p) = L and u(p) E C 3 «z + C(p») n B 4p / 3 (Z)\{(x,y) dist«x, y), z + L) $ p/S}; (C(p»)1.) such that
MnB4p / 3 (Z)\{(x,y) : dist«x,y),z+L) $ piS} (24)
=graphu(p)nB4P / 3 (z)\{(x,y): dist«x,y),z+L) $p/S} and 3
(25)
2: sup pi-I IDiu(p) 1 S 7], i=O
provided that E is small enough depending only on d(z)j4, by 3.1,3.5 and 7.8 we have automatically (26)
7], n, k, {3.
Since for p
~
297
RECTIFIABILITY OF THE SINGULAR SETS
Using (24), (25), (26) with p = ~ we then can select a maximum interval (po,~] C [¥,~] such that there is W E C3(C n B1/S(z)\(Bpo(z) U K(z»j C.L) such that
-n
M
B1/S(z)\(Bpo(z) U (z + K»
= graph(w) n B1/S(z)\(Bpo(z) U (z + K»,
and 3
L~-lIDiwl ~ 71 2 / 3 ,
(27)
;=0 where C E TOfJ with sing C = {O} x Rm (we can take C = q(C(l/S» with q orthogonal such that q(sing C(l/S» = {O} x Rm and IIq - l R ft+kll ~ Cf), and where
K
= {(x,y) : Ixl ~ 41 1(x,y)I}.
By (20), 3.1, (24), (25) it is clear that
so long as f is small enough. Further, from (27) and (24), (25), (26) with p E [po, ~], it follows that w can be extended to give 'Iii E C3(Cn(Bl/S(Z)\(Bpo/2(Z)U (z + K»j C.L) with
(28)
M
n (Bl/S(Z)\(Bpo/2(Z) U (z + K» = graph(w)
_
n (Bl/S(Z)\(Bpo/2(Z) U (z + K»
and 3
(29)
L~-l sup IDi'lii1 ~ C71 2 / 3 ,
;=0 where
K
= {(x,y) : Ixl ~
9
40 1(x,y)I},
On the other hand by the identity (8) in the proof of Lemma 4.4, applied with 'Iii in place of u and with C71 2 / 3 in place of 71, C as in (29), we have (30)
provided 71 is small enough depending on n, k, (3. Now let
LEON SIMON
298
and let w(O') denote the L2(r) function given by w(s)(w) = w(z + sw), w E r. Then by direct integration, the Cauchy-Schwarz inequality, and (30) we obtain (31)
for any po/2 < 0' < T ~ 1/8. Taking T = ~ and using (24)-(26) with P = 1/8, and also (27) again, we then deduce that
for
0'
E [',
H Thus sup p- n -
(32)
2 (
_
JB:'(z)\(BpQ/~(z)U(Z+K»
PE[pQ,tl
Iwl2 ~ GTJ,
and hence by PDE estimates we can improve the estimate (29) to
a
'E ';-1 sup IDiwl ~ G1/ < 1/2/3,
(33)
i=o provided 1/ is small enough depending on n, k, {3. However this contradicts the maximality of the interval [Po, ~l unless Po = d(z)/4. Thus Po = d(z)/4 and, in consequence of (32), (34) Since
sup
[~,tl Z
p-n-2 ( B:'(zH\{(Z,II): Iz- xl 0 such that eM (y ) :::; f3 for each y E B. In particular
300
LEON SIMON
8c(0) ~ f3 for any tangent cone of M at any point y E B, and by Lemma 4.3 we know that {8M(Y) : Y E sing. M n B} is a finite set 0:1 < ... < O:N of positive numbers, where sing. M is as in 2.17. Let Sj
= {z E singM
: 8M(Z)
= O:j},
st = {z E singM : 8M(Z) ~ O:j}. st
Notice that is closed in n by the upper semi-continuity 1.13 of 8M. For any j E {I, ... ,N} and any y E Sj, according to the above discussion, there is p> 0 such that Bp(Y) n is m-rectifiable. Thus, in view of the arbitrariness of y, the set Sj has an open neighbourhood Uj such that
st st n Uj is locally m-rectifiable.
(41)
Of course the st n Uj are also locally compact, because open. Now let
V;
= {z E singM
: 8M(Z)
Then the V; are open in
< O:i+d,
j
st
is closed and
= 0, ... ,N -1,
VN
B
n singM = u,7=o{z
EB
= U,7=oB n
st
n sing M : n Yj
O:j ~ 8M(Z)
is
= n.
n by the upper semi-continuity 1.13 of 8 M,
= 0, O:N+! = 00, sri = sing M, and Uo = 0, we can write
0:0
Uj
and with
< O:j+!}
= (U,7=o(B n st n Uj n V;)) U (U,7=o(B n st\Uj) n Yj). This is evidently a decomposition of B n sing M into a finite union of pairwise disjoint locally compact sets, each of which is locally m-rectifiable; in fact for each j the set (B n st\Uj ) n V; C sing M\ sing. M, and hence has Hausdorff dimension ~ m - 1 by 2.17, and the set B n st n Uj n V; is locally m-rectifiable by (30). This completes the proof of Theorem 2. Proof of Remark 1.14. We have to show that for llm-a.e.
Z
E sing M
there is a unique tangent space for sing M at Z in the Hausdorff distance sense, and also that M has a unique tangent cone at z. For the former of these we have to show that, for llm-a.e. Z E sing M, there is an m-dimensional subspace L z such that for each € > 0 (1)
B1 (0)
n TJz,~ (sing M)
B1 (0)
n Lz
C the f-neighbourhood of Lz
and (2)
C the f-neighbourhood of TJz,~ (sing M)
for all (J' E (0, (J'o) where (J'o = (J'O(f, M, z) .J.. 0 as € .J.. O. Using the notation in the last part of the proof above, let z E Sj be any point where has an approximate tangent space. Then there is an m-dimensional subspace Lz with
st
(3)
RECTIFIABILITY OF THE SINGULAR SETS
301
(Notice such L" exists for Jim-a.e. z E Sj because Sj is locally m-rectifiable.) We show that (1) and (2) hold with this L". In fact the inclusion (2) is evidently already implied by this, so we need only to prove (1). Let Uk .!- 0 be arbitrary, and let C be any tangent cone of M at z with 11",tT., M -t C for some subsequence Uk" By (3) it is evident that the fk neighbourhood of Bl (0) n 11",tT., contains all of Lz n B 1 / 2 (0) for some sequence fk .!- 0, so that, in consequence of the upper semi-continuity 2.3,
st
8c(y) ~ 8c(0) = 8 M (0)
everywhere on L z n B 1 / 2 (0).
Thus by 2.5 and 2.6 we have Lc :J L", and since L z has maximal dimension m, this shows that Lc = L", so C E r with Lc = L". But then by 4.6 we have
Bl (0) n 11",tT., (sing M) c the fk-neighbourhood of L" for some sequence fk .!- O. In view of the arbitrariness of the original sequence 11k we thus obtain (2) as claimed. Finally we want to show that there is a unique tangent cone of C at Jim_ a.e. z E singM. Let Sj = {z E singM : 8M(Z) = aj} as above. For each f > 0, we can subdivide Sj into U~l Sj,i, where Sj,i denotes the set of points Z E Sj such that the conclusions (1) and (2) hold with Uo = t. Provided the original wo, U1 in the definition 7.4 of M are selected with Wo E Sj,i and U1 = 111(f,M,wo,i) ~ t. by (1) and (2) we then have that all points of z E 11wD, tTl Sj,i are contained in the set in the proof of Theorem 2 above. Hence by (31) of the above proof we conclude that there is a unique tangent cone of M at each point z E Sj,i n BtT, (wo) with the exception of a set of Jim-measure ~ fUr. In view of the arbitrariness of f, Wo here (and keeping in mind that we have already established that Sj,i is locally m-rectifiable) this shows that there is a unique tangent cone of M for Jim-a.e. points Z E Sj,i' Since Jim (sing M\(Ui,jSj,i)) = 0, the proof is complete.
Tit
7 Theorems on Countable Rectifiability. Recall that a set is countably m-rectifiable if it can be written as the countable union of m-rectifiable sets. There are some theorems about countable rectifiability of the singular set even without the hypotheses 1.13, 1.13' (Le., without assuming that we are in the top dimension of singularities over the entire class of maps or surfaces under consideration). For minimizing maps such theorems are established in [32]. Here we want to establish such a result for M EM. We are going to prove that sCm) is countably m-rectifiable, where, for a given ME M and m E {I, ... , n -I}, SCm) is the set of points Z E singM such that all tangent cones C of M at z are such that dim sing C ~ m. In fact we shall prove the stronger result that T(m) is countably rectifiable, Where T(m) is the set of points z E sing M such that all tangent cones C of At at z have dim Lc :5 m and sing C = Lc if dim Lc = m. Since trivially SCm) C T(m), this will also prove the above claim about s(m). For each 6 > 0, let Tjm) denote the set of points Z E sing M such that, Whenever C E C with inf.,.E(o,cI) fBI (o)n'1.... M dist 2 ((x, y), C) < 6, then we have
302
LEON SIMON
dim Lc ~ m and sing C
= Lc if dim Lc = m. We claim that
(1)
T(m)
Indeed if z ~ Uf=,l T~h)' then for (0, l/i1 with
C u Vj, and if C E C with f,,.,,,MnBl(0)dist 2 ((x,y),CnB I (0)) < 6 for some (T E (0,6], then, with this
(T,
f"oj,,,MnBd b) dist 2
large j. Since Zj E
Tt;;>,
«x, y), C n BI(O» < 6 for all sufficiently
we have dimLc ~ m, and also singC
= Lc E TJ;>
and
SUPmsn-",-l L~=o IDi Aco I ::; (3 in case dim Lc = m. That is, Z and hence (6) is proved. All the arguments used in the proof of Theorem 4 now carryover to the present setting essentially without change provided we use TJr;;) n S+ in place of S+. (Whenever we needed 2.12 before, we can now use inst'ead (5) above.)
RECTIFIABILITY OF THE SINGULAR SETS
303
Thus We conclude that for each given 6, /3 > 0 and for each z E TI,r;;) with eM(Z) = ec(O) for some C E C with singC = Lc of dimension m, there is p> 0 such that Bp(z) n {w E TI,r;;) : eM(W) ~ eM(Z)} is m-rectifiable, and then the argument in the last part of the proof of Theorem 2 shows that TI,r;;) locally decomposes into a finite union of locally m-rectifiable subsets. In view of (1) and the fact that TIm) = U~lTJ,i), which proves that T(m) is countably m-rectifiable as claimed. REFERENCES
[1]
[2]
[3J [4J [5J [6J [7J [8] [9J
[lOJ [11J
[12J [13J
[14] [15]
F. Almgren, Q-valued functions minimizing Dirichlet's integral and the regularity of of area minimizing rectifiable cur'rents up to codimension two, Preprint. F. Almgren, Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure, Ann. of Math. 87 (1968) 321-391 W. Allard, On the first variation of a varifold, Ann. of Math. 95 (1972) 417-491. F. Bethuel, On the singular set of stationary harmonic maps, CMLA, Preprint # 9226. H. Brezis, J.-M. Coron, & E. Lieb, Harmonic maps with defects, Comm. Math. Physics 107 (1986) 82-100 E. De Giorgi, Frontiere orientate di misura minima, Sem. Mat. Scuola Norm. Sup. Pisa (1961) 1-56. C. L. Evans, Partial regularity for stationary harmonic maps into spheres, Arch. Rational Mech. Anal. 116 (1991) 101-163 H. Federer, Geometrio Measure Theory, Springer, Berlin, 1969. H. Federer, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 76 (1970) 767-771. E. Giusti, Minimal surfaces and functions of bounded variation, Birkhauser, Boston, 1983 M. Giaquinta & E. Giusti, The singular set of the minima of certain quadratic functionals, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984) 45-55 D. Gilbarg & N. Trudinger, Elliptic Partial Differential Equations of Second Order (2nd Edition), Springer, Berlin, 1983. F. Helein, Regularite des applications faiblement harmoniques entre une surface et une variete Riemanninenne, C.R. Acad. Sci, Paris 312 (1991) 591-596. R. Hardt & F.-H. Lin, The singular set of an eneryy minimizing harmonic map from B4 to 52, Preprint, 1990. R. Hardt & F.-H. Lin, Mappings minimizing the LP norm of the gradient,
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LEON SIMON
Comm. Pure & Appl. Math. 40 (1987) 555-588. J. Jost, Harmonic Maps between Riemannian Manifolds, Proc. Centre for Math. Anal., Australian National Univ., 3 1984. S. Luckhaus, Partial Holder continuity for minima of certain energies among maps into a Riemannian manifold, Indiana Univ. Math. J. 37 (1988) 349-367. S. Luckhaus, Convergence of Minimizers for the p-Dirichlet Integral, Preprint, 1991. C. B. Morrey, Multiple integrals in the calculus of variations, Springer, Berlin, 1966. S. Lojasiewicz, Ensembles semi-analytiques t Inst. Hautes Etudes Sci. Publ. Math., 1965. R. E. Reifenberg, Solution of the Plateau problem for m-dimensional surfaces of varying topological type, Acta. Math. 104 (1960) 1-92. E. Riviere, Everywhere discontinuous maps into the sphere, Preprint. R. Schoen & L. Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981) 741-797. R. Schoen & K. Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geometry 17 (1982) 307-336. L. Simon, Lectures on Geometric Measure Theory, Proc. Centre for Math. Anal., Australian National Univ., 3 (1983). _ _ , Cylindrical tangent cones and the singular set of minimal submanifolds, J. Differential Geometry 38 (1993) 585-652. _ _ , On the singularities of harmonic maps, in preparation. _ _, Singularities of Geometric Variational Problems, to appear in Amer. Math. Soc., Proc. RGI Summer School (Utah). _ _ , Proof of the Basic Regularity Theorem for Harmonic Maps, to appear in Amer. Math. Soc.( Proc. RGI Summer School (Utah) _ _II Asymptotics for a class of non-linear evolution equations, with applications to geometric problems, Ann. of Math. 118 (1983) 525:-572. _ _, Theorems on regularity and singularity of harmonic maps ETH Lectures, 1993, to appear. _ _, Rectifiability of the singular set of energy minimizing maps, Calculus of Variations and PDE, 3 (1995) 1-65. J. Taylor, The structure of singularities in soap-bubble-like and soap-filmlike minimal surfaces, Ann. of Math. 103 (1976) 489-539. B. White, Non-unique tangent maps at isolated singularities of harmonic maps Bull. Amer. Math. Soc. 26 (1992) 125-129.
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B. White, Regularity of the singular sets in immisicible fluid interfaces Proc. CMA, Australian National Univ., Canberra 10 (1985) 244-249. STANFORD UNIVERSITY
IN DIFFERENTIAL GEOMETRY, 1995 Vol. 2 ©1995, International Press
SURVEYS
Homology cobordism and the simplest perturbative Chern-Simons 3-manifold invariant CLIFFORD HENRY TAUBES
1 Introduction. Witten predicted [18] that certain products of a certain 2-form could be integrated over products of a compact, oriented 3-manifold to give differential invariants of the 3-manifold. These predicted invariants were first constructed by Axelrod and Singer [2, 3] in the case where the 3-manifold has the rational homology of 8 3 • (A similar prediction in [18] for computing Jones' knot invariants had been partially realized by Bar Natan [4].) Subsequently, Kontsevich [9] gave an alternative realization of Witten's proposed invariants, with the same constraint on the homology of the 3-manifold. (Presumably, the invariants of Axelrod/Singer and of Kontsevich are the same, but the author has not seen a proof that such is the case.) Note that the invariants of Axelrod/Singer and Kontsevich have only been calculated for the 3-sphere (where they vanish). The Axelrod/Singer and Kontsevich invariants are formally related to the 3-manifold invariants of Reshitikin and 'furaev [14]. (The relationship here is presumed analogous to that between Jones, HOMFLY and other knot invariants and the knot invariants of Vassiliev [16], [17]j see [6], [5t [10].) There is no theorem at present which describes the precise relationship between these various 3-manifold invariants. Such a theorem would be useful in light of the fact that the invariants of Reshitikin and 'furaev can be explicitly computedj they have been computed in closed form for lens spaces [8] and Seifert fibered 3-manifolds [13]. This is the first of two articles focusing solely on the simplest of the invariants of Kontsevich, an invariant, 12 , which assigns a number to a 3-manifold M (as constrained above) by integrating the cube of a certain real valued 2-form over M x M. Of particular concern here is the value of 12 on the 3-manifold boundaries of a 4-dimensional spin cobordism which has the rational homology of 8 3 • The results in this article, together with those in the sequel [15], prove that 12 (M) = 12 (M') when M and M' are the boundary components of an oriented, spin 4-manifold W for which: 1. The intersection form on W's second homology (mod torsion) is conjugate to a direct sum of metabolic pairs. 2. The inclusions of M and M' into W induce injections of Hl(·jZ/2). (1.1)
(A metabolic pair is a symmetric, 2 x 2 matrix with zero's on the diagonal.) In particular, the preceding result implies that 12 (M) = 0 when M has the integral homology of 8 3 • These results are restated and proved in [15].
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307
This article makes a large step on the way to (1.1)j the main theorem here, Theorem 2.9, states (in part) that 12 (M) = 12 (M') when M and M' are the boundaries of an oriented, spin 4-manifold W for which the inclusions of M and M' into W induce 1) Isomorphisms on H,lj Q) for p = 0, ... ,4. 2) Injective maps on H1 (·j71./2). (1.2)
In the course of proving Theorem 2.9, 12 (8 3 ) is shown to vanish. Thus, even without the sequel [15], the main theorem here can be used, in principle, to show that 12 vanishes for certain 3-manifolds. (It is possible that 12 == 0 for all
M!) The author hopes that the constructions in this article will prove useful in studying the full set of invariants of Axelrod/Singer and Kontsevich, and this accounts, in part, for the length of the presentation. (The constructions here playa crucial role in [15].) Before beginning the story, the author wishes to thank Robion Kirby and Paul Melvin for their comments concerning this work, and also for their encouragement and support. A debt is owed as well to Dror Bar-Natan for sharing his knowledge of knot invariants. This article is organized as follows: The definition of 12 and the main theorem (Theorem 2.9) are given in the next section. The remaining sections (3-11) are occupied with constructions that are needed for the main theorem's proof. Section 3 is a digression to present certain facts from Morse theory. Section 4 studies the homological constraints which arise in the proof. Sections 5-10 contain the construction of a solution to the homological constraints. The final aspects of the proof of the main theorem are provided in Section 11. 2 The definition and properties of 12 (M). The purpose ofthis section is to give a definition of Kontsevich's invariant, 12 (.), for compact, oriented 3-manifolds that have the rational homology of S3. This section also contains the paper's main theorem about the equality of 12 for a pair of 3- manifolds which occur as the boundary components of a certain kind of 4-dimensional cobordism. a) Topological considerations. Let M be a compact, oriented 3-manifold with the rational homology of S3. Fix a point Po EM. Let t::. c M x M denote the diagonal. Define the subspace (2.1)
I: == .6. u
CPo
x M) U (M x Po).
Lemma 2.1 describes the cohomology of M x M - I:. Before reading Lemma 2.1, be forewarned that a regular neighborhood of I: in M x M is a neighborhood
CLIFFORD HENRY TAUBES
308
of E which strongly deformation retracts (reI E) onto E. It is an exercise to show that such neighborhoods exist. Also, in Lemma 2.1, cohomology is computed with real (IR) coefficients.
LEMMA 2.1. Let E be as defined in (fU). Then 1) H2((M x M) - E) ~ R 2) Let N C M x M be a regular neighborhood of E. Then, restriction gives an isomorphism H2((M x M) - E) ~ H2(N - E). 9) Let i : IR3 ~ N be an embedding which intersects E - CPo,Po) transversely in a single point, i(O). Then i* : H2(N - E) ~ H2(IR3 - 0) is an isomor-
phism.
4)
Hl((M x M) - E)
~
Hl(N - E)
~
O.
Proof. For the first assertion, use Meyer-Vietoris to prove that (M x M) ((Po x M) U (M x Po» has the rational homology of IR6. Then, use Meyer-Vietoris again to compute the cohomology of the remainder when t:J.. is deleted. In fact, this calculation with Meyer-Vietoris shows that M x M - E has the rational cohomology of 8 3 x 52.
Prove the second assertion using the Meyer-Vietoris exact sequence for the cover of M x M by N and M x MUE. (The Kunneth formula gives H2 (M x M) = 0, while restriction injects H 3 (M x M) into H 3 (E).) The third and fourth assertions are left as exercises with Meyer-Vieto-
0
~.
The cohomology of (M x M) - E with rational coefficients is isomorphic to its DeRham cohomology.
b) An invariant. Let C denote the set of pairs (N, cp) where N is a regular neighborhood of E, and where cp : N ~ IR3 is a smooth map with the property that cp-l (0) = !:. Define an equivalence relation on C as follows: Say that (No, CPo) and (N1 ,cpd are equivalent if there is a regular neighborhood N2 C No n Nl and a smooth map (2.2)
=
which obeys q,(0,·) = cp(O) and q,(l,.) CPl and q,-l (0) = [0,1] x E. Let c denote the set of equivalence classes in C. Now, change gears somewhat and pick a smooth, closed 2- form, It, on IR3 - 0 whose integral over the standard unit 2-sphere is equal to 1. For example,
Let cp E C. According to Lemma 2.1, there exists a smooth, closed 2-forIll on M x M - E which agrees with cp. It on N. Fix such a form and call it w", •
HOMOLOGY COBORDISM
309
PROPOSITION 2.2. Let (N, It') E C and choose the following integral converges:
(2.4)
12
=[
lMxM-r.
w", A
w'"
as described above. Then
w'" A w'"
Furthermore, 12 is independent of the choice of w'" to extend It'* Il, and it is independent of the choice of Il. Infact, 12 depends only on the equivalence class of (N, It') in c .
Proof. The integral converges because the integrand has compact support on M x M - N. Indeed, w'" A w'" vanishes on N because w'" on N is the pull back of a form on 8 2 • Now, suppose that (No,It'o) and (NI,It'l) define the same equivalence class in c. Suppose that ILo and III are different choices for Il in Proposition 1.2. Suppose that Wo and WI are closed 2-forms on M x M - E which extend It'o ILo and It'i IJ.I from No and N, respectively. Let N2 C No n NI and "' Il to [0,1] x (M x M - E) as a closed 2-form w. With W defined, compute
°
(2.5)
d(wAwAw)
0= [ l[o,ljx (MxM-J:.)
using Stokes' theorem to express 0 (i.e. Equation (2.5)) as a sum of three terms. (Note that the integrand in (2.5) is compactly supported away from [0,1] x E since w is pulled back from a 4- dimensional manifold on [0,1] X N 2 .) The three terms alluded to above are as follows: The first term is the contribution to Stokes' theorem from {1} x (M x M - E)j it is the integral in (2.4) as computed using the data with subscript ~'t". The second term is the contribution to Stokes theorem from {OJ x (M x ME)j it is the integral in (2.4) as computed using the data with subscript "0". To write down the third term which contributes to the Stokes' theorem computation of (2.5), one must first fix N C N 2 , a smooth, oriented, codimension 1 submanifold that separates E c M x M from M x M - N 2 • With N understood, here is the third contribution to (2.5): (2.6)
812
=[
w /I. w /I.
W
l[o,ljxN
Note that (2.6) is zero because w on [0,1] x N equals 0 on W - crit(f). Also, require of v that it have the following form near p E critk (f) : The pushforward by (t/Jp)-l of v should restrict to a small ball about the origin in lR4 to equal (3.3)
A pseudo-gradient vector field for f will often be called a pseudo- gradient, for short.
CLIFFORD HENRY TAUBES
324
A gradient flow line of a pseudo-gradient v is a map -y, of a closed interval,
I, into W with the following properties: 1) 1= [a,b] with -00 ~ a < b ~ +00. 2) IT a = -00, then -y(a) E crit(f); and if b = +00, then -y(b) E crit(f). 3) IT a> -00, then -y(a) E Mo; and if b < +00, then -y(b) E MI. 4) -y* (at) = v l-r(t) for all tEl (3.4)
(Here, at differentiates the coordinate t to give 1.) IT -y is a gradient flow line of a pseudo-gradient, v, say that -y begins at -y(a) and ends at -y(b). There is a great deal of flexibility in the choice of a pseudo- gradient. And, there are specific constraints which can be imposed on a pseudo-gradient which simplify some subsequent constructions. c) The Morse complex.
With the help of good Morse function f and an appropriate pseudo gradient, v, one can define a finite dimensional complex whose homology is naturally isomorphic to the relative homology H*(W, Mo; Z). (See, e.g. [12].) The complex is written (3.5)
To describe the {Ck} in (3.5), it is necessary to first digress to review the construction of ascending and descending disks: As described in [12], one can use v to define, for each p E critk(f), a pair of open subsets, Bp_ C int(W) and B p + C int(W), which are embedded disks of dimension k and 4 - k, respectively. Here, Bp+ is the ascending disk from p, and Bp_ is the descending disk from p. As a subset, Bp_ is the union {p} with the set of points of int(W) - crit(f) which lie on gradient flow lines which end at p. And, Bp+ is the union of {p} with the set of points in W - crit(f) which lie on gradient flow lines which start . at p. Note that (3.4) implies that 1) 1/Jp(Bp- n Up) = {(Xl,'" ,X4) E ]R4 : Xk+l = ... = X4 = O}, 2) 1/Jp(Bp+ n Up) = {(Xl,· •• ,X4) E ]R4 : Xl = ... = Xk = O}. (3.6)
These disks intersect at one point, p, and there transversally. Otherwise, (3.7)
f
I (Bp_
- p)
< f(P) < f I (Bp+ - p)
HOMOLOGY COBORDISM
325
As W is assumed oriented, an orientation for Bp_ orients Bp+ so that their intersection number, [Bp_] • [Bp+], is equal to {I}. End the digression. One defines C le in (3.5) from the free Z-module, C Ie , on the set of pairs (3.8)
{(P, €) : p E critle(f) and € is an orientation for B p _}
Indeed, set Cle == ~/ ,.." with the equivalence relation (p, €) '" -(p, -f). (The Cle for different choices of pseudo-gradient are canonically isomorphic.) To define the operator in (3.5), it is necessary to make a two part digression. Part 1 of the digression introduces some constraints on the pseudo-gradient v. These are described next.
a
DEFINITION 3.l.Let f be a Morse function on W. A pseudo-gradient v will be called good if the following criteria are met: 1) If p, q E critle(f) and if Pi- q, then Bp+ n Bq_ 0.
=
2)
If p E critle(f) and q E Critlc+l (f) then Bp+ intersects Bq_ transversely. Furthermore, Bp+ nBq_ is a finite union of gradient flow lines, the closure of each starts at p and ends at q.
3) Ifp E critl(f) and q E crit3(f), then Bp+ intersects Bq_ transversally. 4) If Po E Mo and PI E Ml have been apriori specified, then require that Po start a gradient flow line with end at Pl. (By the way, because of their definitions in terms of v's flow lines, descending disks from distinct critical points do not intersect, and likewise, ascending disks.) See [12] for a proof that good pseudo-gradients exist. Henceforth, assume that v is a good pseudo-gradient. Part 2 of the digression considers the intersections of ascending and descending disks. Start the discussion with the introduction of MIe,Ie-l = /-1(4- 1 k -1/8). Due to (3.1), one can conclude that df is nowhere zero along MIe,Ie-l, so this subspace is an embedded submanifold of W. Furthermore, M Ie ,Ic-l is naturally oriented by using df to trivialize its normal bundle. Because of (1) in Definition 3.1, each B p _ intersects MIe,Ie-1 in its interior as a (k - 1) sphere Sp- which is oriented (by df) when Bp_ is oriented. Likewise, Bp+ intersects MIc+I,1e in a sphere, Sp+, of dimension 3 - k which is oriented when Bp_ is oriented. Note that Definition 3.1 implies (in part) the following assertion: If p E critle(f), then Sp- has transversal intersection in MIe,Ie-1 with any Sq+ from q E critle-l (f). With the preceding understood, use [Sp-l . [Sq+ 1E Z to denote the algebraic intersection number of Sp- with Sq+ in MIe,Ie-I. End the digression. Here is the definition of the boundary map a in (3.5):
(3.9)
a(p, €)
==
L:
([Sp-l' [Sq+]) (q, €q).
qEC._ 1
See [12] for a proof that (3.5) with 8 as in (3.9) is a cllain complex whose homology is isomorphic to H
* (W, Mo; Z).
CLIFFORD HENRY TAUBES
326
Note that there is a dual complex to (3.5),
o -+ C*3
(3.10)
~ C*2 ~ C*I -+ 0
which is defined using -land -v when (3.5) is defined from the pair 1 and v. The homology of (3.10) computes H*(W, MI ; Z). Poincare' duality identifies H*(W, M 1 ; Z) with H 4 -*(W, Mo; Z), hence the duality between (3.10) and (3.5).
d) Factoring the cobordism. The purpose of this subsection is to indicate how to factor the cobordism W into two simpler cobordisms. The following proposition summarizes: PROPOSITION 3.2. Let M o , Ml be compact, oriented 9-manilolds with the rational homology 01 S3. Suppose that there is an oriented, spin cobordism, W', between Mo and MI. Then there exists an oriented, spin cobordism, W, between Mo and MI which decomposes as W = WI U WI U W 3, where 1} aWl = -MoUM~,aW2 = -M~UM{, and aW3 = -Mf UMI , where M~ and M{ are compact, oriented 9-manifolds with the rational homology of S3.
2}
W 1 ,2,3 are oriented, spin manifolds.
9} Both WI and W3 have the rational homology of S3. Meanwhile, W 2 has 4}
5}
6}
7}
vanishing first and third Betti numbers. WI and W3 have a good Morse functions with no index 3 critical points. Meanwhile, W 2 has a good Morse function without index 1 and index 3 critical points. If W' has the rational homology of S3, then W above can be assumed to have the rational homology of S3. And, one can assume that M~ = Mi and that W2 is the product cobordism. Let IWI and lw be as given in (2.12). Suppose that CMo or CMl (as in Definition 2.8) is represented by c in ker(lwl). Then lw(c) = 0 too. The intersection forms of Wand W' are conjugate by an element of GI(·, Z).
In particular, Assertions 5 and 6 of the preceding proposition allow one to prove Theorem 2.9's statements concerning 4- dimensional spin cobordisms with the rational homology of S3 between a pair of 3-manifolds with the rational homology of S3 by restricting to the following special case: Special Case: Let M o, Ml be compact, oriented 3-manifolds with the rational homology of S3. Let W be an oriented, spin cobordism between Mo and MI' Assume that W has the rational homology of S3 and assume that W has a good Morse function f with no index 3 critical points. (3.11)
The remainder of this subsection proves Proposition 3.2.
HOMOLOGY COBORDISM
S27
Proof of Proposition 3.2. First of all, let W' be the original spin cobordism between Mo and MI. Then, surgery on W' will produce an oriented, spin cobordism W which has vanishing first and third Betti numbers. The surgery removes tubular neighborhoods of embedded circles and replace them with copies of B2 x 8 2 • (Here, B2 is the unit ball in ]R2.) Given such W, find a good Morse function I on W and a good pseudogradient, Vj and then define the complex in (3.5). Label the critical points of index 1 as {aI, ... , ar }, label those of index 2 as {b 1, ... , br+s+tl, and label the index 3 critical points as {eI,'" , ct}. Here,s = dim(H2 (Wj JR)). (Remember that W has, by assumption, vanishing rational homology in dimensions 1 and 3.) Fix orientations for the descending disks from all of these critical points. With this understood, this set of critical points defines a basis for the complex {CIe} in (3.5). Now, it is convenient to relable the basis for C2 as follows: Since the map 8 EEl 8* : C2 -+ C1 EEl C3 is a surjection, one can relable the critical points {ba} so that (3.12)
8*: Span{br+s+i}~=1 -+ Cs , are both isomorphisms over Q. At the same time, one can require that the projection of Span {b r +i H=1 onto C2 /(8*C1 EEl 8Cs) is an isomorphism. With (3.12) understood, one can use 4.1 in [12] to find a new good Morse function I new which has the following three properties: First, I new agrees with I outside small neighborhoods of the points in crit2(f). Second, I new has the same critical points and pseudo- gradient as I. Third, there exists small € > 0 such that 1) I new( {bI, ... , br }) E (1/2 - 2 €, 1/2 - €) , 2) 3)
Inew({br +1'''' ,br +s }) E (1/2-E,1/2+€), E (1/2 + €, 1/2 + 2E)
I new( {b r +s+1 , •• , br +8 +t})
(3.13) Note that (3.12), (3.13) indicate that W 1) WI == 1-1 ([0,1/2 - ED 2) W 2 == 1-1 ([1/2 - E, 1/2 + E]) 3) Ws == 1- 1 ([1/2 + €, 1])
= WI U W2 U W s , where
(3.14) The boundaries of Wl,2,3 are compact, oriented 3-manifolds with the rational homology of S3. This is guaranteed by (3.12). Meanwhile, the inclusion of any boundary component of W I ,3 into W I ,3 induces an isomorphism of rational homology. This is not true for W 2 ; this W2 has the zero first and third Betti numbers, but the second Betti number of W2 is equal to s.
328
CLIFFORD HENRY TAUBES
Note that the function / new can be used as a Morse function on W1,2,3. On WI, it has no critical points of index 3, on W3 it has no critical points of index 1, while on W2 , it has only critical points of index 2. The preceding remarks prove Assertions 1-5. To prove Assertion 6, suppose, for the sake of argument that eMo is represented by e in ker(lwl). Let ~ be a singular frame for T· Mo in the class e and let f be a smooth frame for T· Mo which agrees with ~ on the complement of a ball about Po. Write T·W' IMo~ T* Mo E9~, where ~, is the trivial bundle, spanned by df IMo. With this understood, ~' extends to a frame (~', df) for T*W' IMo. Note that lWI(e) is the obstruction to extending this frame over W'. Likewise, 1w(e) is the obstruction to extending «(',df) over W. With this understood remark that Assertion 6 will be proved by demonstrating that (e, df) extend over W if it extends over W'. This demonstration requires four steps. Step 1: Fix a frame e' for T* W' which extends
(e, df).
Step 2: Let q C W' be an oriented, embedded circle whose fundamental class is a generator of HI (W'; Z) /Torsion. Suppose a surgery is done on W' to kill the class generated by q. Such a surgery will replace a tubular neighborhood of q in W' with B2 X S3. Because 71'2(80(3» = 0, all framings of T*(B2 x 8 2) are mutually homotopic. A framing of T·(B 2 x S2) rest:i'icts to the boundary where it can be written as he', where h == h( e') is a map from 8 1 x 8 2 to SO( 4). IT h lifts to 8U(2) x SU(2), then the frame (q',df) will also extend over the manifold which is obtained from W' by surgery on q. Step 9: With this last point understood, suppose that h does not lift as required. The strategy is to abandon e' and find a new extension, e" for (q', df) so that the resulting h(e") does lift to 8U(2) x SU(2). Step 4: To construct e", let s : q -t SI be a degree 1 map. Since the restriction map H1(W'; Z) -t H1(q; Z) is surjective, the map s extends as a smooth map from W' to SI. Since Mo has vanishing first cohomology, the map s can be taken to map Mo to point, 1 E Sl. Let j : 8 1 -t 80(4) be a map which generates 71'1 (SO(4» and which takes 1 to the identity matrix. The composition k == j 0 s maps W' to 80(4) and maps Mo to the identity. Thus, e" == ke' defines an extension of (~',df) over W', and h(ke') = h(e)k-l wilrIift to map SI x 8 2 into SU(2) x SU(2). Thus, Assertion 6 follows from this last remark with Step 3. As for Assertion 7, it is directly a consequence of the fact that W is obtained from W' by surgery on a set of circle generators of HI (W'; Z) jTorsion. 0
e) A basis theorem for the Special Case.
Assume here that W is a cobordism which satisfies the assumptions of (3.11). In particular, W has the rational homology of S3, and also W has a good Morse function f with only index 1 and index 2 critical points. Fix a good pseudogradient v for f. Introduce the complex in (3.5) for W. This is a 2-step complex, and the boundarr map 8 : C 2 -t C1 is an isomorphism over the rationals. Let {aI, ••. , a r }
HOMOLOGY COBORDISM
329
label the index 1 critical points and let {b 1 ,··· ,br } label the index 2 critical points. Orient the descending disks from these critical points so that these sets of critical points can be considered as a basis for C1 ,2, respectively. With the basis for C1 ,2 chosen as above, the boundary maps in (3.5) are simply integer valued matrices. That is, Obi = EjSi"j aj, where S == {Si"j} is an integer valued, r x r matrix. Here is a useful observation: The precise form of the matrix S is determined by the choice of good pseudo-gradient v. With this fact understood, one can ask whether there is a choice of peudo- gradient for I which gives a "nice" matrix S. The answer to this question is given by Milnor's basis theorem (Theorem 7.6 in [12]): PROPOSITION 3.3. Let W be a cobordism which satisfies {9.11}. Then W has a good Morse junction, f, with no index 3 critical points and with the following additional properties: There exists a labeling, {a1,··· ,ar } and {b 1, ... br }, for the respective index 1 and index 2 critical points of I. And, there exists a good pseudo- gradient for I and a choice of orientations for the descending disks from I's critical points. And, this data is such that 1) For all i E {I,··· ,r}, 2) Obi = EjSiJ aj, where S == {Si,j} is an upper triangular, integral matrix with positive entries along the diagonal.
9)
For all i E {I,··· ,r - I}, one has I(ai)
> I(ai+d and I(b i ) > I(bi+d.
(3.15) The remainder of this subsection is occupied with proving this proposition. Proof of Proposition 3.3.
Start with a good pseudo-gradient, v, for
f. Fix orientations for the descending disks so that the boundary operator in
(3.5) can be represented as a matrix, T, so that Obi = EjTi,j aj. Note that the matrix T is integral and invertible over the rationals. Now, a fundamental result in algebra (see, e.g. [11]) states that there exists a unimodular, integral matrix V such that V T == T' has only zeros below the diagonal. Let Q== {Qi == E j Vi,j bj}. This is a new basis for C 2 , and oQ = V T a = T'a. With V and T' understood, appeal to Theorem 7.6 in [12] to find a pseudogradient for I, v', for which the resulting descending disks represent the basis b for C2 • For this pseudo-gradient, the boundary operator in (3.5) is given by the matrix T'. By changing the orientations of the descending disks if necessary, one can change the signs of the diagonal elements of T' so that they are all positive. Call the resulting matrix S. The given arrangement of the critical values of f can be insured by making an appropriate, small perturbation. 0 By the way, if the boundary aC2 -+ C 1 is an isomorphism over Z, then the matrix S in Propostion 3.3 can be taken to be a diagonal matrix.
CLIFFORD HENRY TAUBES
330
o· ;
As a last remark, note that the matrix for the adjoint complex, C 1 -+ C2 , will be the transpose of the matrix S in Proposition 3.3. This matrix, ST, will be lower triangular. 0
f) Morse theory on W
X
W
The manifold W x W is a manifold with boundaries and corners. Here it is:
WxW
(3.16) The reader is invited to formalize a "manifold with boundaries and corners", but the picture above should be self explanatory. The good Morse function 1 on W can be used to illuminate (3.16) near the corners. To do so, one must note first that Properties 1 and 2 in (3.1) make it possible to use the pseudo gradient to give W its product structure near oW. To be precise, there is a diffeomorphism,
(3.17)
Ao : 1-1([0,1/8» -+ Mo x [0,1/8)
which restricts to 1-1(0) as the identity and which has >"'01 given by projection to [0,1/8). There is a corresponding
(3.18) Using (3.17), a neighborhood of Mo x Mo in W x W is mapped by >"0 x >"0 to (3.19) (Mo x [0,1/8» x (Mo x [0,1/8»
R:l
Mo x Mo x [0,1/8) x [0,1/8).
Of course, >"0 x >"1, >"1 X >"0 and >"1 X >"1 give similar structure to the other corners ofW x W.
331
HOMOLOGY COBORDISM
With a good Morse function, I, chosen for W, introduce the function F : W x W -+ [-1,1] which sends (x, y) to
(3.20)
F(x, y)
=I(y) - I(x).
This is a function with properties that are listed in the next lemma. The lemma's statement uses the following notation: First, introduce the projections, 1fL : W x W -+ W and 1fR : W x W -+ W which send (x,y) to x and to y, respectively. Second, when v is a vector field on W, introduce the vector fields VL and VR on W x W which are defined so that (3.21)
1) 2)
= v and d1fR VL = OJ d1fL VR = 0 and d7rR VR = V.
d1fLVL
LEMMA 3.4. Let I be a good Morse function lor Wand let v be a good pseudo-gradient for I. Then, the /unction F of (9.20) has only non- degenerate critical points. Furthermore: 1) critn(1) = Uk(Crit4+k-n(1) x critk(1))· 2} The vector field VR - 'ilL is a pseudo-gradient for F which obeys 1- 9 of Definition 9.1. 9) The pseudo-gradient VR -'ilL gives the following descending and ascending disks for (p, q) E cri4+k-n (1) X critk (1) C critn (1):
(3.22) B(p,q)+
4)
= Bp_ x Bq+.
The pseudo-gradient VR -'ilL is nowhere tangent to a boundary or a comer in (9.16).
Proof· The proofs of these assertions are left as exercises. But, for Assertion 3, note for example that near Mo x M o, (AO X Ao)-l (Of (3.19)) pulls back F to send the point ((x, t), (y, s)) in (Mo x [0,1/8)) x (Mo x [0,1/8)) to (3.23)
((AO
X
Ao)-l)* F((x, t), (y,
s»
=
s _. t.
Note, by the way, that (3.22) indicates how to orient B(p,q)_ given orientations for Bp_ and B q+. And, with orientations to the descending disks {B(p,q)_ : (p, q) E crit(F)}, one can consider the analog to the chain complex C in (3.5) as constructed for W x W using the function F and the pseudo-gradient vR -'ilL. The following lemma describes the homology of this complex. 0
332
CLIFFORD HENRY TAUBES
LEMMA 3.5. The analog of the chain complex C in (3.5) as constructed for W x W using F and the pseudo-gradient VR - VL gives a chain complex, C F , which is canonically isomorphic to C· ® C, where C· is the complex in (3.10). The homology of the complex CF is canonically isomorphic to H* (W x W,(W x Mo) U (MI X W)jZ). Notice that the relative homology above is that of the square in (3.16) relative to the union of its bottom and right sides.
o
Proof. This follows from Lemma 3.4 and (3.22). g) The space Z.
As outlined in Section 2k, the first step to proving Theorem 2.9 is to construct an oriented, 7-dimensional manifold Z whose boundary is the disjoint union of Mo x M o, MI X Ml and some number of copies of S3 x S3. The purpose of this subsection is to construct such a Z using the cobordism W and a good Morse function f on W. To begin, construct F from f as in (3.16). Use F to define (3.24)
Z
== F-1(0) = ((x,y) E W
x W: f(x)
= f(y)}.
This subspace Z plays a central role in subsequent parts of the story, and the purpose of this subsection is to describe some of Z's properties. To begin, note that both Mo x Mo and MI x MI lie in Z since f is constant on Mo and also on MI. Near these corners, Z is a manifold with boundary given by the disjoint union of Mo x Mo and MI x MI' See (3.19). Unfortunately, Z is not a manifold everywhere unless f has no critical points. This is because 0 is not a regular value of the function F. Fortunately, the singularities of Z are not hard to describe; they occur at the points of crit(F) n Z, that is, points of the form (P,p) C W x W where p E crit(f). (Remember that the critical points of f are assumed to have distinct critical values.) Furthermore, the neighborhoods of these critical points are relatively easy to describe. The picture is given in the following lemma. The lemma introduces the nation of a cone on a manifold N. This is the space which is obtained by taking [0,1) x N and crushing {OJ x N to a point.
LEMMA 3.6. Let f be a good Morse function on W. Let p E critk(f). Then, a neighborhood of (P,p) in Z is naturally isomorphic to the cone on S3 x S3. In fact with tPP and Up = tPp(]R4) as in (3.2), then the map (t/Jp x tPp)-lmaps Z n (Up x Up) to a subset of]R4 X which intersects a ball neighborhood of (0,0) as the set 0/ (x, y) which obey
r
(3.25)
y~ + ... + y~ + x~+1 + ... + x~
= x~ + ... + x~ + 1I~+1 + ... + y~.
HOMOLOGY COBORDISM
333
Warning: As indicated by {3.25}, the cone on S3 X S3 here is not induced by the obvious product structure on W X W. The product structure which induces {3.25} is the product structure in (3.26)
with B(p,p)± as in {3.22}. Proof. Equation (3.25) is an immediate consequence of (3.2).
o
The manifold (with boundary) Z in Section 2k will be found inside Z; it is obtained by excising from Z. a small ball about each of the singular points (P,p) for p E crit(f). More precisely, one fixes some small r > O. Then, the intersection of Z with Up X Up is mapped by 1/Jp x 1/Jp to the set of (x, y) which obey (3.27) With (3.27) understood, aznup x Up is mapped by 1/Jp x 1/Jp to the set of (x,y) which obey 1) y~ + ... + y~ + X~+l + ... + x~ = r, 2) x~ + ... + x~ + y~+l + ... + y~ = r. (3.28) As the precise value of r here is immaterial (as long as r is small), the precise value will not be specified. There is an alternative approach to defining Z. Here, Z is a "blow up" of Z at the points of the form (P,p) E crit(f). In this case, Z maps to Z by a map 1r. Each point in Z - {(P,p) E crit(F)} has a single point in its inverse image. But, the inverse image of any point (P,p) E crit(F) is the corresponding S3 X S3 c az. This blow up corresponds to resolving the cone point in N == ([0,1) x N)/( {O} x N) with the tautological projection 1r : [0,1) x N ~ N. h) Properties of Z. With Z now defined, here are its salient features: A manifold: Z is a manifold with boundary, (3.29)
Orientation: The manifold int(Z) has a natural orientation. Indeed, W x W has a natural orientation. Then, int(Z) C F-l (0) is open, .and dF '" 0 on int(Z), so the 1 form dF trivializes the normal bundle to int(Z) C W x W. This serves to orient Z. The induced orientation on Ml X Ml C az agrees
334
CLIFFORD HENRY TAUBES
with its canonical orientation, but the induced orientation on Mo x Mo c az disagrees with the canonical orientation. To orient (83 x 8 3 )1" use the inclusion of W ~ ~w C W X W to orient..6. w and hence ~z. The boundary of ~z intersects (8 3 x 8 3 )1' as ~ss(== (~ss)p) Give (~ss)p the induced orientation from ~z. Then, orient the left factor of 8 3 in (8 3 X 8 3 )1' so that the composition of 7rL : ~Ss -+ 8 3 and then the inclusion 8 3 -+ (8 3 x point) C (83 x 8 3 )1' is orientation preserving. Orient the right factor analogously and use the product orientation to orient (8 3 x 8 3 )1'. (Remark that the induced orientation on (8 3 x 8 3 )1' as a boundary component of Z agrees with this orientation if index p is odd, and it disagrees if index p is 2.) Homology: The rational homology in dimensions 0-3 of Z is of some concern in subsequent sections. Consider
LEMMA 3.7. 8uppose that W has the rational homology of 8 3 • Then the rational homology of Z is as follows:
1) Ho(Z) ~)" H1(Z) 9)
~ ~
R H 2(Z)
~
o.
There is a surjection
(3.30) Here L_ is freely generated over IR by
(3.31)
L._ == {Btp,q)_ n Z : (p, q)
E cri4(F) and F(p, q)
> O},
E crit4(F) and F(p, q)
< O},
while L+ is freely generated over IR by
(3.32)
4
== {Btp,q)+
n Z : (p, q)
Note that the intersections which define L.± in (9.91), (9.9~) are all embedded 3-spheres. Also note that the inclusion of Z in W x W gives an isomorphism on 7rl and 7r2·
Proof. Note first that HO,1,2(Z) and HO,1,2(Z) agree, and that
(3.33) This follows using Meyer-Vietoris for the cover of Z by the union of Z and the cones on the (8 3 x 8 3 )1' in «3.25). Next, pick f > 0, but small so that F has only critical points ofthe form (P,p) in F- 1 «-f,f». Let V == F-l« .... f,f)) observe that V strongly deformation retracts into Z. Thus, Hi(V) ~ Hi(Z).
HOMOLOGY COBORDISM
335
To compute Hj(V), observe that W x W can be constructed from V by a sequence
V==V3 Cl/4CVsCV6 ==WXW,
(3.34)
where Vk+l is obtained from Vic by the attachment of disjoint handles, (BIc f< B 8 - 1c ),S, on disjointly embedded (Sk- l x B 8 -k),s in the boundary of Vic. To be precise, V4 contains all of F's index 4-critical points,
V4 == P-I([-1/8, 1/8]);
(3.35)
and V5 contains all index 3,4, and 5 critical points,
V5 == P-I([-3/8,3/8]).
(3.36)
The attaching 3-spheres for the handles that change V3 to V4 are given by (3.31), (3.32). Meanwhile, the attaching 4-spheres for the handles that change V4 to V5 are (3.37)
{B(p,q)_
n p-l (1/8) hp,q)E crit6(F) U{ B(p,q)+
n p-l (-1/8) }(P,q)Ecrits(F)-
The 5-spheres for the attachments that change Vs to V6 should be obvious. The resulting Meyer-Vietoris sequences from (3.34) read, in part, (3.38)
H 3 (L-t) EB H3C[~_) H 3(V4)
Rj
-4
H 3(V3) --+ H 3(V4 ) --+ 0,
H 3(V5)
Rj
H3(VS).
The third assertion in Lemma 3.7 follows from (3.38) and (3.33). The other [J assertions follow by Meyer-Vietoris from (3.34)-(3.37).
4 Homological constraints. In this section, Mo and Ml will both be oriented, 3- dimensional manifolds with the rational homology of S3. And, W will be an oriented, connected, spin cobordism between Mo and MI. Let J : W -4 [0,1] be a good Morse function. Use f to construct the space Z as described in Sections 3g and 3h. The proof of Theorem 2.9 is a five step affair which is outlined in Section 2k. The manifold (with boundary) Z of Sections 3g, h realizes the first step in the proof. The next step in the proof is to construct a subvariety Ez C Z with various properties as outlined in Steps 2 and 3 of Section 2k. The purpose of this section is to reformulate some of these requirements in a purely homological way.
336
CLIFFORD HENRY TAUBES
a) The homology of EM and M
X
M.
In order to understand the homological constraints on Ez, it proves useful to digress first with a homological interpretation of some of the constructions in Section 2. Return then to the milieu of Section 2 where M is a compact, oriented 3-manifold with the rational homology of 8 3 and where EM eM X M is defined by (2.1). The inclusion EM C M X M induces a surjective homomorphism on the respective rational homology groups in dimension 3, with a one dimensional kernel.
(4.1)
aM == [.:lM] - [Po X M] - [M X Po]
This aM bounds (rationally) in M
X
M, and a bounding cyle defines a class,
(4.2) (Here, H.(X, Y) for a space X and subspace Y C Xdenotes the relative homology with rational coefficients.) The Poincare dual of PM is the generator of H 2 (M X M -EM) which figures so prominently in Section 1. End the digression. b) Homological constraints on Ez from wz. Return to the bordism milieu of the introduction. The subvariety EE should have a physical boundary (as a cycle, for example) which is given by (4.3) where (Ess)" is the obvious Ess in the boundary component (8 3 X 8 3 )p of Z. Finding Ez to satisfy (4.3) would satisfy Step 2 in Section 1h. However, there are certain cohomological constraints on a solution to (4.3) which must be satisfied before it can solve the constraints which are implicit in Step 3 of Section 2k, and in particular, Parts 1 and 2 of (2.27). These are expressed by the following lemma: LEMMA 4.1. Let Ez C Z be a subvariety which obeys (/..3). Then, there is a closed 2-form, Wz, on Z - Ez which restricts to any component Y C 8Z - 8Ez to generate H2(y) if and only if H 4(Ez,8E z ) contains a class az which obeys: 1) The image of az in H4(Z, 8Z) is zero.
2)
(4.4)
8az in H3(8Ez) obeys 8az
= aMI
- aMo
+
L
(ass )p.
"Ecrit(f)
(The absence of signs in the last term in (4.4) stems from the convention of Section 3h for orienting the right and left factors of 8 3 in the boundary component (8 3 X S3)" C 8Z.)
The third constraint in (2.27) is the most difficult of all to satisfy. The strategy for satisfying the third constraint in (2.27) has two parts, one homological
HOMOLOGY COBORDISM
337
and the other geometric. For both parts, fix Nz C Z, a regular neighborhood of Ez. The homological issue is to characterize a closed 2-form on N z - Ez which is the restriction from Z - Ez of a closed 2-form Wz from Lemma 4.1. The geometric issue is to find such an w which obeys w A w = O. The following lemma resolves the homological issue: LEMMA 4.2. Suppose that Conditions 1 and !J of Lemma 4.1 are obeyed. Let N z C Z be a regular neighborhood of Ez. A closed 2-form, w, on N z - Ez is the restriction to Nz - Ez of a closed 2-form Wz on Z - Ez as described in Lemma 4.1 if the following occur: 1) The connecting homomorphism from H2(Nz - Ez) to H~omp(Nz) sends w to a multiple of the Poincare dual of az E H4 (Nz, N z n oZ). 2) The restriction homomorphism H2(Z) -t H2(Ez) is surjective.
This lemma is proved below. The last subsection in this section discusses the strategy for finding an appropriate w near Ez with w A w = O. Proof of Lemma 4.1. To prove necessity, start with the observation that the cohomology class in H2(Z - Ez) of the 2-form in question has Poincare dual (4.5)
pz E H5(Z, Ez U oZ).
The requirements in (2.27.1) and (2.27.2) concerning the restriction of Wz to OZ imply the following homological condition on opz (4.6)
opz
= PMl -
PM2
L
+
(pss)p - az,
pEcrit(f)
where (4.7) is a class which obeys (4.4) (so that 02pZ will vanish). To prove the sufficiency assertion of the lemma, start with pz as described. Represent az as a cycle on Ez. By assumption, one has az - T = opz, where T is a 4-cycle on oz, and where pz is a 5-cycle on Z. Note that OT is equal to the right side of (4.6) also. Thus, (4.8)
T - (PMl - PM2
+
L
(Pss )p)
pEcrit(f)
has zero boundary, and so defines a class in H4(8Z). However, this group is zero (H4(8Z) ~ H2(8Z) = 0 (see Section 2). Thus, (4.6) holds for some 5-cycle
338
CLIFFORD HENRY TAUBES
pz on Z. The Poincare dual of pz is a class in H2(Z - Ez) with the required properties. 0 Proof of Lemma 4.2. The question of extending a closed 2-form on N z Ez over Z - Ez is described by part of the Meyer-Vietoris sequence for the cover of Z by (Z - Ez) U Nz. The relevent part is: (4.9)
H2(Z) -+ H2(Z - E z ) EB H 2(Ez) -+ H2(Nz - Ez) -+ H3(Z)
The last arrow in (4.9) factors through the inclusion induced map H~omp(Nz) -+ H 3(Z). So, if the image of w in H~omp(Nz) is Poincare dual to a multiple of Uz as a class in H4(Nz, Nz n 8Z), then the image of win H3(Z) is zero if the image of Uz in H4(Z,8Z) is zero. This is the first condition in Lemma 4.1. Thus, under Condition 1 of Lemma 4.2, the class w maps to zero in H3(Z). When Condition 2 of Lemma 4.2 holds, then w must be in the image of the restriction homomorphism from H2(Z - Ez) because of the exactness of (4.9). 0 c) Satisfying Lemma 4.1's constraints. The second constraint in Lemma 4.1 will be satisfied by construction; as it is essentially a restatement of (4.3) with orientations taken into account. The first constraint in Lemma 4.1 is more subtle. Here is a strategy for finding a solution: The variety Ez will be constructed from a union of varieties, (4.10)
Each variety on the right side of (4.10) will carry a fundamental class. (Here, a variety is a union of embedded submanifolds. If the constituent submanifolds are oriented, then the variety has a fundamental class which is the sum (in the relevent homology group) of the fundamental classes of the constituent submanifolds.) And, for a particular integer N > 0, the class Uz will be given
as (4.11) In (4.10), (4.11), az and EL,R are honest submanifolds; these will be defined in subsequent subsections. Meanwhile, E± will be honest varieties unless N = 1 in (4.11). The construction of E± is quite lengthy and starts in the next section with the completion in Section 10. But, see subsections 4/, 9 below. With (4.11) understood, the first constraint of Lemma 4.1 will be solved with the help of Lemma 4.3, below. (The statement of this lemma reintroduces L.~ from (3.31), (3.32).) LEMMA 4.3. Suppose that W has the rational homology of S3. Let V C Z be a union of dimension 4 sub manifold with boundary such that 8V c. {) Z. Suppose
339
HOMOLOGY COBORDISM
that each component of V cames a fundamental class. Then [V] E H4(Z, 8Zj 1R) vanishes if: 1) [8V] = 0 in H 3 ({JZj 1R). £) V has zero intersection number with any component x C (It- u 4). (The intersection number of V with an embedded, 9-dimensional submanifold of Z is defined to be the sum of the intersection numbers of the components of V.) Proof. Poincare duality equates H4(Z, 8Z) with H3(Z). Intersection theory makes this explicit, as the intersection pairing between H4(Z,8Z) and H3(Z) becomes, under Poincare duality, the dual pairing between H3(Z) and H3(Z), Now, use this fact with Assertion 3 of Lemma 3.7. 0
d) The subspace az Let a w
c W x W denote the diagonal.
Clearly, aw
c
Z. Let a z denote the intersection of aW with Z c Z. (Alternately, if Z is thought of as the blow up of Z, then a z can be defined as the inverse image of aw under this blow up.) Note that az is a submanifold with boundary in W, and
(4.12) The orientation of W defines an orientation for a w and thus for a z . The orientation of (ass)p is induced from the orientation of a z in Section 3h as a boundary component ()f az. With this understood, one has:
LEMMA 4.4.
Let [az] E H4(Z,8Z) denote the fundamental class of az.
Then
(4.13)
8[az] = -raMo]
+ [aMI] +
E
[(ass)p].
pEcrit(f)
as a class in H3(8Z). Proof. This is left as an exercise. As a final remark, note that (4.14)
l'his is a consequence of Condition 1 in Definition 3.1.
o
CLIFFORD HENRY TAUBES
340
e) The submanifolds ER,L' By assumption (see 4 of Definition 3.1), there is a gradient flow line for the pseudo-gradient v which starts at Po and which ends at Po. Let "I denote this line. Define 1) ER b x W) n Z, 2) EL (W x "I) n Z.
=
=
(4.15) Here are the properties of these spaces:
4.5. Both ER and EL are embedded submanifolds (with boundary)
LEMMA
of Z. Also,
1) 8ER = (Po x Mo) U (PI x MI). E) 8EL = (Mo x Po) U (MI x PI)' 9)
Let 7rL and 7rR denote the respective right and left factor projections from W x W to W. Then 7rR : ER -+ Wand 7rL l EL -+ W are both diJJeomorphisms.
4)
ER·n!::J. Z
= EL n!::J. z = ER n EL = b x "I) n !::J.z. Furthermore, this subspace ("I x "I) n!::J.z has a neighborhood U C Z with a diffeomorphism (of manifold with boundary) 1/Ju : U ~ [0,1] X lR.3 X lR.3 which obeys
(a) (b) (c) (d) (e) (f) (g) 5)
6)
1/Ju(h x "I) n !::J.z) = [0,1] x (0,0). 1/JU(ER) = [0,1] x {O} X lR.3 • 1/JU(EL) = [0,1] X lR.3 X {O}. 1/Ju(!::J.z) = [0,1] X !::J.RS, 1/Ju(Mo x Mo) = {O} X lR.3 X lR.3 • 1/JU(MI x M l ) = {I} x lR.3 x lR.3 . The interchange map (z, z') -+ (z', z) on Z is mapped by 1/Ju to
(t,x,y) -+ (t,y,x). Both ER and EL have empty intersection with the components of L._ U4 of (9.90), (9.91). Orient ER and EL by 7rL and 7rR, respectively. Then
(4.16)
8[EL]
= -[Mo x Po] + [MI x pd.
The remainder of this subsection is occupied with the proof of this lemma. Proof. Since "I is a flow line of v, it has a parametrization (4.17)
1.: [0,1]-+ W
with (-y* J)(t) = t. This implies that the function F of (3.16) restricts without critical points to "I x W and to W x"l j therefore, both ER and EL are submanifolds of Z.
HOMOLOGY COBORDISM
341
Assertions 1 and 2 of the lemma follow because 'Y is assumed to miss crit(f). To prove the third assertion, use 'Y to view ER as the graph of fin [0,1] x W, where'TrR restricts as the obvious projection to W. The proof of Assertion 3 for E L is analogous. To prove Assertion 4, note that 'Y, being embedded, has a neighborhood U"( C W with a diffeomorphism tP., : U"( ~ [0,1] X Ji3 which obeys f(tP:;l(t,x» = t and tP.,("(t» = (t,O). (Use the implicit function theorem to construct such a tP"( .) Take U = U., X U'" and take tPu = tP., X tP"( . The verification of (a)-(g) 0 follow immediately.
f) The varieties E±. With ~z and ER,L defined in the preceding section, the solution (Jz of (4.11) to Lemma 4.1's constraints is missing still [E_] and [E+]. Indeed, the class [~z] - [ER] - [EL] is a class in H 4 (Z, aZ) whose boundary is equal to
(4.18)
-(JMo
+ (JM1 +
L
[~S3]p,
pEcrit(f)
which is only a part of the right side of (4.4). As remarked earlier, the construction of E± is quite lengthy. To simplify matters, the decomposition given in Proposition 3.2 will be invoked to break the discussion into two parts so that the cobordism W can be assumed to obey the conditions of (3.11). That is, W will be assumed to have the rational homology of S3 and W has a good Morse function with no index 3 critical points. The construction of E± for W given by (3.11) is started in the next section with a digression to describe certain constructions on such W. The construction of E± for (3.11) is completed in Section 9. With W understood to be given by (3.11), here is a rough description of E±: Fix a good Morse function f : W ~ [0,1] with no index 3 critical points. Let aI, ... ,ar and b1 ,' •• ,br label the index 1 and index 2 critical points of f. Now fix a good pseudo-gradient, v, for f, and fix orientations from the descending disks from crit(f) such that the conclusions of Proposition 3.3 hold. That is, with the orientations implicit, the points aI, ... ,ar and bl , ... ,br define a basis for C 1 and C2 , respectively. And, with respect to this basis, the boundary map in (3.5), C 1 ~ C2 , is represented by an upper triangular matrix, S, with positive entries on the diagonal. A pair E±, of subvarieties (with boundary) of Z will be constructed with 8E± C 8Z. The variety E_ is obtained as the intersection with Z of a subvariety of W x W; this subvariety is constructed by performing various surgeries on multiple copies of products of the ascending disks from points in critl (f) with the descending disks from the points in crit2 (f). Meanwhile, the variety E+ is obtained as the intersection with Z of a different subvariety of W x W. In this case, the subvariety is constructed by surgery on multiple copies of the product of the descending disks from crit2 (f) with the ascending disks from critl (f).
a:
342
CLIFFORD HENRY TAUBES
The varieties E± will be naturally oriented and seen to define classes [E±] C H4(Z,8Z). The boundaries of these classes are
L: [S31,,_,
8[E-1 = N
(4.19)
"Ecrit(f)
8[E+1
=N
L:
[S3],,+,
"Ecrit(f)
where (4.19) has introduced the following shorthand: When p is a critical point of I, use [S3],,_ to denote [S3 x point] E H3((S3 x S3),,), and use [S3],,+ to denote [point x S3} E H3((S3 X S3),,),. Here, the orientations on (S3 x point) and (point x S3) are defined in Section 3h. (The diagonal in (S3 x S3)" is oriented as a component of the boundary of Az and then the right and left factors of S3 in (S3 x S3)" are oriented by using the canonical identification of S3 with ~S3.) The [E±], of (4.19) will be constructed to have zero intersection pairing with the classes in L± of (3.30). This will insure that Oz of (4.11) satisfies both requirements of Lemma 4.1. (See Lemma 4.3.) g) Constraints from Wz A Wz
= O.
With Ez in (4.10) constructed so that both requirements of Lemma 4.1 are satisfied, there is a 2-form on Z - Ez which is a candidate for the form Wz in Step 3 of Theorem 2.9's proof. The issue then arises as to whether Oz can be found to satisfy the conditions in (2.27). The construction of a closed 2-form which satisfies,the conditions of (2.27) is carried out in Section 10. However, to motivate some ofthe intervening contortions, here is a rough summary of the difficulties: Remark 1: As long as E± in (4.10) have empty intersection with Mo x Mo and with MI x M 1 , then there is no obstruction to finding Wz which obeys (2.27.1). (See Lemma 2.1) Remark 2: The remaining requirements of (2.27) are harder to satisfy. In particular, the second requirement in (2.27) will require that for each p E crit(f), (4.20)
1) 2)
E_ n (S3 x S3)" E+ n (S3 x S3)"
= S3 X x" = x" X S3.
This requirement and (4.19) are incompatible unless N = 1 or unless E± are singular. Together, (4.19), (4.20) force the use subvarieties for E± instead of submanifolds. Given (4.20), the second constraint in (2.27) can also be satisfied. (See Lemma 2.1 again.) Remark 3: The first condition of Lemma 4.2 is not easy to satisfy with a. 2-form w which obeys w A w = O. In the case where N = 1 in (4.11) (so E± are manifolds) the strategy will be to find a regular neighborhood Nz C Z of Ez and a map
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343
(4.21) which obeys 't'Zl(O) = I:z and which pulls back the generator of H~omp(1R3) to a non-zero multiple of the Poincare dual in H~omp(Nz) to (1z E H 4 (Nz, N z n aZ). In this case, (4.22) with J1. as in (2.3). In the case where N > 1 in (4.11), the preceding strategy will be modified. When N > 1 in (4.11), then 't'z, as in (4.21), will be defined only in a neighborhood of Az U EL U ER C I:z, and Wz will be defined near Az U EL U ER by (4.22). But, near the remainder of I:z (i.e. near most of E±), the form Wz will be defined somewhat differently. (The basic difference being that Wz will be defined locally as the pull-back of a closed 2-form from a space of dimension less than 4. However, the space in question will not always be 52. In some places, the space will be the compliment in 51 of N + 1 distinct points.) This strategy for constructing a closed, square zero solution to Condition 1 of Lemma 4.2 requires Az, EL,R and the constituent submanifolds of E± to have trivial normal bundles in Z. (See Remark 4, below.) The success of this strategy also requires that the mutual intersections of Z, EL,R and E± have a canonical form. (See Remark 5, below.) Remark -I: The normal bundle of Az in Z is trivial if and only if Z is a spin manifold. Indeed, H 4 (Az) = 0 and therefore, an oriented 3-plane bundle over Z is classified by its 2nd Stieffel-Whitney class. Since Z has trivial normal bundle in W x W, the Stieffel-Whitney classes of the normal bundle to Az in Z are the same as those of the normal bundle to Az in W x W. The latter are the restrictions of the Stieffel-Whitney classes of the normal bundle of Aw in W x W. And, this last normal bundle is naturally isomorphic to the tangent bundle of W. Finally remember that W is said to be spin if the 2nd StieffelWhitney class of its tangent bundle vanishes. The normal bundles to EL,R are trivial, since they are isomorphic to the normal bundle to the path 'Y in W. A constituent submanifold of E+ (or E_) has a normal bundle in Z. IT care is taken in constructing E±, then these normal bundles will be trivial too. Remark 5: The construction of a square zero w to satisfy the first condition of Lemma 4.2 seems to require that E± do not intersect each other or Az and EL,R in a complicated way. Infact, the E± that are finally constructed will have empty intersections with EL,R, while (4.23) where {filr=l are disjoint, embedded paths in l:1z. In fact, fix the label i E {I, ... ,r} and let a == ai and b == bi be the i'th pair of index 1 and index 2 critical points of the Morse function f as described by Proposition 3.3. Then, Vi
CLIFFORD HENRY TAUBES
344
will start at the point (Xa, Xa) E (S3
(S3
X
X
S3) a and will end at the point (Xb, Xb) E
S 3h.
Assertion 4 of Lemma 4.5 describes the intersections amongst EL,R and il z . Assertion 4 of Lemma 4.5 and (4.23) (with some conditions on normal bundle framings) insure that the intersection of E± with ilz has the appropriate form for the construction of a square zero w to satisfy the first condition of Lemma 4.2. Remark 6: The second condition in Lemma 4.2 will be satisfied by taking care to construct E± to have vanishing H2. Note that Z, ilz and EL,R all have vanishing H2. (See Lemmas 3.7, 4.4 and 4.5, respectively). Care must also be taken to insure that E± do not intersect each other or ilz and EL,R in a complicated way. Infact,
LEMMA 4.6. Suppose that Ez is given by (4.10) with ilz and EL,R as described in Sections 4d and 4e, respectively. Suppose that E± c Z are varieties which have empty intersection with EL,R and which intersect each other and ilz as described in (4.29). Suppose, in addition, that H2(E±i Q) = O. Then H2(Ezi Q) = 0 and the homomorphism H2(Z; Q) -t H2(Ezi Q) of Lemma 4.2 is surjective by default.
Proof. Because the intersections of ilz , EL,R and E± with each other are a union of line segments (which have vanishing Hl), the Meyer-Vietoris exact sequence shows that H 2 (Ez) is isomorphic to the direct sum of H 2 0 for (.) == il z , EL,R and E±. By assumption H2(E±) O. Meanwhile, H2(il z ) ~ 0, since ilz is the compliment in ilw ~ W of a finite union of disjoint (open) 0 4-balls. And, H2(EL,R) ~ 0 since EL.R ~ W.
=
Remark 7: In summary, the construction of E± for the case of (3.11) will proceed with care taken with: 1) Normal bundle framings. 2) 3)
Intersections with ilz,EL,R and with each other. Keeping H2(E±) equal to zero.
(4.24)
5 Disk intersections for the Special Case. The construction of E± for W given by (3.11) starts in this section with a digression to describe certain constructions on such W. The constructions here serve to modify the ascending disks from index 1 critical points and also descending disks from index 2 critical points. With W understood to be given by (3.11), begin the discussion by fixing a good Morse function! : W -+ [0,1] as described by Proposition 3.3. As in Proposition 3.3, let al,'" ,ar label 1's index 1 critical points and bl ,'" ,b,. label the index 2 critical points.
HOMOLOGY COBORDISM
345
Fix a good pseudo-gradient, V, for f, and fix orientations from the descending disks from crit(f) so that the conclusions of Proposition 3.3 hold. That is, with the orientations implicit, the points aI, ... ,ar and b1, ... ,br define a basis for Cl and C2 , respectively. And, with respect to this basis, the boundary map in (3.5), a : C1 -+ C2 , is represented by an upper triangular matrix, S, with positive entries on the diagonal. The matrix S gives a certain amount of algebraic information about the intersections of the descending disks from crit2 (f) and the ascending disks from Critl(f). That is, the intersection of the descending disk from bi and the ascending disk from aj is a discrete set of How lines which start at aj and end at bi. Each such How line carries a sign, ±1. And, the matrix element Bi,j computes the sum of these ±l's. In particular, Proposition 3.3 insures that the algebraic intersection number of the descending disk from bi and the ascending disk from aj is zero if i > j. However, even when i > j, the point aj may lie in the closure of the descending disk from bi. This is an unpleasant fact which must be circumvented in order to facilitate certain constructions in the subsequent subsections. The purpose of this subsection is to modify the descending and ascending disks so as to make this eventuality irrelevent. The expense here is to replace the disk with a more complicated submanifold of W. a) Past and future. The purpose of this subsection is to introduce some terminology which will arise in the modification constructions below. To begin, focus on a subset U C W. Define the past of U, written past(U), as follows: A pointx E past(U) if there is a gradient How line 'Y : [a, b] -+ W and times t, t' E (a, b] with t' ~ t and with 1) 'Y(t) = x,
2) 'Y(t') E U. (5.1) Define the future of U, written fut(U), as the subset of points x in W for which there is a gradient flow line which obeys (5.1) but where t, t' E [a, b) and t' ~ t. Note that past(U)n fut(U) = U. For example, if pEW is not a critical point, then past(p) is the set of points which are hit before p on the gradient flow line through p. However, if p E crit(f), then past(p) = Bp_. b) Tubing descending disks froIll crit2(f). This subsection begins the modification process; it describes a construction, tubing, which modifies the descending disk from an index 2 critical point bi so
that the closure of the modified submanifold is disjoint from any index 1 critical Point aj for j < i. To make the tubing construction, focus first on some index 2 critical point b == bi and a particular index 1 critical point a = aj for j < i. A descending
CLIFFORD HENRY TAUBES
346
disk from the index 2 critical point b will intersect a neighborhood, Ua of a in a finite set of components. Each of these components contains the intersection of Ua with a gradient flow line which starts at a and ends at b. To be precise, let V C B b - n Ua be a component. After a small isotopy, one can find Morse coordinates for Ua so that
(5.2)
1/Ja(V)
= {(Xl,X2,X3,X4); X2 ~ 0 and X3 = X4 = o}.
With (5.2) understood, the flow line between a and b which lies in V is given in the Morse coordinates by intersecting 1/Ja(V) with the ray {(Xl,X2,X3,X4) : X2 > 0 and Xl = X3 = X4 = o}. To consider the full intersection of Bb- with a neighborhood of a in W, it is convenient to first intersect 1/Ja(Ba+) with a small radius sphere in lR" about the origin. Call the result S+i in Morse coordinates, this S+ is a small radius 2-sphere in the 3-plane where Xl = O. The intersection Bb- n S+ is transverse, and is a finite number of points, Bb- n S+ = {ea }. Because j < i and the matrix S is upper triangular, the 2-sphere S+ has zero algebraic intersection number with B b _. This means that the points {e of Bb-'s intersection with S+ can be paired so that each pair contains one point with positive intersection number and one with negative intersection number. Write this pairing as Q }
(5.3) Since S+ is a 2-sphere, the two points of any pair Gan be connected by a path in S+. These paths can be drawn so that paths coming from distinct pairs in Bb_nS+ do not intersect. The paths should also be drawn to avoid intersections of S+ with any other descending sphere from crit2(J). Let {(/J}~=l be the set of paths just defined. The value of f on S+ is some constant, fo > f(a). Then, introduce M == f-l(Jo) nua • This will be a smooth 3-manifold given by
(5.4)
1/Ja(M+)
= {(Xl.'"
,X4 :
-X~
+ X~ + X~ +- X~ = fo}.
With M+ understood, thicken each ( E {(/J} to a thin ribbon in M+i call this ribbon ( ~ I x I, where I == [0,1]. (The ribbon should be thin so that it's only intersection with a descending disk from crit2(J) is with 8(.) Thus parameterized, I x {1/2} == (, while 81 x I is embedded in Bb- n M+. To be explicit, parameterize as ((T) E S+ for T E [0,1]. Then, to a first approximation, ~ should be parameterized by (T, T') as (5.5)
1) 2)
Xl
= fJ/2 £(2T'-1),
(X2,X3,X4)
= (1 + £2 (2T' _1)2)1/2 (Tn
for small £ > O. Let 710- denote the past of (0,0) i it is part of a gradient flow line which starts on Mo. Let 710+ denote the past of (0,1), part of another gradient flow line starting on Mo. -
HOMOLOGY COBORDISM
347
The union 110 == 1Jo- U(O,·) U77o+ is a piece-wise smooth curve in Bb-. Here is a picture of 1Jo and past(1Jo):
{.
/
~ I ~I, / '70-
\
'f+
(
Past(11o)
(5.6) Let 1Jl - denote the past of (1,0) and let 771+ denote the past of (1, 1). Set == 1Jl- U (1,·) U '71+. This is a piece-wise smooth curve in Bb-. With the preceding understood, here is a surgery on Bb-: Delete from Bbthe set past(1Jo)U past(77d to get a manifold with piecewise smooth boundary 1Jo U1Jl, and then glue on to this boundary image «()U past«((·, O»U past«((·, 1). Call the resulting space B~_. See the following piCture: '71
Pa.st(~, 0)
(5.7)
348
CLIFFORD HENRY TAUBES
The surgery just described is the tubing construction on a cancelling pair of intersection points of Bb- with S+ Effect this tubing construction for all the pairs in (5.3) which comprise Bb_'S intersection with S+. Because the surgeries are constructed using gradient flow lines, the surgeries from different pairs in (5.3) do not interfere (or intersect) each other. After all n surgeries are performed, the result is a piecewise smooth submanifold of W whose closure misses the critical point aj. This submanifold can be smoothed after a small perturbation and will henceforth will be assumed smooth. Effect the same tubing construction for all pairs of intersection points for all aj with j < i. Use Bu- to denote the result of doing this surgery. (Note: Because the surgeries are defined using gradient flow lines, the surgeries which come from different index 1 critical points do not interfere nor intersect with each other.) Finally, effect this same tubing construction for all bi in crit2(J). Note that this can be done so that the resulting set of submanifolds {B1b- : b = bi}i=l are disjoint in W. (The point here is that the paths ( and the ribbons (in (5.5) have only the two boundary points of ( as intersection points with descending disks from Crit2(J). The rest of the tubing construction uses only gradient flow lines-and so won't create intersections with descending disks.)
c) Normal bundles. Let b = b. E Crit2(J). The submanifold Bb- C W is oriented as the negative disk from the degree 2 critical point b = b•. As an oriented submanifold of W, Bb- has a canonical trivialization of its normal bundle (up to homotopy). Simply flow the trivialization of the normal bundle of B b- at b along B b- using the pseudo-gradient v. The preceding subsection described the construction of a submanifold Bufrom Bb- by doing surgery on embedded arcs in Bb- with endpoints on Bb- n Mo. The resulting 2-dimensional submanifold can be seen to be orientable, and it inherits a canonical orientation from Bb-. (Note that each surgery that is performed on B b - is constrained to lie in a 3-dimensional ball in W. One dimension of this ball is the pseudo-gradient flow direction, the other two dimensions can be parameterized by the ribbon coordinates on ( in (5.5).) As B 1b - is not closed in W, the normal bundle-to B1b- will be a trivial bundle, and the claim is that there is an essentially canonical trivialization up to homotopy. The point is that in constructing B1b- from Bb- one does a large number, say N, of essentially identical, non-interfering surgeries. So, one need only check that the canonical normal trivialization of Bb- extends over anyone of these surgeries to give a normal trivialization of the postoperative manifold which agrees with the normal trivialization of B b - away from the area of surgery. That such is the case is easy to check, since each individual surgery can be performed inside a 3- dimensional ball inside of W.
HOMOLOGY COBORDISM
349
d) Tubing ascending disks from critl (f).
Let a == ai C critl (f). The closure of the ascending disk from a will typically intersect many of the points in Crit2 (f). The purpose of this subsection is to modifies the ascending disk so that the closure of the resulting submanifold of W is disjoint from {bj};>i' This modification procedure will also be called tubing. To begin the tubing construction, focus attention first on an index 2 critical point b == bj with j > i. Introduce the Morse coordinates, f(P'). Consider L(p,p')+ == (Bp_ x B p'+) n Z E 4. This is the boundary of the subset of Bp_ x B p'+ where F ~ O. (The latter is a manifold with boundary.) Orient Bp_ x By+ with the product orientation and then agree henceforth to orient L(p,y)+ with the induced orientation as the boundary of the subset where F ~ O. Consider now the 3-sphere L(p' ,p)_ == (Bp' + x Bp_) n Z E L.-. This is the boundary of the subset of B p'+ x Bp_ where F ~ O. (The latter is a manifold with boundary.) Orient By + x Bp_ with the product orientation and then agree henceforth to orient L.(y,p)_ with the induced orientation as the boundary of the subset where F ~ O. LEMMA 6.2. Add the following to the conclusions of Lemma 6.1: The submanifolds {Yi,;±} have tmnsversal intersections with the 3- spheres in L.± and Yi,j- n L.- 0 and Yi,H n 4 = 0, where L.± are given by (3.31), (3. 32}. Furthermore, the {Yi,i±} can be oriented so that 1} The intersection of Yi,;- with L(p,p')+ E 4 is empty unless p :::: ai or p' = bj • If p = ai and p' = ak, then the intersection number is -S;,k. If p = bk and p' = bj , then this intersection number is Sk,i'
=
2)
The intersection of Yi,H with L(p' ,p)_ E L.- is empty unless p' :::= bj or p = ai. If p' = b; and p = bk, then the intersectian number is Sk,i' If p = ai and p' ak, then the intersection number is Sj,k.
=
3} (6.2)
= (Si,;) ([S3]a_ + [S3]b_), 8[Yi,i+] = (Si,i) ([S3]a+ + [S31b+)'
8[Yi,i-]
Here, Si,i > 0 is given in (3. 15}. (For p defined subsequent to (.+.19).)
=a
or b, the classes [S3]p± are
The proof of this lemma is deferred to Subsection 6d, below. b) The construction of [E±J.
With the orientations of Lemma 6.2, the submanifolds {Yi,j±} of Lemma 6.1 will define homology classes in H4(Z,8Z) and linear combinations of these classes will produce classes [E±l which fit into (4.11) to solve the constraints of Lemma 4.1. To be precise here, introduce the matrix S of (3.15) and the
HOMOLOGY COBORDISM
355
integer valued matrix T == det(S) S-1. Note that T == (Ti,j) is upper triangular (when i > j, then Ti,j = 0) with Ti,i = det(S)/Si,i' With T understood, introduce
(6.3)
[E1-] == LTi,j [Yi,j-] and [E1+] == LTj,i [Yi,H]' i.,j
i,j
(In (6.3), the sums are over all pairs i,j with 1 ~ i ~ j ~ r.) Here are the salient features of these classes:
LEMMA
(6.4)
6.3. Define the classes [Ea] by {6.9}. Then 8[E1_]
= det(S)
L
[S3]p_.
pE crit(f)
8[E1+l = det(S)
L
[S3l p +'
pE crit(f)
Furthermore, [Eal have zero intersection pairing with the classes which are generated by the 9-speres in k± of (9.91) and {9.92}. It follows from this lemma that Lemma ..p is satisfied if the classes [E±l in (4.11) are set equal to [Eal from {6.9}. In this case, (4.11)'s integer N must equal det(S). (In later constructions, it proves convenient to take [E±l in {4·11} to be some multiple of [Eal from (6.9}.) Proof. Consider first the properties of [E1 -]. It follows from Assertion 1 of Lemma 6.1 and Assertion 1 of Lemma 6.2 that 8[E1_] obeys (6.4). This is because the boundary annihilates all terms in (6.3) save those for which i = j. Then, (6.4) follows from (6.2) and the fact that Ti,i = det(S)/ Si,i' According to Assertion 2 of Lemma 6.2, [E1-1 is represented by the fundamental classes of submanifolds with empty intersection with the classes from k_. To study the intersection pairing between [E1-] and a class from ~, fix integers m and n with 1 ~ m < n ~ r. Let a == am and let a' == an. Consider the pairing between [E1-l and the class of L(a,a')+' Using Assertion 3 of Lemma 6.2, one finds that this number is equal to (6.5)
LTm,kSk,n, k
which is zero because m =f. n and T is proportional to S-1. Next, let b == bm and let b' == bn and consider the pairing between [E 1 -] and the class of L(b,b/)+. Using Assertion 3 of Lemma 6.2 again, one finds that this pairing is equal to (6.6)
356
CLIFFORD HENRY TAUBES
which is also zero, because m '# n and T =det(8) 8- 1 • Thus Lemma 6.3 is proved for [EI-J. The prooffor [EHl is analogous and is left to the reader. 0 c) Proof of Lemma 6.1
Fix i and j such that 1 ~ i ~ j ~ r and let a == ai and b == bj. For Assertion l's proof, note that Yi,j- n int(Z) will be a submanifold of int(Z) if F's restriction to Bl a+ X B1b- has zero as a regular value. This will follow if f's restriction to B 1a+ has disjoint critical values from its restriction to B lb _. With an arbitrarily small isotopy, of B1b- near f-IU(a», one can insure that f(a) is not a critical value of f on B lb _. Likewise, an arbitrarily small isotopy of B 1a+ near where f = feb) will insure that feb) is not a critical value of f on B la +. With this understood, a small isotopy of Bl a+ which is the identity near a will insure that the critical values of f on Bl a+ are disjoint from those of f on B 1b-. Argue as follows to prove that Yi,i- is closed: The closure of Blb- in W adds only the descending disks from {akh~j. However, f(B la+) ~ f(ai) > f( {akh>i) (see Assertion 2 of Proposition 3.2). Therefore, where (niJ) ~ 3/8, the closure of (B1a+ X B1b-) n Z adds nothing unless i = j, and then, only the point (a,a) is added. Likewise, Bl a+ is not closed in W, but its closure adds only ascending disks from {bkhi.) However, f(B1b=) ~ f(bj ) < f({bkh<j) because of Assertion 2 in Proposition 3.2. Therefore, where (niJ) ~ 3/8, the closure of (B 1a+ X B 1b-) n Z adds nothing except when i = j, in which case only the point (b, b) is added. The preceding proves that (B1a+ X B lb -) n Z is closed. A similar argument proves that l'i,i+ is closed. Note that the preceding argument proves Assertions 2 and 3 also. To prove Assertion 4, consider first the neighborhood Ua of a in W as described by the Morse coordinates (3.2). The submanifold Blb- intersects this ball in at least 8 i ,i components; and a typical component, say V, has the following form: There is a unit vector v with coordinates (0, V2, V3, V4) and (6.7) (The unit vectors (i.e. v) are distinct for distinct components of B 1b - n Ua .) Equation (3.6) describes Bl a+ near a since near a, it is identical to B a+. Consider next the neighborhood Ua x Ua of (a, a) in W x W, and use the coordinates of (3.25). One sees that near (a, a), each component of Bl a + X B 1b - has the form B 1a+ X V, where V C B1b- is given by (6.7). Thus, the intersection Bl a+ X V with Z near (a, a) is given by the set of points «Xl,X2,X3,X4), (Yl,Y2,Y3,Y4» E]R4 X lR" where 1) Xl = 0 2) (Y2,Y3,Y4) = tv for t > 0, 3)
t2
= Y~ + x~ + x~ + x~.
HOMOLOGY COBORDISM
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(6.8) Note that this set intersects (8S
X
8 S )a
c az as 8 s
x Pv, where
(6.9) Equations (6.7) and (6.8) establish the first part of Assertion 3 concerning the intersection of Yi,;- with (8S X 8 S )a' An analogous argument shows that the intersection of Yi,;- with (8S x 8 S )b has the following form: The coordinate chart Ub describes a neighborhood of b in W. A component, V, of the intersection of Bl a+ with Ub is given as
(6.10) and v E
]R2
with
1v 1= I}.
Use Ub x Ub to describe a neighborhood of (b, b) in W. The intersection of B 1a+ X B1b- with this neighborhood will be a union of components, each of the form V X B lb - with V as above. With this understood, V x B 1b - intersects Z as the set of points in JR4 x ]R4 of the form ((Xl, X2, Xs, X4), (YI, Y2, Ys, Y4») where (6.11)
°
1) (Xl, X2) = t v for t > and 2) Ys = Y4 = 3) t 2 = y~ + y~ + x~ + x~.
°
1v 1= 1
The preceding equation demonstrates that V x Blb- intersects (S3 x S3)b C 8 3 x Pv, where Pv = (rVI,rV2,0,0) . The proof of Assertion 5 of the Lemma 6.1 follows essentially the same arguments which prove Assertion 4. The details for Assertion 5's proof are omitted. Consider now the proof of Assertion 6: Both B 1a+ and B lb - are orient able (as described in the previous section), and so their product is orientable. Then, the restriction of dF to the product trivializes the normal.bundle of Yi,;- in B 1a+ X Blb- and similarly that of Yi,j+ in Blb- X B 1a +. To prove Assertion 7, remark that both B 1a+ and B1b- were constructed with canonically trivial normal bundles. Thus, their product has a canonical (up to homotopy) trivialization of its normal bundle in W x W. With this understood, remember that Z is cut out of W x W as part F-1(0), while Yi,;- is cut out of B 1a+ X B1b- as part of F-1(0), so the trivialization of (B 1a+ X Blb_)'s normal bundle in W x W defines, upon restriction to F- 1 (0), a trivialization of the normal bundle to Yi,;- in Z. Once again, the argument for Yi,i+ is analogous and omitted. To prove Assertion 8, first remember that B 1a-+< and Blb- are constructed from Ba+ and Bb-, respectively by surgery. The surgery on B b - occurs near where f = 1/4, while the surgery on B 1a+ occurs near f = 1/2. This implies that Yi,j- can be seen as the result of a surgery on the 4-sphere which is the intersection of the descending disk from F's index 5 critical point (~, b) with F- 1 (1/8). The surgery is on embedded SO x B 4 ,s in said 4-sphere. The number
az as
35S
CLIFFORD HENRY TAUBES
of these surgeries is the combined total of the surgeries which make Bla+ from Ba+ and Blb- from Bb-. Each such surgery increases the rank of Hl(·jZ) by one, but leaves H2 (.j Z) = o. Assertion 9 follows from (4.1S) and (S.16). 0
d) Proof of Lemma 6.2. Consider first that the 3-spheres in L± do not come near the critical points (P,p) of F. This follows from Proposition 3.2. Therefore, an (arbitrarily) small isotopy of B 1a+ or of B1b- will result in transversal intersections between Yi,i± and any of the spheres in L.±. Remark next that the intersection of Yi,j- with some L(p',p)_ is non-empty only if Bla+ n B p'+ "I 0 and also B1b- n Bp_ '" 0. The former is empty if p and p' have index 2, while the latter are empty if p and p' have index 1. To prove Assertion 1, one should consider orienting Yi,j- as follows: Orient B 1a+ X B1b- with its product orientation. Then, note that Yi,j- is a codimension zero part of the boundary of the subset of B 1a+ X Blb- where F ~ o. Give Yi,jthe induced boundary orientation. Use 0 to denote said orientation. With the orientation 0, the intersection number between Yi,j- and some L(p,p'l+ E 4 is equal to the coefficient in front of (P,p') in the expression for the 8(a, b) in the complex C F of Lemma 3.S. (Note that B 1a+ X Blb- is homologous to the descending S-disk from (a, b).) The computation of this coefficient is straightforward and leads to Assertion 1. (The fact that the intersection in question is empty unless p = ai or p' = bj follows from the fact that when a and a' are index 1 critical points of j, then B 1a+ n B a,- = 0 unless a = a'. Likewise, when band b' are index 2 critical points of j, then B1b- n Bb'+ = 0 unless b = b'.) The proof of Assertion 2 is analogous. Here, the orientation 0 for Yi,H is defined by considering Yi,H as a co dimension zero part of the boundary of the subset of B1b- X B la+ where F ~ O. Consider now the proof of Assertion 3. There is a proof along the lines of the proof of Assertion 1, but a direct proof is had by the following argument: Let a == ai and b == bi. An intersection point, q, of (B a + n M 3/ S ) with (B b- n M 3 / S ) corresponds to one boundary component of (Ba+ x Bb-) n Z in (83 X 8 3)a and, likewise, to one boundary component in (8 3 x 8 3 )&. (And vice-versa.) The orientation of these boundary components relative to the given orientations of (8 3 )a_ and to (8 3 )b_ will be found equal, but opposite the local intersection number at q of (Ba+ n M 3 / S ) with (Bb- n M 3 / S ) in M 3 / S • 8tep 1: This step compares the local intersection number at q with the orientation of the corresponding boundary component of (Ba+ n Bb-) n Z in (8 3 X 8 3 )a. To begin, take the Morse coordinates near a of (3.2) so that Ba+ = {x == (Xl,X2,X3,X4) : Xl = O}. Orient Ba+ by {h8384 E A3TBa+. A neighborhood, U C M 3 / S of M 3 / 8 's intersection with Ba+ is isotopic to {x : -x? + x~ + x~ + x~ = R2} for some R > T. This U is oriented at q E (0, R, 0, 0) E U by -81~84. Now q lies in B a +, hut suppose that q is also a point of intersection Bb- and B a+. Suppose further that the local intersection number at q of (B a+ n M 3 / S )
359
HOMOLOGY COBORDISM
with (Bb- n M3 / s ) is equal to E = ±1. Without loss of generality, Bb- can be assumed to intersect a neighborhood of U as {x : X2 > 0 and X3 = X4 = o}. To obtain the correct intersection number at q, it is necessary to orient B,,_ using -E 81 th. (Note that df = dX2 at q, so (Ba+ n M3 / s ) is oriented near q by as84, while (B,,_ n M 3/ s ) at q is oriented by -E8l _ Then, their intersection at q has local orientation -E~as84 which agrees or disagrees with the orientation -8l as84 of M 3/ s depending on whether E = ±1.) With the preceding understood, it follows that Ba+ x B,,_ is oriented near (q, q) by -fth83848~ 8~; here the prime indicates a vector field from the second factor of W in W x W, while the absence of a prime indicates a vector field from the first factor of W. Now, at the point (q, q), the I-form dF = dx~ - dX2; this implies that f (th + 8~)83848L orients (B a+ x B,,_) n Z in (83 x 8 3 )a near (q,q). Finally, the boundary ofth~ component of (Ba+ xB,,_)nZ in (83 X 8 3)a which corresponds to the point q is oriented by contracting this last frame with -dx~ - dX2 which yields -Eas848r. This disagrees with the orientation of (S3 )awhen E = +1 and it agrees with said orientation when E -1. Step 2: This step compares the local intersection number at q with the orientation of the corresponding component in (8 3 X 8 3 )" of the boundary of (Ba+ x Bb-) n Z. To begin, take the Morse coordinates of (3.2) around b. Then, B,,_ = {x : X3 = X4 = o}. Orient B b- by 8 1 th. If. neighborhood, U, of the point q in M 3 / s is isotopic to the subset given in Morse coordinates as {x : -x~ - x~ + x~ + x~ = .... R2}. Here R » rand q is the point (0, R, 0, 0). The orientation of M3 / 8 is determined from the fact that df at q is -dx2 • Thus, 81 83 84 orients M 3/ s . Meanwhile, a neighborhood of q in Ba+ can be assumed given by the set {x : Xl = 0 andx 2 > o}. This part of Ba+ is oriented by fthas84. (Thus, (Ba+ n M 3 / s ) is oriented at q by - f as 84 while -81 orients (B b- n M 3 / s ) at q. Their intersection gives E81 as84 for the orientation of M 3 / s as it should.) The orientation for Ba+ x Bb- near (q,q) is thus given by fthas848r8~. The I-form dF at (q, q) is given by -dx~ + dX2, and this means that f (82 + 8~)83848i orients the part near (q,q) of (Ba+ nBb-) n z. With this last point understood, it follows that - f as848~ orients the part of 8«Ba+ n B b-) n Z) in (S3 x 8 3 h which corresponds to q. Note that this orientation disagr~es with the given orientation of (S3)b_ when f = +1, but it agrees when f = -1. In particular, note that this anti-correlation with the local intersection number at q is the same as that for components of 8«Ba+ x Bb_) n Z) in (S3 X S3)a. It follows from the preceding calculations that (6.2) holds if the orientation - 0 is used on {Yi,j-}. A similar argument shows that the second line of (6.2) is correct if the {Yi,H} are also oriented with -0. The details here are left to the reader.
=
e) Push-oft's.
The next task is to provide a representative of each [E1±l as the fundamental class of a smoothly embedded submanifold (with boundary), E1± C Z. Here,
oE1 ±
C
oZ.
The construction of E 1 ± requires the introduction of a procedure, called
360
CLIFFORD HENRY TAUBES
push-off, for making copies of embedded submanifolds. The following digression described the push-off procedure. Start the digression by considering the following abstract situation: Let X be a compact manifold with boundary, and let Y C X be a compact submanifold with boundary, which intersects ax transversally as ay. Let Ny -t Y denote the normal bundle to Y in X. (Note that Ny restricts to ay as the latter's normal bundle in ax.) Suppose that Ny admits a section, s, which never vanishes. Let e : Ny -t X be an exponential map which maps Ny 18Y to ax. (See (2.13).) Together, e and S and a real number A '" 0 define a map, (6.12)
e(AS(')) : Y
-t
X,
whose image is disjoint from Y. IT A has small absolute value, then the image, Y', of (6.12) will be an embedding of Y into X, where ay' is an embedding of
ay into ax. This image, Y', is called a push-off of Y. Here are some properties of the push-off: (.) Y' is disjoint from Y, but smoothly isotopic to Y. (The obvious isotopy is to consider A -t 0 in (6.12). This isotopy will isotope ay' to ay in ax.) (.) IT Y comes with some apriori orientation, then Y' has a canonically induced orientation which makes [Y] = [Y'] in H*(X,aX). (.) Let V C X be a submanifold which intersects Y transversally. Then V will also intersect Y' transversally if A in 6.12) has sufficiently small absolute value. (.) Let V C X be a closed submanifold with empty intersection with Y. Then V n Y' = 0 if A in (6.12) has sufficiently small absolute value. (.) IT Y has a framed normal bundle, then this framing naturally induces a framing of the normal bundle to Y' . (6.13)
Note also that one can define any finite number of disjoint push-offs of Y by using different values of A in (6.12). Alternately, one can use different sections {Sl,'} of Ny with fixed A as long as the {Sj} are no-where vanishing and no two are anywhere equal. In the sequel, assume the following conventions: (-) Any pair of distinct push-offs of the same submanifold are mu,tually disjoint. (-) Suppose that the normal bundle Ny is trivial, and that an apriori trivialization has been specified. (Call it the canonical trivialization.) In this case, agree that all push-offs of Y will be defined by using for s in (6.12) a constant linear combination of basis vectors for the canonical trivialization. (-) When the precise choice of exponential map or parameter A or section s in (6.12) are irrelevent to subsequent discussions, their presence will not be
HOMOLOGY COBORDISM
361
explicitly noted. (But, keep the preceding convention on the section s when the normal bundle to Y has been trivialized.) (6.14) (The last two conventions in (6.14) allow one to speak of a push-off of Y with-out cluttering the conversation with a list of irrelevent (but necessary) choices.) End the digression. f) E1± as submanifolds.
The purpose of this last subsection is to define [E1±] of (6.3) as the fundamental class of a closed, embedded submanifold (with boundary) of Z. Consider first [E1 -]. This [E1-] is a sum of fundamental classes of the {Yi,j-}. The first observation is that each Yi,j- n Yn,m- = 0 unless m = j. This is because the various {B 1b - he crit2(f) are mutually disjoint. There may be non-empty intersections between Yi,j- and Yk,j- when i '" k. These can be avoided if the following convention is used: Remember that each B1b- has trivial normal bundle in W. And, remember that said normal bundle has a canonical trivialization up to homotopy. For each b E Crit2(J), choose a trivialization of the normal bundle of B lb - which is in the canonical homotopy class. Then, fix j and when i < j, define Yi,j- as in (6.1) but where B1b- is replaced by a push-off copy. For each such i, use a different push-off copy. This will make Yi,j- disjoint form Yk,jwhen i '" k. Now, generalize this process of separating the {Yi,j-} as follows: Reintroduce the matrix T = (Ti,j) which appears in (6.3). For each pair (i,j) with 1 ~ i ~ j ~ r, let a ai and b bj . Take I Ti,j I distinct push-off copies of B1b- and use these in (6.1) to define I Ti,j I distinct push-off copies of Yi,j_. It will prove convenient to require that all such push-off copies are disjoint from the flow line 'Y of Part 4 in Definition 3.1. (One can make all such copies in past(U), where U C Wis an open subset which contains crit(J) and whose past and future are disjoint from 'Y • See (5.16).) Since the various {Blb- he crit2 (f) are mutually disjoint, one can make all of these push-offs so that each copy of Yi,j- is disjoint from each copy of Yk,lwhen (i,j) '" (k,l). With the preceding understood, consider:
=
=
PROPOSITION 6.4. Define E 1- C Z as an oriented sub manifold of Z (with boundary) as the union over all pairs (i,j) (with 1 ~ i ~ j ~ r)) of the I Ti,j I push-offs of Yi,j- as defined above. Take these copies with the following orientations: Orient the copies of Yi,j- as i11 Lemma 6.2 if Ti,j > 0; and oriented them in reverse if T .. ,j < O. Define El+ C Z as a submanifold to be the image of E 1 - under the switch map on W x W which sends (x,y) to (y,x). (This map preserves Z.) Then these oriented submani/olds can be assumed to have the following properties: 1) The fundamental clas.ges oj EH obey (6.3).
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CLIFFORD HENRY TAUBES
2) El± intersect OZ transversely in OEl±' 3) El± have empty intersection with Mo X Mo and MI X MI. 4) El± have trivial nonnal bundles in Z, and said nonnal bundles have canonical trivializations up to homotopy.
5) H2(El±; Z) = O. 6) El± have empty intersection with
ER,L
of (4.15).
The proof of this proposition is left to the reader. 7 The second pass at E±. Assume here that W obeys the constraints of (3.11). If E± in (4.10) is E 1± of Proposition 6.4, then the resulting I::z satisfies Steps 1 and 2 plus Part 1 of Step 3 in Section 2k's outline of the proof of Theorem 2.9. However, the completion of Step 3 requires modifications of El±. The problem is that El± intersect the various (S3 x S3)p C oZ too many times, and they intersect each other too many times, and they intersect f:l. z too many times. The change of El± into E± is a multi-step process which begins in this section and ends in Section 10. Then, Section 11 constructs a 2-form Wz to satisfy (2.27). This section starts the process by modifying El± to make a submanifold, E2±, with simpler intersections with the (S3 x S3)p C oZ. a) The submanifold
E~_.
To begin the modification process, fix i E {I,· .. , r} and, as usual, set a :: a, and b :: bi' Make 2det(S) additional push-off copies of Blb-. Make these copies so that they are disjoint from all other push-off copies of {Bib' _ : b' E crit 2 (f)} which have so far been constructed. Use these 2det(S) push-off copies of BIb..., to make 2 det(S) copies of B 1a+ X B1b- and then 2 det(S) copies of Yi,,- as describe in (6.1). Orient the first det(S) of these Yi,i- canonically, and orient the remaining det(S) of these copies opposite to their canonical orientation. The first det(S) copies of Yi,i- (the ones with the canonical orientation) will be called the special Yi,i-. Define Ei_ to be the union of Proposition 6.4's E 1 - with the (oriented) submanifold which is comprised of the union of the preceding 2det(S) copies of oriented Yi,i-. Notice that this Ei_ still obeys the conclusions of Lemma 6.3 and Proposition 6.4. b) Tubing near (a,a) Consider now the intersection of E~_ with (S3 (6.9), this intersection is given as
X
S3)a: As described in (6.8),
(7.1)
where
A~ C
8 3 is a finite set of points. Each point in A~ comes with a sign more plus signs than minus signs. This means that
{±I}J and there are det(S)
HOMOLOGY COBORDISM
363
the set A~ can be decomposed as Ao UTo, where the points in To can be paired so that the signs in each pair add to zero. It proves useful to take some care in defining the set Ao. Here is how: To begin, note that the intersection of Bb- n j-l(3/8) with Bo+ n j-l(3/8) is transversal, and has intersection number 8 i ,i. Pick a point in this intersection where the local intersection number is positive. Such a point lies on a gradient flow line, 1'(= I'i) which starts at a and ends at b. The intersection of I' x I' with (83 X 8 3 )0 is a point, Po x Po, where Po E 8 3 • With Po singled out, note that the intersection of any push-off copy of ¥i,1 with (83 X 8 3)0 contains a unique (8 3 x p~) where p~ is the push-off of Po. (There is a canonical isotopy between the push-off copy and the original (shrink A to zero in (6.12), and under this isotopy, p~ moves to Po.) In particular, each of the det(8) special copies of ¥i,i defines such a point p~, and these det(8) points are the points that comprise
Ao· As remarked above, the points in To can be paired up so the signs of each pair sum to zero, (7.2) For each pair {e a , e~} in (7.2), E 1 - induces orientations on 8 3 x ea and 8 3 x e~, and these orientations are opposite. Now, for each pair, {e, e/} on (7.2)'s right side, embed [0,1] into 8 3 to have boundary {e, e/ }. (Do this in such a way that the embedded intervals from distinct pairs do not intersect.~ The associated 8 3 x [0,1] has boundary (8 3 x e) U (83 x e/) and the orientations here agree with those which are induced by E 1 _. Hence, 8 3 x [0,1] C (8 3 X 8 3 )0 can be surgered to E 1 _ along their common boundaries, (8 3 x e) U (8 3 x e' ). The result is a topological embedding in Z of a smooth, oriented manifold with boundary, where the boundary embeds (smoothly) in az, (The embedding has "comers", these being the components of 8 3 x {ea,e~} where the surgery took place.) The point is that this new manifold has two less boundary components then E~ _. Here is a picture:
83
X
[0,1]
(7.3)
Make the preceding construction for each pair on the right side of (7.2). The result is a topological embedding of a surgered EL. (The "comers" of the
364
CLIFFORD HENRY TAUBES
embedding are the components of S3 X T a.) The embedding of this surgered Ef_ intersects (S3 X S3)a in S3 X Aa (where it is the same as Ef_) and also in a copy of S3 x [0,1] for each pair on the right in (7.2). (The copies of S3 x [0,1] for distinct pairs will not intersect if one takes care to insure that the embedded [0, 1]'s from different pairs do not intersect.) Now note that the copies of S3 x [0, 1] can be isotoped normally off (S3 X S3)a (push radially outward in the coordinates of Lemma 3.6 so that the resulting embedding of the surgered E~_ intersects (S3 X S3)a in S3 X Aa. And, note that all of the" corners" in the resulting embedding can be readily smoothed so that the result is an embedded submanifold of Z. The following diagram illustrates:
smoothed
(7.4)
The preceding construction can be done at all a E critl (f). The result is a submanifold, Ef'_ C W. Note that Ef'_ has a minimal number of intersections with any (S3 X S3)a C az as its intersection is equal S3 X Aa"a set of det(S) push-off copies of S3 X Pa. Note also that Ef'_ agrees with Ef_ away from {(S3 x S3)a}aE critl(/). c) Tubing near (b, b)
=
Let b bi E critl (f). Consider now the intersection of Ef'_ with (sa x S 3 h: As described in (6.8), 6.9), this intersection is given as (7.5)
S3
X
A'b'
where A~ c S3 is a finite set of points. Each point in A~ comes with a sign (±1), and there are det(S) more plus signs than minus signs. This means that the set A~ can be decomposed as Ab U T b, where the points in Tb can be paired so that the signs in each pair add to zero. It proves useful to take some care in defining the set A b • Here is how: The flow line 1-'(= I-'i) which starts at a and ends at b. The intersection of I-' x I-' with (S3 x S 3 h is a point, Pb x Pb, where Pb E S3. With Pb singled out, note that the intersection of any push-off copy of Yi.i with (S3 x S 3 h contains a unique (S3 x Ji,,) where P~ is the push-off of Pb. (There is a canonical isotopy between the push- off copy and the original (shrink A to zero in (6.12), and under this isotopy, P~ moves to Pb.) In particular, each of the det(S) special copies of Yi.i
HOMOLOGY COBORDISM
defines such a point
Pb'
366
and these det(S) points are the points that comprise
Ab. As remarked above, the points in Tb can be paired up so the signs of each pair sum to zero,
(7.6) For each pair {e a , e~} in (7.5), Ef/_ induces orientations on S3 x e a and S3 x e~, and these orientations are opposite. With this understood, one can repeat the tubing construction as described in the previous subsection (see (7.4), (7.5» to surger E~/_ near (b, b) and then isotope the result to obtain an embedded submanifold of Z which intersects (S3 x S3h in S3 x A b. FUrthermore, this last construction can be done simultaneously near all (b, b) for b E crit2(!)' Use E 2 - to denote the resulting submanifold of Z. d) The intersection of E2± and ER,L' The next four subsections describe various properties of E 2 ±. The purpose of this subsection is to prove
LEMMA 7.1. The submanifolds E 2 ± C Z can be constructed as described above so that they do not intersect EL,R of (4.15).
Proof. Let U C W be an open neighborhood of critl (f) and let U ' C W be an open neighborhood of Crit2 (f). Then E2± can be made (as described above) so that they are supported in Z's intersection with (fut(U) x past(U' The latter set is disjoint from ER,L if U and U' are not too big; this is because the flow line 'Y misses 1's critical points. 0
».
e) The intersection of E2± with 6. z : Fix i E {I,· .. ,r} and let a == ai and b == bi. By construction, E 2 - intersects (S3 X S3)a in S3 X Aa. It intersects (S3 x S3h in S3 x Ab • Here, Aa and Ab are sets of det(S) points. Now, there is a natural way to pair the points in Aa with those in Ab and here it is: When p E Aa and p' E Ab are partners, then (P,p) and (P',p') are the endpoints of a transversal component of E 2 - n 6. z which is an embedding of [0, 1]. Such a pairing exists for the following reasonS: If p E Aa, then S3 x p is a component of the intersection of a push-off copy of Yi,i- with (8 3 X 8 3 )a. By design, there exists a unique p' == p'(P) E Ab for which 8 3 x p' is a component of the intersection of the same push- off copy with (8 3 x 8 3)b. This is another definition of the pairing between Aa and A b • To finish the story, remark that the afore-mentioned push-off copy of Yi,i- is (B 1a+ X B~b_) n Z, where B~b_ is a push-off copy of Blb_. And, both p and p'
366
CLIFFORD HENRY TAUBES
lie on a push-off copy, J." C B 1a+ n B~b_' of a chosen flow line, p,(= f..ti), which starts at a and ends at b. Finally, (f..t' X J.") intersects Z transversally in 4z and (f..t' X f..t') n Z is an embedded interval in !::t.. z and a transversal component of ~_ n!::t..z.) With the preceding understood, one sees that
(7.7) where ri is the union of det(S) push-offs (in !::t.. z ) of (f..ti X f..ti) n !::t.. z , and where C C int(!::t.. z ) is compact. Infact, after an (arbitrarilly small) isotopy of the push-offs of the {B1b- : b E Crit2(f)} (with support away from crit(f», one can arrange for the intersection in (7.7) to be transversal. In this case, C is a disjoint union of embedded circles in int(!::t..z).
f) Normal framings. Consider now the normal bundle to E 2 _. Of particular interest in subsequent sections is the fact (see Lemma 7.2, below) that E 2 - has trivial normal bundle. Also of interest is the behavior of a framing of this normal bundle on 8E2 _ and along the components of {ri} from (7.7). Two digressions are required before Lemma 7.2: The first digression defines the notion of a product framing of the normal bundle of a submanifold in Z: This is a framing of the normal bundle with the property that each basis vector is annihilated by the differential of either 7rL or 7rR. The same definition works to define the product framing of a submanifold of W x W. A second digression is required to set the stage for a discussion of the normal framing near 8~_ and {rd. To start, consider i E {I,." ,T}. As usual, let a ai and b bi. Let f..t C ri be a component and define p,r/ by requiring (p, p) f..t n (S3 X S3)a and (p' ,p') J.' n (S3 x S 3h. Associate to J.' the subset of E 2 -
== =
=
(7.8) Note the following: Let p,' C r i be any other component. Then, E 2 - near the J." version of (7.8) is naturally defined as a push-off of E 2 - near the f..t version of (7.8). (Near the J.'-version of (7.8), ~_ is a push-off copy of an open neighborhood of (B 1a+ X Blb_)nZ. And, near the f..t'-version, E 2 _ is a different push-off copy of the same open neighborhood. Infact, each of these push-off copies is constructed as B1a+x (push-off copy of B 1b -). These last observations give a natural method of comparing a given normal framing of E 2- along the f..t and J1.' versions of (7.8). See (6.13). End the second digression.
LEMMA 7.2. The sub manifold ~_ has trivial normal bundle in Z. Furthermore, the normal bundle to E2- has a /raming with the following properties:
HOMOLOGY COBORDISM
367
Let i E {I, ... , r} and let J.I. E rio Then the frame is a product frame along (7.8) and it restricts as a constant frame along S3 X P and S3 X p'. Furthermore, let J.I.' c r i be a different component. Then the push-off which identifies E 2 - near the J.I. and J.I.' versions of (7.8) will identify the restriction of the frame to the J.I. and tt' versions of (7.8). Proof. Because E 2 - is constructed by surgering E I _ and the latter is a union of (6.1)'s {Yi,j _ }, the proof starts with a description of the normal bundles to (6.1)'s {Yi,J±}' To begin, consider i,j E {I,,,, ,r} such that i ~ j. Let a ai and b bj. Then Bl a+ X Bu- C W x W has trivial normal bundle with a natural product framing. This implies that Yi,j- in (6.1) has a natural product framing of its normal bundle in Z. (See Lemma 6.1.) Consider now i = j and the induced normal framing of a component of
=
=
D
oYi,i-.
LEMMA 7.3. Let c denote either ai or hi. Let S3 x p be a component of OYi,i- n (S3 x S3)e. Then the product normal framing of Yi,i- in Z induces a product normal framing of S3 x p in (S3 x S3)c and this induced normal
framing is homotopic through product framings to the constant normal framing as defined by choosing a fixed basis for T S3 Ip and using the projection 7T'R to write the normal bundle in question as S3 x TS 3 Ip. Proof. Consider first the case c = ai. Here, p is described by (6.9). Think of the vector v (V2, V3, V4) as a point in the unit 2-sphere#about the origin in the 3-plane spanned by the coordinates (Y2, Y3, Y4)' With this understood, then (6.9) implies that a product normal frame to Yi,i- restricts to (S3 x p) C oYi.ito have the form oZlIe2,e3), where the vectors e2,3 E TS 2 lv, and where OXI is tangent to the Xl axis. In particular, this is a normal frame for S3 x p in (S3 x S3)e. Furthermore, it is homotopic through product frames to the trivial frame because 7T'3(SO(2)) = 1. (In fact, the vectors e2,3 depend only on the YI coordinate. ) Next, consider the case where c = bi: Here, S3 x p is described in (6.11). Think of the vector v (Vl,V2) in (6.11) as a point in the unit circle in the plane X3 = X4 = O. Then, a product normal frame from Yi,i- restricts to (S3 x p) C OYi,i- to have the form (el,oY3,OY4)' where el E TS2 Iv and where 0Y3 ... are tangent to the Y3 and Y4 coordinate axis, respectively. This frame is evidently homotopic through product frames to the constant frame; simply homotope el to a constant length vector. End the digression. D
=
=
To complete the proof of Lemma 7.2, remember that E 2 - was constructed from E 1 - by taking a pair, S3 x e and S3 x e /, in the same boundary component and gluing to them a boundary S3 x I. Here I is an embedded interval in S3 with boundary {e, e'}. According to Lemma 7.3, the induced normal framing on any boundary component is homotopic to the constant framing; and so there is no obstruction to connecting the normal framing on S3 x e to the normal
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framing on 8 3 x e' over the interval 8 3 xI. The following diagram illustrates the procedure:
t
(7.9) The aforementioned argument shows that E 2 - has a framing for its normal bundle. But, the argument above also shows that there is a framing for the normal.bundle of E 2 - which agrees with Lemma 7.3's product framing for E l near (7.8) for any i E {I,··· .r} and any J.t E rio (Remember that near (7.8), E2- and E l - agree.) This last observation plus Lemma 7.3 imply the final two statements of Lemma 7.2. g) Further properties.
Define E 2- as above. Then, define E2+ C Z to be the image of ~_ under the switch map which sends (x, y) C Z to (y, x). The following proposition lists the salient features of E2±: PROPOSITION 7.4. Define E 2 ± as above. These submanifolds can be constructed and oriented so that the following hold: 1) The fundamental classes of E2± obey (6.9). 2) E2± intersect 8Z transversely in 8E2± 9) E2± have empty intersection with Mo x Mo and Ml x M l . 4) If p E crit(f), the intersection of E 2 - with (83 x 8 3 )p is 8 3 X Ap where Ap C 8 3 is a set of det(8) points. Similarly, the intersection of E2+ with (83 x 8 3 )p is Ap X 8 3 • 5) The normal bundle of E 2 - are described by Lemma 7.2 and the normal bundle of E2+ is described by Lemma 7.2 if (7.8) is replaced by its switched version, (p x 8 3 ) U (P' X 8 3 ) U J.t. 6) H2(E2±iZ) = o. 7) E2± have empty intersection with ER,L of (-4.15).
Proof. The only assertion which is not already proved is Assertion 6. To prove Assertion 6 for E 2 - , remark first that E 1 _ has vanishing H2. (See Proposition 6.4.) Then, note that E 2 - is constructed from E 1 -, by gluing various
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copies of S3 x I onto boundary (S3 x SO)'s. This sort of surgery will decrease HO or increase HI, but it can not change H2. 0
8 The third pass at E±. The submanifolds E 2± of the preceding section intersect the diagonal as described in (7.7), with C C int(~z) being a finite union of embedded circles. The purpose of this subsection is to modify some number of like oriented, push-off copies of E 2 ± so that the result, E 3 ±, intersects Ilz as in (7.7) but with C = 0. To be precise, consider: PROPOSITION 8.1. There are oriented submanifolds (with boundary) E 3 ± C Z with the following properties: 1) E3+ is the image of E 3 - under the switch map on Z sending (x, y) to (y,x).
2) 3) 4) 5)
6)
7)
8)
The fundamental classes [E3±l are equal to N [E1±l for some integer N ~ 1. Here, [El±l are described by (6.3) and Lemma 6.3. E3± have empty intersection with Mo x Mo and MI x MI. E3± have empty intersection with EL,R of (4.15). If p E crit(f), then the intersection of E 3- with (S3 x S3)p has the form S3 x Ap, where Ap is a set of N points. Similarly, the intersection of E3+ with (S3 x S3)p is Ap X S3. E 3 - n Ilz = U'=l r, , where r, c Ilz is as follows: There is a flow line 1" which starts at and ends at b,. With the canonical identification of Ilw with W understood, r, is the union of N like oriented, disjoint, push-oil copies of a closed interval, I elL'. And, each of these N push-oils of I starts in (Aa x Aa) n Ilz and ends in (Ab x Ab) n Il z . Both E3± have trivial normal bundles in Z. The normal bundle of E 3has a framing, (, which restricts to a product normal framing on a neighborhood of (U'=l r i) U {S3 X Ap }PE crit(f). Furthermore, this framing ( restricts to {S3 x Ap}PE crit(f) as a constant framing. The normal bundle to E3+ in Z is described by applying the switch map to the preceding. H2(E3±; Q) O.
a,
=
(Compare with Proposition 7.4.) The rest of this section is devoted to the construction of E 3 _. The first subsection below (8a) introduces some of the basic tools. Subsections 8b - 8e apply the tools from 8a to the proof of Proposition 8.1. The final subsections, 8f - 8h, contain the proofs of three propositions that are stated in 8a. a) Deleting circles.
In comparing Propositions 8.1 and 7.4, one sees that the essential difference between E 2 - and E 3 - is that the intersection of both are described by a form of (7.7), but that E 3 - n Ilz has no compact components. With this understood, remark that E 3 - will be constructed from some number of like oriented, disjoint,
CLIFFORD HENRY TAUBES
370
push-off copies of E2- by surgery, with the point of the surgery to eliminate the unwanted compact components of the intersection with ~z. Of course, this must be done so as not to destroy any of desired properties of ~_-i.e., Assertions 2-5 and 7, 8 of Proposition 8.1. In abstraction, the problem is to remove circles which are components of the transversal intersection between two four dimensional submanifolds inside a seven dimensional submanifold. Here is the model: MODEL: Let X be a connected, oriented 7-manifold, and let A, B C X be oriented, dimension 4 submanifolds which intersect transversally. Let 0 C X be an open set and let 0' == (A n B) n O. Suppose that 0' is compact; a disjoint union of oriented, embedded circles.
(8.1) Given the model, here are the problems: PROBLEM 1: Find an oriented, dimension 4 submanifold A' C X with the following properties: 1) A' n (B n 0) 0. 2) A - (A n 0) A' - (A' nO). 3) [A] [A'] in H4(X, X - 0).
= =
=
PROBLEM 2: Find A' as in Problem 1 with H2(A'; Q) PROBLEM 3: Assuming that A - (A
= O.
n 0) has trivial normal bundle, find
A' solving Problems 1 and 2 with trivial normal bundle. And, given, apriori, a frame (for A' as normal bundle over A- (AnO), extend (over A' as a normal bundle framing. (8.2) These three problems will arise a number of times in the subsequent two sections and will be solved under various assumptions on A, B and O. The solution to Problem 1 begins with the following basic surgery result: PROPOSITION 8.2. Let X, A, B, and 0 be as described in (8.1) and in Problem 1 of (8.2). If the class, [0'], of 0' is zero in HI (B n 0; Z), then there is a solution to Problem 1.
Problem 2 can be solved when extra conditions are added:
8.3. Let X,A,B, and 0 be as in (8.1) and Problems 1 and (8.2). Assume that [O']=OinHI(BnOjZ). The map H~omp(Aj Q) -+ H 3 (A; Q) is injective. And, assume either HI(O'jQ) -+ HI(AjQ) is injective, or else assume B n 0 is connected and [0'] '# 0 in HI (Aj Q), Then, there exists A' C X which solves Problems 1 and is such that H 2(A / j Q) ~ H2(A; Q). Thus, Problem 2 is solved by A' if H2(A; Q) = O.
PROPOSITION
2 of a} b} c} d)
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Remark that Condition b of this proposition will be true automatically if
A n 0 is the interior of a manifold with boundary, A, whose boundary, 8A, obeys H2(8Aj Q) = O. To solve Problem 3 of (8.2), it is necessary to digress first to define a '1./2 valued invariant for homologically trivial, normally framed circles in an oriented 4-manifold with even intersection form. (This is invariant is well known to 4manifold topologists.) To start the digression, let B denote the oriented 4-manifold. To say that B's intersection form is even is to say that the self- intersection number of any embedded, orient able surface in B is an even number. (Note that B need not be compact.) Let u C B be the finite union of disjointly embedded, oriented circles which represents the trivial element in HI (Bj '1.). The invariant in question, XB,u(-), assigns ± 1 to the various homotopy classes of framings of the normal bundle to u in B. (If u is a single circle, then there are precisely two normal framings up to homotopy since 1fl(SO(3» ~ '1./2.) To calculate XB,u, first choose an oriented surface with boundary, ReB, such that 8R = u. An oriented frame (== (el,e2,ea) for the normal bundle to u in B' will be called an adapted frame when the vector ea is the inward pointing normal vector to R along 8R. LEMMA 8.4. Let B, u and R be as described above. Let' be an oriented, normal frame for u C B. Then, is homotopic to an adapted frame.
Proof. On a component, C, of (J, two normal frames differ by a map from SI to SO(3). With this understood, note that 1fl(SO(3» ~ Z/2, so there are two homotopy classes of normal frames along C. Two normal frames for which ea is the inward normal to R differ by a map from SI to SO(3) which factors through a map from SI to SO(2) C SO(3). With the preceding understood, the lemma follows because the induced homomorphism from 1fl (SO(2» to 1fl (SO(3» is 0 surjective.
The important feature of an adapted normal frame is that an adapted normal frame allows one to make an unambiguous definition of the mod(2) selfintersection number, (R· Rh, of R. Here is how: Take a section of R's normal bundle in B which agrees with el on 8R. Perturb the section away from 8R so that it has transverse intersection with the zero section. Then, count the number of such intersection points mod(2). One can also define R . R E Z by counting intersections with sign, but only the mod(2) intersection number is required for the definition of XB,u' LEMMA 8.5. If two adapted frames are homotopic in the space of all normal /rames for u, then the corresponding values of (R . Rh agree.
Proof. Adapted, normal frames to a given component C C u can be found which differ by a degree one map to 80(2) and are such that the corresponding
CLIFFORD HENRY TAUBES
372
push-off's of R are identical save for a small open set near a point in O. With this understood, one need only check the lemma for the case where R is a planar 2-disk in See, e.g Section 1.3 of [7]. 0
r.
It follows from Lemmas 8.4 and 8.5 that the surface R defines a map, XB ,tT (.), from the set of homotopy classes of normal frames of u C B to '1-/2. By definition, XB,tT(() assigns to, the number (R· Rh that is computed by using an adapted frame which is homotopic to ,. Consider the dependence of XB ,tT (-) on the surface R:
LEMMA 8.6. Suppose that B has even intersection pairing in its second homology. Then XB ,tT (-) is the same for all surfaces R bounding u.
Proof. Let ( be a framing of the normal bundle to u in B. Let R 1 ,2 C B be a pair of surfaces which bound u. The task is to show that Rl . Rl = R2 . R2 mod(2). One can assume, with no loss of generality, that ( is adapted to R 1 • Since 1fl (8 2 ) ::::l 0, the surface R2 can be isotoped, with u fixed, so that e3 is the outward pointing normal vector to R 2. With this understood, Rl and R2 can be joined together along u to obtain a 0 1 immersion of a closed, oriented surface, R, in B. (The lack of smoothness occurs across u.) The surface R may not be embedded because Rl and R 2 , though individually immersed, may intersect each other. Any way, with a small isotopy of Rl (away from 8Rt), the jntersections of Rl with R2 can be made transverse. An embedded surface in B has a well defined self-intersection number. An immersed surface has a well defined intersection number also. In this case,
(B.3) The number in (8.3) is the intersection number for the embedded surface that is obtained by resolving all of the double points of R. Given that (B.3) is the self intersection number of an embedded surface in B, the assumptions in Lemma B.5 require that (8.3) be an even number. Thus Rl . Rl = R2 . R2 mod(2) as required. (Here is how to resolve a double point of an immersed surface: In local coordinates the transveral intersection of the two sheets of the surface is described by the zeros in C2 = r of the equation (B.4)
Zl Z2
= o.
The resolution of the intersection point replaces the solution to (B.4) with the solution to the equation Zl Z2 = E. Here, E E C is small but not zero.) 0 With the invariant XB,u(') of Lemma 8.6 understood, end the digression. fiyr,; i~ " IlQlution to {8.2)'s third problem:
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PROPOSITION 8.7. Let X,A,B, and 0 be as in {8.1} and Problems 1 and f2 of {8.2}.- Assume that Conditions a, b and either cor d of Proposition 8.9 hold. Suppose that B has even intersection form and that A c X has a trivial normal bundle. Let ( be a given frame for A's normal bundle in X. 1) The restriction of ( to (J' (AnB) nO defines a normal frame, (" e I", to (J' in B.
=
2) 9}
=
ffxB,,,«(,,) = 0, then there is a solution, A' c X, to Problem 1 such that the normal frame ( over A - (A n 0) extends over A'. Thus, if H2(Aj Q) = 0, then A' solves Problems 1-9 of {8.2}.
The proofs for Propositions 8.2, 8.3 and 8.7 are given in Subsections 8f-8h.
b) The proof of Proposition 8.1. Let E~ _ denote the disjoint union of some number N ~ 1 disjoint copies of E2 _. The goal is to apply Propositions 8.2,8.3 and 8.7 to remove the compact (circle) components, C, of the intersection of E~_ with t!.z. With this goal understood, Proposition 8.2, 8.3 and 8.7 will be considered with the following identifications: Take
(8.5)
X = int(Z) ,
A
=
int(E~_),
B = int(t!.z).
Take 0 to be the compliment in int(Z) of the closure of a regular neighborhood of
(8.6)
=
Here, -, {t. ({ti x {ti) n t!.z with {ti as in Section 7b. This regular neighborhood should contain {ri} in (7.7) of E 2 - nt!.z, and it should also contain the push-off copies of {ri}which comprise the interval components of E~_n t!.z. Needless to say, 0 should contain the compact components of E~_ n int(Z). With this choice of X, A, B and 0, the assertions of Proposition 8.1 will follow from Proposition 7.4 if the hypothesis of Propositions 8.2, B.3 and B.7 can be verified for a suitable N. (Remember that E~_ is comprised of N pushoff copies of E 2 _.) Note: With regard to Proposition 8.7, the normal framing, (, of any push- off copy of E 2 - c EL should be the normal framing of E 2 which is described by Lemma 7.2. Subsections Bc-8e verify that there exists N ~ 1 for which the hypothesis of these three propositions are satisfied. 0
c) Removing circles in E~_
n t!.z
•
The purpose of this subsection is to verify that there exists an integer Nl ;::: 1 which is such that the hypothesis of Proposition 8.1 can be verified when E~_ is any multiple of Nl push-off copies of E 2 _. The discussion begins with a digression to study the first homology of B nO. (Equations (8.5) and (8.6) define B and 0.) The projection 7rL (or 7rR) identifies
CLIFFORD HENRY TAUBES
374
B with int(W). This projection identifies B n 0 with the compliment in W of U crit(J) U (Ur=lIJ.i) U 'Y, where 'Y is the How line a regular neighborhood of in 4 of Definition 3.1. Now consider (1 C B n 0, a finite union of embedded, oriented circles. After a small isotopy, the circles in (1 can be arranged to have empty intersection with the descending disks from crit2(J). With this isotopy understood, the pseudo-gradient How will isotope the circles in (1 so that the resulting circles, (11, lies in the open submanifold W3 == {x E W : 3/4 < f(x) < I}. That is, f«(11) is larger than any critical value of f. The pseudo-gradient How defines a diffeomorphism between W3 and M1 x (3/4, 1). By assumption, M1 is a rational homology sphere, which means that the homology class, [(1d, of (11 is zero in H 1 (W3 - W3 n 'Yi Q). (Note that W3 n'Y = PI x (3/4,1).) Alternately, one can conclude that
aw
(8.7) for some integer Nl ~ 1. This means that Nl push-off copies of (11 bounds an embedded surface in W3 - W3 n 'Y. (Orient all Nl push-off copies of (11 identically. ) Thus, Nl push-off copies of (1 will bound an embedded surface in B n O. End the digression. To verify Proposition 8.2's hypothesis for E~_, consider the discussion of the preceding subsection where (1 is equal to 0 in (7.7). This choice of (1 determines the integer Nl in (8.7). If Nl = 1 in (8.7), then Proposition 8.1 can be directly applied to A == ~_ so that the result, A', intersects l:l.z C Z as described by (7.7) but with 0 = 0. However, the case NI > 1 in (8.7) can not be ruled out. In the case that N1 > 1, let m ~ 1 and let E~_ denote the disjoint union mNt disjoint, push-off copies of E 2 -, all oriented as E 2 -. (Use the normal framing of Section 7f when making these push-offs.) With E~_ understood, observe that (8.8)
where 0' in (8.8) is, by design, m N1 disjoint, push-off copies of 0 from (7.7). In (8.8), each ri is the union of mNI det(S) push-offs (in l:l.z) of IJ.i == l:l.z n (JJi x IJ.i). By construction, the homology class of 0' in H t (B n OJ Z) is zero. (Because [0'] = mNt [C] and the class of 0 is Nt-torsion.) With the preceding understood, then Proposition 8.1 can be applied with X, A, B and 0 as described by (8.5) and (8.6) so long as the number N is a multiple of Nl in (8.7). d) Constraining H2.
Proposition 8.2 constructs a submanifold A' C Z from some number N > 1 push-off copies of E 2 _. (Here, N must be a multiple of Nl from (8.7).) This
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A' is constructed so that it misses EL,R and a form like ~_ near 8Z. And, the intersection of A' with liz is the union Ui=tri, where r i is the union of N det(S) push-off copies of the path -1'I.• H Proposition B.l's E 3 - is this A', then A' will have to have vanishing 2nd cohomology. That is, A' must be a solution to Problem 2 in (B.2). Proposition B.3 will be used to solve Problem 2 in the case at handj this is the subject of the present subsection. The task here is to verify that the conditions of Proposition B.2 can be met for A E~_ n X with E~_ some number, N, of like- oriented, push-off copies of E 2_. (Note that Assertion 6 of Proposition 7.4 asserts that H2(Aj Q) = 0.) Taking Conditions a - c in order, remark that the previous subsection has established that Condition a is satisfied when N is divisable by a certain integer (N1 of(B.7». Condition b is satisfied because of Assertion 4 of Proposition 7.4. That is, A has closure a manifold with boundary, and the boundary is a number of copies of S3. Since H2(S3) = 0, the required injectivity holds. Condition c is established by the following lemma:
=
LEMMA B.B. Let C denote the union of the compact components of ~_nliz. The inclusion of C into E 2 - induces a monomorphism from HI (Cj 1'.) into HI (E2 - j 1'.).
Proof. Remark that ~_ is obtained via ambient surgery (in Z) on various embedded (SO x B 4 )'s in disjoint unions of {Yi,i- : i ~ j E {I,··· ,r}} (see (7.3». As remarked earlier, Yi,i- can be viewed as the result of ambient surgery (in F- 1(I/B)) on various embedded (SO x B4)'S in the 4-sphere which is the intersection ofthe descending 5-disk from (ai, bi) with F-I(l/B). For a given So x B4, the SO x {OJ is a pair of algebraically cancelling intersections of said descending 4-sphere with the ascending 4-disk from some (ai, ak) or (b k , bj ) in crit4(F). Thus, E 2 - is obtained from a disjoint union of embedded 4- spheres in F-I(I/B) by ambient surgery on embedded (SO x B 4 )'s. It follows from the preceding that Hl(E~_) is a summand of some number of 1'.'s. And, it follows that a union, 0', of oriented, embedded circles in E 2 - injects its first homology into Hl (E2 - j Q) if: 1) An added I x S3 which intersects 0' has intersection number ±1 with {point} x S3 . 2) Each component of 0' intersects at least one I X S3. (8.9)
In the present circumstances,
0'
is the union of the compact components of
n D..z. To understand 0', remember that E'l- is constructed from E 1 - by ambient surgery. The reader can check that this surgery is disjoint from any compact components of El- n D..z. Indeed, the surgery from E 1 - to E 2 - takes
E~_
376
CLIFFORD HENRY TAUBES
place on push-off copies of {Yi,i-}i=l' but the compact components of El-n.6. z are the components of the various push-offs of Ui<j(Yi,j- n .6. z ). Thus, the compact components of E 2 - n .6. z are of two types: A Type 1 component is a compact component from E 1 - n.6. z . And, a Type 2 component is created by the surgery which changed E 1- into &_. (The latter are made in the surgery on the various push-off copies of {Yi,i- H=l') To understand the Type 1 components, use 7rL or 7rR to identify .6. w with W and this intersection is identitified with B 1a+ n B1b-. (Here, a = ai and b = bj.) To see the latter, start with Ba+ n Bb-. This is a disjoint union of flow lines which start at a and end at b. The surgery which changes Bb- to B1beffects the intersection with B a+. The effect is to surger the flow lines near a. See the following picture:
(8.10)
A similar picture occurs near b when Ba+ is surgered to produce B 1a+. The resulting intersection B 1a+ n B lb - differs from Ba+ n B b- in that the ends of the flow lines in the latter have been tied together near a and near b to produce a compact intersection with some number, ni,j, of components. (This 'fti,j is at least one, but no more than half of the number of components in Ba+ n Bb-.)
377
HOMOLOGY COBORDISM
See the following (very schematic) picture: surgery to Bb-
) Ba.
Surgery to Ba+
11111111
(8.11)
= B16-n B a+
The effect of the preceding picture for the intersection with tlz of one pushoff copy of Yi,i- is as follows: Each surgery ties together an end of one flow line (for f's pseudo-gradient) in tlz with the nearby end of a se~ond flow line in tlz the tie being across the associated I x S3. (Here, the canonical identification of tlw with W is taken implicitly.) See below: Added by surgery
(8.12)
Thus, each copy of Yi,i- in E 2 - (for i < j) produces ni,i components in u. And, (8.9) is satisfied for each such copy. As (8.9) is obeyed for each copy of Yi,i-' and as the surgeries on the different copies of Yi,i are independent, it follows that (8.9) is satisfied by the set of all Type 1 components. That is, (8.9) is satisfied by the union of the compact components of E 1 - n tl z . Consider now the Type 2 components. To understand these components, remember that E 2 - was constructed from E 1 - by ambient surgery on various push-off copies of {Yi,i-}' The surgeries do not connect a push-off of Yi,i- with one of l'i,j- if j # i.
CLIFFORD HENRY TAUBES
378
Fix i E { 1, . .. ,r}. The union of push-off copies of a given Yi, i intersects Llz in the union of the corresponding push-offs of the set of flow lines which is Ba+ n B b_. (Use a ai and b bi). As far as these copies of Yi,i- are concerned, E 2 - is constructed from them by surgery on embedded (SO x B4)'s. The result surgery changes the afore mentioned intersection with Ll z ; each such surgery near (a, a) ties the ends near (a, a) of two of flow line copies across the added I x S3. There is a similar effect near (b, b). See (8.12). It follows from the preceding picture that (8.9) holds for all of the Type 1 flow lines also, and since the added (I x S3)'S which effect Type 1 flow lines are disjoint from Type 2 flow lines, the lemma is established in total. 0
=
=
e) Prescribed framing. The purpose of this subsection is to establish that the conditions of Proposition 8.7 can be met for X,A,B and 0 of (8.5) and (8.6) if the number N (of copies of E 2 - in E~_) is an even multiple of the integer Nl in (8.7). Here, the framing ( of the normal bundle in Z to each push-off copy of E 2 - c E~_ is described by Lemma 7.2. To begin, observe first that B has vanishing rational homology in dimension 2, so the condition on B's intersection form is trivially satisfied. Also, as A is some number of push-off copies of E 2 -, it has trivial normal bundle in Z. Next, remember that an integer Nl > 1 has already been found which has the following properties: If N 2': 1 is divisible by Nl and if A is taken to be N pushoff copies of E 2 - (all like oriented), then a is the boundary of an embedded, oriented surface ReB. With the frame a as described above, let (IT lIT' The final question is the value of XB,IT«(IT)' Here is the answer: If N is an even multiple of N 1, then XB,IT «IT) = O. This assertion follows from the following lemma:
=(
LEMMA 8.9. Let X be an oriented 4-manifold with even intersection form. Let ai, a2 C X be compact, oriented, embedded l-manifolds which are disjoint. Let (1 and (2 be normal frames for al,2, respectively. Let a = al U a2 and let e by the normal frame for a which is given by ( IlTl,2= (1,2, Then XB,IT«() XB,lTl«I) + XB,1T2«(2).
=
Proof. By assumption, al bounds an embedded, oriented surface, Rl eX. Likewise, a2 bounds a similar surface, R 2. If Rl and R2 are in general position, then R2 n a2 = 0 and vice- versa. Meanwhile, Rl will intersect R2 transversally in a finite set of points. Resolve the double points in Rl U R2 (as in (8.4» to obtain a compact, oriented, embedded surface, ReX, with boundary a. Now, no generality is lost by assuming that (1,2 are adapted frames for R 1 ,2, respectively. In this case, ( will be an adapted frame for R. Then, R· R is given by (8.3), and the lemma follows. 0
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379
f) Proof of Proposition 8.2. The proof starts with a digression for some constructions on a neighborhood of B in X. To start the digression, define N -+ B to be the normal bundle to B in X. Fix an exponential map,
e:N-+X
(8.13)
which maps N I., into A. Put a smooth fiber metric on N with the property that e embeds the set
(8.14)
N' == {v E N:I v
1< 2}
onto a neighborhood N C X of B. Agree now to identify N with N' using e. Introduce the 2-sphere bundle 8 -+ B,
(8.15)
8 == {v E N
:1 v 1= I}.
Identify 8 with its image by e in X. This 8 is the boundary of a tubular neighborhood of B in X,
(8.16)
T == {v EN
:1 v I~ I}.
End the digression. The proof proper of Proposition 8.2 starts by remarking that u, by assumption, is the boundary of a smooth, oriented, embedded surface (with boundary), ReB n O. Find such an R for which int(R) has no compact components. Because u has co dimension 3 in B, one can require that int(R) nu = 0. (The local model for R near u is given by taking u to be the line Xl = X2 = X3 = 0 in r, and R the half plane Xl = X2 = 0 with X3 ~ 0.) Let 8 R == 8 IR . This is a smooth oriented 4-manifold (with boundary) which is embedded in X. The boundary of this 4-manifold is 8., == 8 I.,. Note that 8., C A is the boundary of T., == T I." the embedded image of u x B3 onto a tubular neighborhood of u in A. With the preceding understood, introduce the following surgery on A:
(8.17)
A~
== (A - int(T.,)) U SR.
Note that A~ C X is a Co embedding of a smooth, ori~nted manifold; the embedding has a corner at StT where SR and A - int(TtT) overlap. See the
CLIFFORD HENRY TAUBES
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following picture:
~///~R
"0'
(B.18) A neighborhood of this corner of A~ is embedded in TR. Smooth A~ in TR along the corner, and one obtains a smoothly submanifold, A' C X which solves Problem 1 in (8.2). Indeed, the first two requirements are met by construction. As for the third, remark that [T IR] defines a 5-dimensional cycle in 0 whose boundary is [A']- [A]. 0
g) Proof of Proposition B.3.
Consider first the proof of the proposition under the Assumption a-c. Let A' be as described above (see (8.17)). Then, H2(A') can be computed using the following homology exact sequences for the pairs (A,Tu) and (A',SR):
(B.19)
1) 2)
Hl(A) -+ Hl(Tu) -+ H2(A,Tu) -+ H2(A) -+ H2(Tu) . Hl(A') -+ Hl(SR) -+ H2(A ' ,SR) -+ H2(A') -+ H 2(SR) -+ H 3 (A' ,SR)'
(Use rational coefficients please.) In Sequence 1, the first arrow is surjective because of Assumption c. And, H2(Tu) :::: 0 because Tu is a tubular neighborhood of a disjoint union of circles. Thus (B.20)
To analyze the second sequence of (8.19), note that its first arrow is surjective. This is because R is path connected, thus forcing 8 R to be a topologically trivial 2-sphere bundle. For the same reason, H2 (8R) :::: H2 (aTu). Meanwhile, excision identifies H* (A' ,8R) :::: H* (A, Tu). Thus, with (8.20), the second sequence in (8.19) implies
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(8.21) Poincare' duality plus Assumptions b and c of Proposition 8.3 imply that the last arrow in (8.21) is surjective, thus establishing an isomorphism between H2(A) and H 2 (A'). Now consider Proposi~ion 8.3 with Assumptions a, b and c'. Remark here that when u is connected (i.e. just one circle), then Assumption c' implies Assumption c. With this fact understood, here is the task ahead: Under Assumptions a, band c', find an ambient (in 0) surgery on A so that the result, A, has the following properties: (8.22)
1) [AJ = [A] in H4(X, X - OJ Z). 2) H2(A.) ~ H2(A). 3) H:omp(A) -+ H 3 (A.) is injective. 4) A intersects B inside 0 in a single compact component which is not trivial in HI (A, Q) but which bounds in B n O. A solution to (8.22) will validate Proposition 8.3. Here is an algorythm for constructing A: To begin choose a pair of components, Cl ,2 C u. Fix Pl E Cl and P2 E C2 • By assumption, B n 0 is path connected, so there is a path in B (a smoothly embedded interval), T, which starts at Pl and ends at P2. Make sure that int(T) has empty intersection with A. Also, arrange T so that it is not tangent to A along its boundary. The choice of PI as the starting point and P2 as the ending point orients T. Let V -+ T denote the normal bundle to T in B. This is an oriented 3-plane bundle over T. Note that TCl IPI C V IPI ~ IR3 and also TC2 C V Ip2~ R3 are oriented lines. As 8 2 is path connected, there is an oriented, dimension 1 sub-bundle Va C V whose restriction to PI is TCI and whose restriction to P2 is the line TC2 , but oriented in reverse. With Vo understood, remark that Va EB N -+ T is an oriented 4-plane subbundle of the normal bundle to T in O. Also note that this bundle restricts to PI and as T A IP1' and it restricts to P2 as the 4- plane T A 1P2' but with its orientation reversed. The normal bundle to T in X is isomorphic to V EB N. Fix an exponential map eT : V EBN -+ X which restricts to N as e in (8.13), which restricts to map V into B, and which maps VO IP1,2 into Cl ,2, respectively. (Thus, e.,. Iv is an exponential map for T in B.) Put a fiber metric on the bundle Va EB N such that e.,. embeds the subspace of vectors v with norm less than 2. Let 8 C Vo EB N denote the radius 1 sphere bundle, and identify 8 with its embedded image under e T • Let T C Yo EB N denote the radius one, 3-ball bundle .. Identify T with its embedded image under e T • Note that T IPI is a tubular neighborhood in A of PI, while T 1P2 is a tubular neighborhood of P2 in A.
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Now define the following surgery on A:
(8.23)
Ao
=
(A - int(T IPI UT Ip2)) U 8.
This is a Co embedding of a smooth manifold, AI, in X. Here, the embedding is smooth away from 8 IPI W8 Ip2' where there is a corner. Near this corner, Ao is embedded in Vo Ell N. Smooth out the corner in Vo Ell N and the result is an embedding of Al into X:
, ,
- - - rA
(8.24) With Al understood, consider its properties with respect to (8.22): First, the homology classes of Al and A agree are equal in H4(X,X - 0). This is because Al and Ao define the same class and the 5-manifold T defines a cycle in 0 with boundary [Ao] - [A]. Second, H2(Al) = 0, because Al has been obtained from A by surgery on an embedded 8 0 x B4 (i.e. T IPI UT Ip2). Third, H~omp(Al) ~ H3(At} is injective. Both are either equal to their A counter-parts, or are obtained from their A counter parts by the addition of 1 generator which is dual to the 3-sphere 8 IPI. (Prove this with Meyer-Vietoris.) Fourth, the intersection of Al with B in 0 has one less component then that of A with B in O. This is because the surgery from A to Al has surgered C1 to C 2 by removing an embedded 8 0 x Bl from C 1 UC2 (i.e. (TnVo) IPI u(TnVO)Pl); the missing 8 0 x Bl is replaced by 8 n Vo. (See (8.24).) Note that the homology class in HI (B n 0) of 0"1 == Al n B n 0 is the same as that of 0" == An B n O. Indeed, 0"1 defines the same homology class as 0"0 == Ao n B n 0, and Tn Vo as a 2-cycle in B n 0 has boundary [0"0] - [0"1]. If 0" had only two components to start with, then set A == Al and stop, because (8.22) has just been verified for this A. If 0" had more than two components, iterate the preceding procedure by renaming Al == A and 0"1 == 0". The iteration stops with A which obeys (8.22). 0
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h) Proof of Proposition 8.7. The construction of A' from A for solving Problem 1 (and Problem 2) of (8.2) via Propositions 8.2 and 8.3 is described in the preceding two subsections. The construction involves two types of ambient surgeries in X. The first type of surgery gives the smoothing, AI, of Ao in (8.23). The second type of surgery gives the smoothing, A', of the ~ in (8.17), (8.18). Type 1: Extending frames for surgery on SO x B4. Consider the smoothing, At, of Ao in (8.23). Let U c A be an open neighborhood of T Ipl uT 11'2. 0
LEMMA 8.lD. Let ( be a normal frame for A in X. Then there is a normal frame for Al in X which agrees with ( on A - u.
Type 1: Extending frames over surgery on Sl x B3. The assumption here is that A' is obtained from A by smoothing the surgery A~ in (8.17). (See (8.18) too.) Let ( be a normal frame for A in X, and let U C A be a neighborhood of T.,.. LEMMA 8.11. The normal frame ( on A - U extends as a normal frame over the smoothing, A', of (8.17) if and only if ( I.,. is homotopic to an adapted frame for which (R . R) mod(2) = o.
H B has even intersection pairing, then according to Lemma 8.6, the Z/2 number (R· R) mod(2) is the invariant XB,.,.«( I.,.). Thus, Lemmas 8.10 and 8.11 with the constructions in the two preceding subsections prove Proposition 8.7. The remainder of this section is occupied with the proofs of the preceding two lemmas. Proof of Lemma 8.10. The strategy is to first define a normal frame, (1, for int(S) in X. Having done so, the final step proves that there are no obstructions to connecting (I to ( on A - U. To construct (1, first fix a normal frame, (e1' e2, e3), for the normal bundle (V) of Tin B. One can arrange such a frame so that e3 is tangent to the sub-line yo. Then, (e1,e2) orient VIVo. Use the exponential map er : V EB N -+ X to identify a neighborhood of the zero section of V $ N with a neighborhood of T in X. With this identification understood, then the normal bundle to int(S) in X is spanned by (I == (e1,e2,e~), where e~ E T(Vo EB N) Is restricts to the fiber over x ETas the inward pointing normal vector to S I., in (Vo EB N) I.,. With (1 understood, consider connecting (I to, near as. To make such a connection, introduce the tangent vector, v to T. Let lL denote a lift of v to VEBN. At PI or P2, the triple (e1, e2, v) defines a normal frame. for A in X. The normal frame ( for A in X can be homotoped inside U so that it agrees with (et,e2'v) at PI and P2 and equals (el,e2,lU on a neighborhood, U' , of T Ipt.2
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with compact support in U. Thus, 2/3 of the normal f,rame (, i.e. (el,e2), have been extended over A 1 • The compliment in the normal bundle to A1 of the 2-plane span of (el, e2) is an oriented line bundle which is framed by!l. on U - U' and by e~ on int(S). There is no obstruction to framing this compliment by a frame which agrees with !l. on U - U' and with e~ on the compliment in S of an apriori specified neighborhood of as. 0 Proof of Lemma 8.11. The strategy for extending (on A - U as a normal frame for A' will be to construct a normal frame, (1, for SR in X, and then consider whether ( and (1 can be joined. To construct (1, introduce the normal 2-plane bundle, V, to R in B. Since R is oriented, V is an oriented bundle and so trivial because inteR) has no compact components. Let el,e2 be a frame for V. Note that V E9 N -+ R is the normal bundle to R in X. Choose an exponential map eR : V E9 N -+ X which restricts to N as e in (8.13). Use eR to identify a neighborhood of the zero section in the bundle V E9 N with a neighborhood of RinX. Let e3 be the inward pointing normal vector to SR C TR eN IR. Then the triple (1 == (el, e2, e3) span the normal bundle to S R in X. With the normal frame for S R understood, consider its extension to a normal frame for A on the compliment in U of a neighborhood U' of Tq. For this purpose, introduce v to denote the inward pointing normal vector field to R along u. Lift v to a vector field !l. on (V E9 N)q. Since A near u is identified by (8.13) with a neighborhood of the zero section of N Iq, it follows that a normal frame for A near u is given by the triple (e1e2,!l.). Furthermore, there is no obstruction to joining this frame on the compliment of a neighborhood U' of Tq with the frame (1 == (e1, e2, e3) on the interior of SR. (The pair (e1, e2) define 2/3 of the extension, and Q and e3 define the same orientation for the complimentary line.) With the preceding understood, then one can conclude that the normal frame ( extends from A - U to A' if the restriction of ( to u is homotopic to the normal frame (el, e2,!l.). Now the latter frame is an adapted frame and, by construction, R· R = 0 for (e1,e2,!l.). Thus, (Iq is homotopic to (e1,e2,u) if and only if (Iq is homotopic to an adapted frame for which the corresponding R . R is even. (See Lemma 8.5.) 0
9 The fourth pass at E±. The submanifold E3- of the preceding section intersects Az as described by Assertion 6 of Proposition 8.1. Let f) : Z -+ Z denote the switch map which sends (x,y) to (y,x). Since E 3 +. = f)(E3_), the intersections of E 3 - with Az are also intersections of E 3 + with A z . Unfortunately, there may be compact components to E 3 - n E 3 + which occur in Z - A z . Such extra components are troublesome and must be eliminated, and their elimination is the goal of this section. As will be seen, surgery on E 3 ± will result in oriented submanifolds (with hnlln~nJY1 ~~+ ~ which have the following properties:
[
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PROPOSITION 9.1. There are oriented, embedded submanifolds (with boundary) E4± C Z with the following properties: 1) There is an open neighborhood U C Z of Az U az such that E4+ n U and E4- n U are images of each other under the switch map on Z. 2) The fundamental classes [E4±] are equal to N [El±] for some integer N ~ 1. Here, [El±] are described by (6.3) and Lemma 6.3. 3) E4± have empty intersection with Mo x Mo and Ml X MI. 4) E4± have empty intersection with EL,R of (4.15). 5) If p E crit(f), then the intersection of E4- with (S3 x S3)p has the form S3 x Ap, where Ap is a set of N points. Similarly, the intersection of E4+ with (S3 x S3)p is Ap X S3.
n Az
= ui=lri, where ri C
!:l.z is as follows: There is a flow line and ends at bi' With the canonical identification of Aw with W understood, r i is the union of N like oriented, disjoint, push-off copies of a closed interval, I C /Ji. And, each of these N pushofJs of I starts in (All X Aa) n!:l.z and ends in (Ab x Ab) n!:l.z. Likewise, EH n!:l.z = Ui=lr,. 7) E 4- n E4+ = Ui'=l r~, where r~ C Z is the union of r i with N - 1 like oriented, push-off copies of r i in Z - !:l.z. 8) Both E4± have trivial normal bundles in Z. The normal bundle of E4has a framing, (, which restricts to a product normal framing on a neighborhood 0/ (Ui'=lri) U {S3 x Ap}pE crit(f). Furthermore, this framing ( restricts to {S3 x Ap}PE crit(f) as a constant framing. The normal bundle to EH in Z has a framing which restricts to EH n U as the image 0/ ( under the switch map. 9) H2(E4±; Q) = O.
6)
E4-
/Ji which starts at
ai
(Compare with Proposition 8.1. The only essential change is in E 4 - n E H • But note that the integer N, the points {Ap} and the line segments {rd which appear here may be different from those which appear in Proposition 8.1.) The rest of this section is devoted to the construction of E4±.
a) E 3 - n E 3+ on Z - Az. Isotope E3+ in Z - (!:l.z U aZ) so that its intersection on Z - !:l.z with E 3 - is transversal. (Still use E 3+ to denote the after isotopy submanifold.) This intersection is now a finite union of disjoint, embedded, oriented circles, a, with N - 1 like oriented, push-off copies of {ri } in Z - !:l.z. (These pusho1£s of {ri} are disjoint from a.) There is a natural inclination to remove the circle components, a, by apply the techniques from the previous section (Propositions 8.2, 8.3 and 8.7). However, either
(9.1) or not; but [aJ is never non-trivial in one and trivial in the other. (Because, before perturbing E 3 +. one was the image of the other under the switch map.)
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IT (9.1) holds, then one can not (directly) apply Proposition 8.2 to remove the intersection u. IT (9.1) does not apply, then Proposition 8.2 can be applied, but not (directly) Proposition 8.3. So, whether or not (9.1) holds, some preliminary work must be done before the techniques from Section 8 can be employed. Consider the case where (9.1) holds.
=
LEMMA 9.2. Let 0 Z - (~z U 8Z U EL U ER)' There is an ambient surgery (in 0) of E 3 - which results in an oriented submanifold (with boundary) E~_ C Z with the following properties: 1) Assertions 2-8 of Proposition 8.1 hold when E~_ is substituted for E 3 _. 2) There is a tubular neighborhood, Ut:. C Z, of ~z which intersects E~_ n E~+ as u'=lr~, where q is the union of r i with N - 1 disjoint, like oriented, push-off copies of rio 9) E~_ n E~+ intersects Z - Ut:. as a disjoint union, u', of oriented circles which obey (a) [u'] "I 0 E Hl(E~_; Q). (b) [u'] = 0 E H_ 1 (E3+; Q).
(9.2) (Note that there is no E~+; the symmetry under the switch map will be broken here.) The proof of this lemma will be given shortly. Consider the case where (9.1) is false. The goal here is to modify E3+ as described in the following lemma:
=
LEMMA 9.3. Let 0 Z - (8Z U ~z U EL U ER)' There is an ambient surgery (in 0) of some number n ~ 1 of like-oriented push-offs of E3+ which results in an oriented, embedded submanifold, (with boundary) E~+ C Z with the following properties: 1) Let E~_ denote the union of n like-oriented, push-off copies of E s _. The Assertions 2-8 of Proposition 8.1 hold when E~± are substituted for Ea. 2) There is a tubular neighborhood, Ut:. C Z, of ~z which intersects E~_ n E~+ is U'=l q, where q is the union of r i with N - 1 disjoint, like oriented, push-off copies of r i. 9) E~_ n E~+ intersects Z - Ut:. as a disjoint union, u', of oriented circles which obey (a) [u'l"l 0 E Hl(E~_j Q). (b) [u'] = 0 E H-l(E3+;Q)'
(9.3)
This lemma will also be proved shortly.
HOMOLOGY COBORDISM
Proof of Prop.osition 9.1.
387
=
When (9.1) is true, take E~_ from Lemma 9.2 and set E~+ E3+. When (9.1) is false, take E~± from Lemma 9.3. Use 0" to denote the intersection in Z - (8Z U Ua U EL U ER) between E~±. Use (9.2b) or (9.3b) in the respective cases, to find an integer N3 ~ 1 with the property that N3 [0"1 = 0 E H 1 (E3+jZ). Take 2 N3 push-off copies of E~_, all with the same orientation as E 3-, and let A denote the interior of the resulting union. Take 2 N3 push-off copies of E~+, all oriented as E~+, and let Bo denote the interior of the resulting union. Let X int(Z) and let be as before. Now, Proposition 8.2 can be invoked using Bo for B, but not Proposition 8.3 because Assumption c has not been shown to hold, and because Assumption c! will be false because Bo will not be connected. However, there is a surgery which remedies this problem: 0
°=
°
=
°
LEMMA 9.4. Let B o, A, X and be as described above. Then, there is an ambient surgery in A on some finite number of embedded (80 x B4) 's in Bo such that the result, B is path connected. This surgery does not change either H 1 ('j Z) or H2 ('j Z). Finally, if ( is a normal frame for Bo in X, and ifU C Bo is a neighborhood of the (80 x B4) 's, then ( IBO-U extends smoothly over B as a normal frame for B in X.
°-
The proof of this lemma is given below. With Lemma 9.4 understood, Propositions 8.2, 8.3 and 8.7 can be applied using X, 0, A and B as described above. Use E 4 - to denote the closure in Z of the promised solution, A', to Problems 1-3 in (8.2). Relable E4+ to denote the closure in Z of B. The pair E4± will satisfy the requirements of Proposition 9.1.
b) Making [0"] '" 0 E Hl(E~_jQ). This subsection is concerned with the construction of E~_ of Lemma 9.2. This construction requires a preliminary digression to introduce another surgery tool. The digression concerns the abstract model of (8.1). Proposition 9.5, below, summarizes the digression. The statement of Proposition 9.5 requires the following remarks to set the stage: When 8 -+ A is an embedded 2-sphere, let 1)s -+ 8 denote the normal bundle to 8 in A. Suppose that P C X is an embedded 3-dimensional ball with boundary 8. (The local model here takes 8 to be the plane X3 = ... = X7 = 0 in ]R7 and then P is the half plane X3 ~ O,X4 = ... = X7 = 0.). Use Np to denote the normal bundle to P in X. The natural inclusion (9.4)
O-+1)s-+Npls
plays an important role in Proposition 9.5. PROPOSITION 9.5. Let A,B,O,X and 0' be as described in {8.1}. Assume that: a} H~omp(Aj Q) -+ H3(A; Q) is injective.
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b)
[a] = 0 E Hl (A; Q). Suppose that there exists an embedded 2-sphere S C (A n 0) - a which bounds an embedded 9-ball P cO, and:
c)
The fundamental class of S is homologically trivial in H2 (A; 10)·
d)
int(P)
nA
= 0.
e) P intersects B transversally. f) With respect to some orientation on P, [P n B] g) (Np Is)/vs ~ S is a trivial 2-plane bundle. 1)
i
0 E Ho(P; Z).
Let U C X be an open neighborhood of P. There exists an embedded, oriented, 4-dimensional submanifold A' C X with the following properties: A' intersects B transversally in a', and [a'] i 0 E HdA';IQ).
= H 2 (A, 10).
2) 9)
H 2 (A'; 10)
4)
A'
5)
[a'] = [a] E Hl (B; 10). Ftt.rthermore, If A has trivial normal bundle in X, then so does A'. And, if ( is a frame for the normal bundle to A in X, then, IA-U extends over A' as a frame for the normal bundle to A' in X.
6)
H:omp(A'; Q) ~ H3 (A'; Q) is injective.
=A
on X - U and [A]
= [A']
in H4(X,X - 0).
This proposition will be proved in the next subsection. Consider now its application to (9.2).
Proof of Lemma 9.2. The lemma will be proved by applying Proposition 9.5. For this purpose, take X to be int(Z) and then define 0 Z - (8Z U UlJ. U EL U ER). Take A int(E3_) and, likewise, take B int(Ea+). Given that the assumptions of Proposition 9.5 hold when (9.1) is true, one should take E~_ to be the closure in Z of the sub manifold A' of Proposition 9.5. As for the validity of the assumptions of Proposition 9.5, remark that Assumptions a - b are satisfied by construction; see Proposition 8.1. (Assumption b holds since E 3 - is a manifold with boundary whose boundary is a union of 3-spheres; and H2(S3) = 0.) The remaining assumptions of Proposition 9.5 will be verified with the exhibition of a 2-sphere SeA n 0 with the requisite properties. To find the appropriate 2-sphere, it is important to remember that E 3 - was constructed from E~_. Here is a brief summary: The compact components of EL n dz bound a connected, oriented, embedded surface (with boundary) R C il z . Let N ~ dz be the normal bundle. A fiber metric was chosen for N, and an exponential map e : N ~ Z was chosen so that e mapped the radius 2 ball fiber bundle in N diffeomorphically onto its image in Z. (This ball bundle in N was identified using e with a neighborhood of ilz in Z.) Also, e was constrained to map N laR into E~_. Next, the radius 1 sphere bundle, SR C N IR was introduced, as well as the radius 1 ball bundle, TR C N IR. Finally, E 3 _ was defined to be the result of smoothing the corner in the surgery
=
=
=
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(9.5) By the way, no genetality is lost by assuming that the surgery in (9.5) occured. Indeed, if E 2 - = E3- and no such surgery occurs (in which case E 2 _n int(~z) has no compact components), then there are isotopies of El± so that the resulting submanifolds obey the conclusions of Proposition 9.1. Or, if E 2 - = E 3 - , then one could add two push-off copies of Y1,l- to Ef_ of Section 7a, one oriented positively and the other oriented negatively. Then, the tubing construction of E 2 - will insure that E 2 - n int(~z) has a compact component. With (9.5) understood, remark that E3+ is obtained from 9(E3-) by an isotopy. Let R' C R denote the complement of a (small) collar of 8R. Near R',E3- is 8R IR' and this coincides with 9(E3-) near R'. (Thus, 9(E3-) and E 3- dQ not intersect transverally.) Choose the isotopy to obtain E3+ from 9(E3-) so that E3+ near R' is a sphere bundle 8R+ eN IR' of radius greater than 1. Pick a point x E R' and let
(9.6) This is a homologically trivial, embedded 2-sphere in E 3 _. This 8 z bounds the 3-ball Tz C N Iz which is the unit ball in the fiber of N at x. Notice that this 3-ball has empty intersection with E 3+ since the latter intersects the fiber at x in a sphere of radius larger than 1. However, the ball T z intersects ~z transversally in a single point, namely x. This intersection with ~z will now be traded for N transversal intersections with E3+. (This is the same N as in Proposition 8.1.) The technique used here is called "connect summing with a transverse sphere" (see, e.g. Chapter 1 of [7]). This technique proceeds as follows: Fix p E crit(J) and then fix a point q E 8 3 - Ap. Observe that the sphere 8 3 x q C (83 X 8 3 )p intersects ~z transversally once (at q x q), and it intersects E 3 + transversally N times, at q x Ap. Orient S3 x q and all N intersections with E 3+ will have the same sign. (See Assertion 5 of Proposition 8.1.) Because q ¢ Ap , this S3 x q will have empty intersection with E 3 _. Take this S3 x q C (S3 X 8 3)p and push it off 8Z so that it is an embedded submanifold, Y C int(Z). Push if off only slightly, so that Y still intersects E3+ in the N points of the push-off of q x Ap , and so that Y intersects ~z in the push-off of q x q. Also, do not let Y intersect E 3 _. Remark that N is an oriented vector bundle, and thus Tz is an oriented 3ball. Orient Y so that its intersection number with ~z is the opposite of that for T z n ~z. Now, one can "tube" Y to T", to obtain a new 3-ball, P C Z with the following properties: 1) 2)
3)
=
8P S"" P n (Az U az U EL U E R ) = int(P) n E 3 - = 0,
0,
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4)
P n E3+ is N distinct points, all homologically the same in Ho(P).
(9.7) Abstractly, P is the connect sum of Y and T z . Realize this connect sum in Z by choosing a path, T, between x and x, Y n!:::.z in int(!:::.z). Make sure that T avoids the paths {ri}i=l of Assertion 6 in Proposition 8.1. Modify the exponential map e: N -t Z so that e maps N Izl into Y. Let S,. eN Ir denote the radius 1/8 sphere bundle. Let T~ C N 1:1: and TZI eN Izl denote the radius balls of radius 1/8. Then Y is obtained by smoothing the corners of the surgery
=
(9.8)
«Y - T~/) u (Tz
- T~))
u Sr.
Here is a picture:
-
!:::.z
(9.9)
_
!:::.z
(9.10)
=
The 3-ball P and the 2-sphere S Sz satisfy Assumptions c - J of Proposition 9.5. To apply Proposition 9.5 to prove Lemma 9.2, it is only' necessary to check that Assumption f of Proposition 9.5 is satisfied. For this purpose, let 7r : Sz -t x denote the projection. Then, the normal bundle to Sz in A is isomorphic to 7r*TR Iz. Meanwhile, the normal bundle to P in X along Sz is the same as the normal bundle to Tz in X along Sz which is isomorphic to 7r*TD.z I",. Thus, the quotient (Np Is)/vs ~ 7r*«TD.z Iz)/(TR 1:1:)), which is lli' 0
tlfiyinl. reouireo,
HOMOLOGY COBORDISM
391
c) Proof of Proposition 9.5. Since [S] is homologically trivial in A, Vs -t S is a trivial 2- plane bundle. Since P is a 3-ball, so N P -t P is a trivial 4-plane bundle. And, because (Np Is)/vs is a trivial bundle, there are no obstructions to extending Vs over P as a 2-plane sub bundle of N p. Use v -t P to denote this 2-plane bundle. Fix an exponentional map ep : Np -t X with the following properties: First, ep should restrict to Vs as a map into A. And, ep should restrict to map Np IpnB into B. Fix a fiber metric on Np with the property that ep embeds the interior of the unit ball bundle in Np onto a neighborhood of P in X. Then, use ep to implicitly identify the unit ball bundle in N p with its ep-image. With the preceding understood, let So C v denote the I-sphere bundle of radius 1/4. Let to C v denote the ball bundle of radius 1/4. (Thus, aCto I int(P) = So I int(P)') Introduce the surgery
(9.11)
Ao
= (A -
to Is) u So,
which is a Co embedding of a smooth, oriented 4-manifold, A' into X. This embedding is smooth away from the corner at So Is and it can be smoothed inside v to produce a smooth embedding of A' into X. Note that A' can be arranged to agree with A on the compliment of any apriori specified open neighborhood U C X of P. With A' understood, consider the various assertions of Proposition 9.5: To prove Assertion 1, consider that (9.12)
u'
= (A' n B) n 0 = So IpnB U
u.
This is a transversal intersection because P intersects B transversaly. The fact that lU'l '" 0 in Hl(A'j Q) follows via Meyer- Vietoris and the fact that [pnBj '" 0 in Ho(P). (Use the Meyer- Vietoris sequences for the decompositions A = (A - So Is) U (to Is) and also A' = (A - So Is) u so.) Meyer-Vietoris also proves Assertion 2, namely H2(A' j Q) = H2(Aj Q) . (Use the same sequences as above.) Assertion 3 is true because one can interpret to as a cycle, and this cycle obeys ato = [A] - [A']. Assertion 4 is true because the circles in So IPnB bound the discs to IPnB. To prove Assertion 5, the strategy will be to find a framing of the normal bundle to int(so) in X which is compatible with the framing ( on A - S. To begin, introduce the notation ll. to denote the vector field along So C to which points radially inward on each 2- ball fiber of to -t P. With ll. understood, the normal bundle to So in X is isomorphic to (Np/v)ffi Span(ll.). The bundle Np/v -t P is a trivial 2-plane bundle (because P is a ball), and so it has a global frame, (el,e2)' Thus, (el,e2,1l.) is a frame for the normal bundle to int(so) in X. Now consider the normal bundle to A along S. For this purpose, let e3 denote the inward pointing tangent bundle to P along S _ (So
e3
spans the nonnal
CLIFFORD HENRY TAUBES
392
bundle to 8 in P). Then, along 8, the normal bundle to A in X is isomorphic to (Np/v) Is Ell Span(e3)' Thus, Np Is Ell Span(e3) -+ 8 is isomorphic to the normal bundle of 8 in X. Let es : Np Is Ell Span(e3) -+ X be an exponential map which maps Vs into A and which maps v Is Ell Span(e3) into P. Use es to identify a neighborhood of 8 in X with a neighborhood of the zero section of the bundle Np Is Ell Span(e3) -+ 8. Because 11'2(80(3)) = 0, there are no obstructions to homotoping the given normal frame 2 to be distinct. Now, for each a ~ 2, mimick the surgery in (8.23) to make ~ ambient connect sum of Bo,a with BO,l' Since the {Pa} are distinct, these connect sums can be made with out interfering with each other. Use B to denote the result, after smoothing near the corners. The verification that B does the job is left to the reader as an exercise. (For the framing issue, see Lemma 8.10.) 0
10 The last pass at E±. It is the purpose of this section to explain how to make E± from E4± of the preceding section. The metamorphasis from E 4 ± to E± will be called melding. This melding operation only changes E4± in a neighborhood of tl. z U az, and the neighborhood in question can be as small as desired. In particular, in this neighborhood, E4± should be the image of E 4 - under the switch map e : Z -t Z which sends (x, y) to (y, x). (See AsseJ,"tion 1 of Proposition 9.1). Furthermore, in this neighborhood, E 4- (hence, E4+) should consist, locally, of N parallel push-off copies (see Assertions 5 and 6 of Proposition 9.1). The effect of the melding will be to push all of these parallel copies together on some smaller neighborhood of tl.z U az. The cost of the melding is that E± will not be a manifold (unless N = 1 in Proposition 9.1).
396
CLIFFORD HENRY TAUBES
a) E 4 - near Llz U 8Z.
Consider E 4 _. The following construction will be done r times, once for each pair in {(ai, bini=l' These r versions can be done simultaneously, so fix attention on one index i, and simplify notation by setting a == ai and b == bi. The set r i C E 4 - n Llz is a set of N embedded .intervals which connect the N components of 8 3 x Aa = E4- n (83 x 8 3 )a with the N components of 8 3 x Ab = E 4 - n (8 3 x 8 3 h. Near (10.1) E 4 - consists of N components (sheets), {Ya }:;=l' Here is a picture:
(83
X
8 3 )0
/
----'-----,.....
-\
I
.......--,---~'
(10.2)
The sheets {YQ}a~2 are push-off copies of Y l • As for Y l , it is an embedded image in Z of the compliment in the open, unit 4-ball of the interiors of a pair of disjoint 4-balls, B±, ofradius 1/8, respectively centered at (±1/4, 0, 0, 0). Note that the boundary of B_ is mapped to 8 3 X Pa C 8 3 X Aa, and the boundary of B+ is mapped to the corresponding 8 3 x Pb C 8 3 X A b • Furthermore, Yi n Llz is the segment of the Xl axis between (±1/8, 0, 0, 0) E 8B±. Here is a picture ofYI :
0---0 (10.3)
397
HOMOLOGY COBORDISM
As remarked, the {Ya-Jo>2 in (10.2) are push-off copies of Yi. To be precise here, remember that YI' hM a framing, , == (et, e2, e3), to its normal bundle, NYl, which is a product framing that restricts to both (83 X Pal and (83 x Pb) as a constant framing. (See Assertion 8 of Propositio 9.1.) Fix an exponential map,
(10.4)
which maps NYl's restriction to (83 X Pal into (83 x 8 3 )a, and which likewise maps Ny1's restriction to (8 3 X Pb) into (83 x 8 3 h. Fix a metric on NY1 which makes the frame , orthonormal, and fix € > 0 such that (10.4) embeds the interior of the radius 2€ ball bundle onto a neighborhood of YI in Z. Use e in (10.4) to identify the interior of this ball bundle with its image in Z. With the preceding understood, the copy Yo of YI can be taken as the image of the section So : YI -+ NY1 that is given by
(10.5)
b) The meld. With the preceding picture E 4 - near (10.1) understood, here is the meld: Fix a function (3 : [0,1] -+ [0,1] which has the following properties:
(10.6) 1)
(3 (3
== 1 on [5/8,1]. == 0 on [0,1/2].
2) 3) {3 is nondecreasing. As described in (10.3), identify Y I with a subset of the unit ball about the origin in ]R4. Use x to denote the Euclidean coordinate in and, by restriction, a point in Yi. By the way, note that the assignment to x E Yi of the number {3(1 x D defines a smooth function on Y I which vanishes in a neighborhood of
r,
(10.7)
B_ U {(Xl, 0, 0, 0) : -1/8:$
Xl
:$ 1/8} U B+.
For a ~ 2, define the deformation, Y~, of Yo as follows: Y~ is the image in Ny of the section s~ which sends X E Yi to (10.8)
s~(X)
== (a -1) N- I (3(1
X I) Eel'
Notice that y~ agrees with Yo on the compliment of a regular neighborhood of (10.1) in Z; hut that Y.! coincides with Yt on a smaller neighhorhood of
398
CLIFFORD HENRY TAUBES
(10.1). Here is a picture for a: ~ 1:
(10.9) In (10.9), the shaded region marks where Y~ and Y1 intersect. Here is a picture of all the {Y~}Q~l:
(10.10) Use E_ to denote the result of applying the preceding meld operation to E 4 in a neighborhood of (10.1) for each i E {I", . ,r}. As for E+, remember that EH coincided with 9(E4-) near each of the r versions of (10.2). This neighborhood can be assumed to include the regions that are depicted in (10.2). With this understood, set E+ == EH outside of the 9-image of the regions in (10.2), but inside the 9-image of each region in (10.2), declare
(10.11) c) Properties of E±. The following proposition describes some of the salient features of E±:
10.1. Construct E± c Z as described above. Then: There is an open neighborhood U C Z of D..z u az such that E+ n U and E_ n U are images of each other under the switch map on Z. 2} The fundamental classes [E±l are equal to N [E1±l for some integer N ~ 1. Here, [El±l are described by {6.8} and Lemma 6.8. 3} E± have empty intersection with Mo x Mo and Ml x M 1 , .I} E± have empty intersection with EL,R of (4. 15}. PROPOSITION
1}
HOMOLOGY COBORDISM
399
5) If p E crit(f),jhen the intersection of E_ with (S3 x S3)p has the form S3 x x P ' where xp is a single point. Similarly, the intersection of E+ with (S3 x S3)p is xp X S3. 6)
E± n t:::.z
=
Ui=l Vi, where Vi C t:::.z is as follows: There is a flow line I'i which starts at ai and ends at bi . With the canonical identification of t:::.w with W understood, Vi is a closed interval in a push-off copy of I'i. And, Vi starts at (XCl,X CI ) E (S3 x S3)CI and Vi ends at (Xb,Xb) E (S3 x S3h. Here, a ai and b bi'
=
7) 8)
E_ n E+ = Ui=l Vi. H2(E±i Q) = O.
=
The proof is straightforward and left to the reader. (See Proposition 9.1. Also, use Meyer-Vietoris to compute H2(E±).) Here is a picture:
(10.12)
11 Completing the proof. The purpose of this last section is to complete the proof of Theorem 2.9. The strategy here will be as follows: Suppose that Mo and MI are cobordant via a spin 4-manifold, W, with the rational homology of S3. Factor the cobordism as in Assertion 5 of Proposition 3.2 into two pieces, WI n W 3 • Both WI and W3 are given by (3.11). Here, W l is a cobordism from Mo to a rational homology sphere M, while W3 is 'lo cobordism from M to MI. In both cases, the manifold with boundary, Z. (= ZI,3), has been defined, and Sections 4d, 4e and 10 describe the variety I:z. C Z. (The latter require a choice of base pointp eM.) Let Z ZlUZ3 and I:z I:Zl UI: zs , where the common boundary components in both cases are identified (these being M x M in the former and I:M in the latter).
=
=
400
CLIFFORD HENRY TAUBES
With Z and ~z understood, the proof plan from Section 2k will be complete with the completions of Steps 3 and 4 in Section 2k. These steps are consider~d below. Completing Step 3 requires the construction of a 2-form Wz on Z - ~z which satisfies (2.27). This 2-form will be constructed first on the compliment of ~z in a regular neighborhood Nz C Z of ~z. The extension to Z - ~z will be made by appeal to Lemma 4.2. Step 4 of Theorem 2.9's proof (from Section 2k) will be completed during the construction of wz. Let NZl == NZnZl and define Nzs analogously. The 2-form Wz on Nz - ~z will be constructed first on NZl - EZl and second on Nzs - ~zs' The case of NZl -EZl is considered in Subsections 11a-h and that of Nzs -~3 is considered in Subsection 11i. These two constructions are matched in Subsection 11j where Step 4 of Section 2k is verified. Section 11k completes the proof of Theorem 2.9 with a discussion of the conditions in Lemma 4.2. To avoid cumbersome notation, the subscript ''1'' will be dropped in Subsections a-h. Thus, in these sections, Z will denote Zl, Ez will denote EZll etc. a) Preliminary remarks.
Construction of Wz on Nz - ~z is accomplished in two steps. The first step defines Wz near t::.z U EL U ER by using (4.22), but where cpz is a map which is defined only on a regular neighborhood, N', in Z of t::.z U EL U E R . This cpz has cpzl(D) = ~z n N'. The second step constructs Wz on Nz - N'. The construction of cpz occupies Subsections b - I, below. The construction of Wz on Nz - N' occupies Subsections 9 and h.
b) Near EL n ER. The purpose of this subsection is to construct cpz near EL n ER. To be precise, a neighborhood U C Z of EL n ER will be described with a map (11.1)
cpu: U -t IR3
which obeys (11.2)
Then, cpz I U will be declared equal to cpu. A digression on framings begins the construction of CPu. To start the digression, fix a frame for T M Ip. This frame can be thought of as a frame for the normal bundle to p in T M. Use the pseudo- gradient flow to extend this frame as a normal framing to the flow line 'Y C Z. This normal framing to 'Y induces a framing of T Mo Ipo. End the digression. Parameterize that flow line 'Y so that IC'Y(t)) = t. Let N'"( -t 'Y denote the normal bundle to 'Y in W and select an exponential map e'"( : N'"( -t W which maps N'"( Ipoop into Mo,M, respectively. Require that
HOMOLOGY COBORDISM
f
(11.3)
0
e'Y(t)
401
= t.
Use this exponential map and the normal framing with the afore- mentioned parameterization of'Y to define a diffeomorphism, 1/J-y, from a neighborhood, 01' C W of'Y onto [0,1] x B, where B C 1R3 is a ball-neighborhood of the origin. Using the preceding identification, build 1/Ju (u'Y X u'Y) Iz of the neighborhood U Z n (o'y x 01') of EL n ER with [0,1] x B x B. This U and 1/Ju obey the conclusions of Assertion 4 in Lemma 4.5 except that B C 1R3 should everywhere replace 1R3 and B x B should everywhere replace 1R3 x 1R3 . With the preceding understood, define cpz on U to be the composition of the map 1/Ju with the map from [0,1] x 1R3 x 1R3 which sends (t,x,y) to ~o(x,y) with ~o given by (2.15). Note that cpz lu agrees with Proposition 2.5's map cP when restricted to a neighborhood of Po x Po in Mo x Mo, or to a neighborhood of p x pin M x M. Note also, for reference below, that the map cpz lu is invariant under the switch map e : Z ~ Z which sends (x,y) to (y,x).
=
=
c) Near EL U ER'
The purpose of this subsection is to construct cpz near EL U E R . To begin, remark that the normal bundle N R ~ ER to ER in Z is naturally isomorphic to rrlN'Y. Thus, said normal bundle has a natural framing. Take the dual to this natural framing to frame the dual bundle, NR and choose an exponential map eR : NR ~ Z. The framing of NR and eR together define a map, CPR, from a neighborhood of ER in Z into 1R3 which has ER as the inverse image of zero. (See (2.14).) Choose this exponential map so that it sends NR IMoXMo into Mo x Mo and likewise sends NR IMxM into M. (The exponential map e'Y : N'Y ~ W of the preceding subsection induces such an exponential map in a natural way.) On ER n U, the differentials of the maps CPR and cpu are scalar multiples of each other, and so there is a homotopy of CPR near U which has it agree with cpz lu on U and so extend cpz lu to map a neighborhood of ER in Z to 1R3 with the correct inverse image of zero. See the Step 2 of the proof of Proposition 2.5 for the details. With cpz lu now extended over E R , extend it further over ER U EL by using the switch map e : Z ~ Z. Use cpz IRL to denote this extended map. Note that cpz IR,L can be made so that its restriction to a neighborhood of CPo xMo)U(Mo xPo) in Mo xMo agrees with the map cP for Mo in Proposition 2.5. Likewise, its restriction to a neighborhood of (p x M) U (M x p) in M x M can be arranged to agree with the analogous cP for M d) Near E± n
~z.
The intersections between E± and between these varieties and ~z form a set of disjoint line segments, {Vi}i=I. (Note that E± are manifolds near these line segments.) The purpose of this subsection is to define the map cP z near each Vi·
CLIFFORD HENRY TAUBES
402
To start, fix i E {I,··· ,r}. Let a == ai and b == bi • Note that Vi has end points (xa, xa) C (S3 X S3)a and (Xb, Xb) C (S3 x S 3 h. Also, the identification, using 7rL or 7rR, of t:J..z with a subset of W identifies Vi with a subinterval in a pseudo-gradient flow line which starts at a and ends at b. Remember that E_ is the result of melding E 4 - of Proposition 9.1. This means, in particular, that (S3 X S3)a U Vi U (S3 x S 3 h has a neighborhood, Ui C Z, such that E_nUi is the same point set as a component, Y, of E 4 _nt:J.. Z . Meanwhile, E+ n Ui = 9(Y). Remark next that E4- has, according to Assertion 8 of Proposition 9.1, a special normal framing, (. And, EH has a special normal frame, (', which restricts to Ui as the image of ( under the switch map a. The pair of frames «(, (') restrict to Vi to frame the normal bundle Ni ~ Vi of Vi in Z. Notice that e fixes t:J..z and the differential of a (denoted a*) acts on Ni and interchanges Span«() with Span('). Fix an exponential map e : Ni ~ Vi with the following properties:
(11.4) 1) . e :Span«() ~ E_. 2) e :Span«(') ~ E+. 3) At (Xa,xa),e maps Ni into (S3 X S3)a. 4) At (Xb,xb),e maps Ni into (S3 x S 3 h. 5) a 0 e = eo a·. Together, the map e and the frames «(, (') define a map (11.5) with the following properties:
(11.6) 1) 2) 3) 4) 5) 6) 7) 8)
°
There is an open ball B C IR3 about and 1/J embedds 1/J is the identity on Vi x (0,0). 1/J-l(E_) = {(t,x, 0) E Vi x IR3 X IR3}. 1/J-l(E+) = {(t,O,y) E Vi x IR3 X IR3}. 1/J-l(t:J..Z) = {(t,x,x) E Vi x IR3 X IR3}. 1/J«x a,xa) x IR3 x IR3) C (S3 X S3)a. 1/J«Xb,Xb) x IR3 x IR3) C (S3 x S 3 h· 1/J(t,x,y) = 8(1/J(t,y,x».
Vi
x B x B.
Given the preceding, define the map cpz on a neighborhood of declaring that (11.7)
(cpz
0
1/;)(t, x, y)
==
~o(x, y),
Vi
in Z by
HOMOLOGY COBORDISM
where
~o
403
is given in (2.15).
e) Near (S3 x S3)p. The next step is to define the map cpz near Z's boundary components {( S3 X S3)p he crit(J). So, fix i E {I,···, r} and let p denote either ai or bi. A neighborhood, Vp C Z, of (83 x S3)p is diffeomorphic to the product (S3 x S3)p X [0,1) as a manifold with boundary. Furthermore, there is no difficulty in finding such a diffeomorphism so that (11.8)
1) 2) 3)
4)
E_ n Vp = (S3 X xp) x [0,1). E+ n Vp = (xp X S3) X [0,1). t::.z n Vp = t::.ss x [0,1). The switch map acts by Sex, y, t) = (y, x, t).
In Vp, the variety E_ is smooth and it agrees with a component, Y, of E 4- n Vp. Also, E+ n Vp = 8(E_ n Vp). Also, E4- has the normal frame, (, which restricts to Y as a constant frame. And, E4+ has the normal frame (' which restricts to 8(Y) as 8 * (. Use these constant frames to define frames for the dual bundles to the normal bundles of Y and 8(Y) in Z. Then, use these frames for the conormal bundles to extend cpz I Ui of (11.7) to a neighborhood of E± n Vp by mimicking Step 2 in the proof of Proposition 2.5. (Exploit the product structure on Vp in (11.8).) Meanwhile, T*S3 has its singular framing which gives (see Proposition 2.7, Definition 2.8 and Lemma 2.11) the canonical homotopy class of singular framing for which the value of 12 (S3) is zero. As in Step 3 of Proposition 2.5's proof, use this framing to obtain a singular framing of the normal bundle to t::.S3 x [0,1) in Z. (Remember that (S3 x 8 3)p has two obvious projections to 8 3 , these are given by the product structure in (3.26) and are denoted 11"::1::. To be explicit, introduce the coordinate sytem 'l/J p of (3.2) and introduce Up == 'l/Jp(JR4). Note that Up x Up is a neighborhood of (P,p) in W x W. With this understood, 11"± are the restrictions to (S3 x S3)p of the maps from Up x Up to JR4 which are given by
(11.9) when p E critl (f)j and by (11.10)
1I"_(x,y)
== (Yl',Y2,X3,:C4) and 1I"+(x,y) == (Xl,X2,Y3,Y4)
when p E crit 2(J). Then, the map 11"* *'d'fi T*S3'. h h al bundle to Ass in (S3 x 83)".) + - 11"_ 1 entl es Wlt t e norm
404
CLIFFORD HENRY TAUBES
As in Step 3 of the proof of Proposition 2.5, use the induced framing of the normal bundle to ~~ x [0,1) in Z and the product structure of Vp as described in (11.8) to extend j and 8i,i > O.
Note: If Hl(Mo;Z) = 0, then the basis {u;} == {T;} is allowed. Let / : M ~ [0,1] be the good Morse function witp only index 2 critical points. One can arrange for such an f to have one critical level, /-1(1/2). A choice of pseudo-gradient for the function f defines the descending 2-disks, {B,,- : p E crit(f)} , from the critical points of /. Each B,,- is an embedded 2-disks in /-1 ([0,1/2]) to which / restricts with a single maximum, p. Orient these disks and they give a basis for H 2 (W, Mo; Z). Using Milnor's basis theorem (Theorem 7.6 in [6]), one can find: 1) (2.3)
2) 3)
A good Morse function / on W with critical value 1/2 and only index 2 critical points. A labeling{b1, "0, br } of crit(f). A pseudo-gradient, v, for /.
And, these are such that the given basis {uiH=1 for H 2 (W, Mo; Z) is given by (2.4)
Here, [B,,-] E H 2 (W, Mo; Z) is the fundamental class for an appropriate choice of orientation for B,,-. b) Factoring the cobordism.
It proves convenient to factor the cobordism W into a linear chain of simpler cobordisms. The following proposition describes the process: PROPOSITION 2.1. Let M o,1 be a pair of compact, oriented 3-manifolds, each with the rational homology of 8 3 • Let W be a compact, connected, oriented 4 dimensional cobordism between MQ and MI. Assume that the intersection
421
METABOLIC COBORDISMS
form for W obeys (1.3) and assume that W has a good Morse function with only indea; 2 critical points. Then W can be decomposed as
(2.5) where Wj C W is a compact 4-dimensional submanifold with two boundary components, Fj and Fj+1' which are embedded, 3-dimensional sub manifolds of W. These have the following properties: 1) For each j, Fj has the rational homology of 8 3 • 2) Fn+1 Mo and F1 M 1. 3) For each j, Wj n Wj-1 Fj . 4) For each j,H2 (Wj ;Z) ~ EB2Z and the intersection matria; is conjugate by GL(2; Z) to H(mj) for some mj E Z - to}. 5) For each j, Wj has a good Morse function which has only two critical points, both with indea; 2.
=
=
=
The remainder of this subsection is occupied with the proof of this proposition. Proof. The first step is the construction of the W j , and the second step verifies their properties. Step 1: Because of (1.3), the number r of critical points of f must be even. Given this point, fix small € > 0 and modify f slightly so that for j E {I, ... , r/2}, the critical points (b2j , b2j-d have critical value 1/2 - €. (j /r). Thus, (b r , br-d have the smallest critical value, while (b 2 , bt) have the largest critical value. Set F r / 2+1 M o, and for j E {2, ... , r /2}, let
=
Fj
(2.6)
=f-1(1/2 -
€.
(j - 1/2)/r).
Note that each Fj is a smooth, oriented submanifold which splits W into two pieces. For 2 ~ j ~ r /2 f let V; C W denote the closure of the component of W - Fj which contains Mo. Set W r / 2 Vr / 2 and for 1 < j < r/2, set
=
(2.7) For j = 1, define W1
Wj
=W -
=V; -
int(V;
+ 1).
int(V2) and define F1
= M 1.
Step 2: Consider now the properties of the {Wj} and {Fj }: First of all, Assertions 2 and 3 of Proposition 2.1 follow by construction. As for Assertion 5, note that Wj is a submanifold with boundary in W which contains no critical points of f on its boundary, and which contains only the critical points b2 j, b2 j-1 of f in its interior. Thus, a rescaling of f on Wj will yield a good Morse funhion on Wj to verify Assertion 5 of Proposition 2.1. The proofs of Assertions 1 and 4 of Proposition 2.1 require a digression to construct representative cycles for the generators {Til of H 2 (W; Z). The cycle for a given Ti will be the fundamental class of a submanifoldTi C W. To start the digression, remember that H 2 (Wj Z) is assumed to have a basis {Tj}j=l in which the intersection form is given by (1.3). And, remember that
422
CLIFFORD HENRY TAUBES
the image of Ti in H 2 (W, Mo; Z) is given by EbE crit(f)Si,j(b) . Uj(b) , where Uj(b)s shorthand for [B b_] with b bj • (This introduces the indexing function j(.) : crit(J) -+ {1, ... ,T} which is defined so that j(b) j when b bj.) Finally, remember that the Morse function, and its pseudo-gradient have been assumed chosen so that the matrix (Si,j) obeys (2.2). The submanifold representative T i , for Ti can be recovered from (Si,j) and {Bb- : b E crit(f)} by the following construction: Let M1 / 4 ,-1(1/4). Note that M1 / 4 is diffeomorphic to Mo. Note as well that CbB b_ n M 1 /4 is an embedded circle which is naturally oriented given that B b- is oriented. Thus, Cb- determines a homology class, [Cb-] E HI (M1 / 4 ; Z). Meyer-Vietoris (Eq. (2.1)) implies that
=
=
=
==
(2.8) Construct push-offs of each Cb- by taking a push-off copy of the corresponding Bb- and intersecting with M1 / 4 • Let!!:.i c: M1/ 4 denote the oriented 1dimensional submanifold which is the union, indexed by b E crit(f), of ISi,j(b) I push-off copies of Cb-, oriented correctly when Si,j(b) > 0 and oriented incorrectly otherwise. According to (2.8), this !!:'t bounds an oriented surface with boundary, Ri C M 1 / 4 , which is such that int(~) n!!:.i = 0. With ~ understood, represent Ti by the fundamental class of a subvariety TI which is defined to be the union of ~ with the union, indexed by b E crit(f), of ISi,j(b) I push-off copies of Bb-, oriented correctly if Si,;(b) > 0 and oriented incorrectly otherwise. Smooth the corners of TI near -, J.t. to obtain an embedded surface, Ti C W. This Ti is naturally oriented and its fundamental class representJ5 the class Ti. End the digression. To return to the proof of Proposition 2.1, and, in particular, the proof of Assertion 1. By construction, H 2 (V;, Mo; Z) is generated by {[Bb-] : j(b) ~ 2j -I} and thus is a free group. Since Mo is a rational homology sphere, H 2 (Mo; Z) = 0 and therefore (1.3) (with V; replacing W) asserts that H 2 (V;; Z) is also free; by construction, its generators are UTi] : i ~ 2j - I}. The intersection form of V; is the restriction of the form for W to {[Ti] : i ~ 2j - I}. This is a sum as in (1.3) and is non-degenerate over Q. The non-degeneracy of the intersection form of V; over Q implies that F j is a rational homology sphere. To prove Assertion 4, note that H2 (Wj , Fj+l; Z) is freely generated by {[Bb_n Wj] : j(b) = 2j-1 or 2j}. Since Fj+1 is a rational homology sphere, H 2(Fj +l; Z) = 0 and so the (Wj, Fj+l) analog of (1.3) implies that H 2(Wj; Z) is free ofrank
2. Furthermore, the intersection form on H 2 (Wj; Z) must be non-degenerate because the boundary of Wj has no rational homology. In fact, the inclusion of Wj into V; induces an injection H 2(Wj ; Z) -+ H 2(V;; Z) with image the generators [T2j-l] and [T2j ]. This implies the statement in Assertion 5 concerning the intersection form on H2 (Wj; Z). Here is why H2(Wjj Z) injects into H 2(V;j Z): One must prove that the submanifolds {T2j, T2j-t} are homologous to submanifolds which lie in Wj.
METABOLIC COBORDISMS
423
This happens if T2 j and T2 j-l have zero intersection number with all B b- for b = bi and i > 2j. Indeed, if T == T 2 j, T 2j - l has zero intersection number as described, then the intersection points of T with each such Bb- can be paired as ± pairs. (One point with positive intersection number, and one with negative.) Then, surgery on these embedded So's in T will yield a new surface, T', (with larger genus) which is homologous to T and which has no intersection with Bbwhen b == bi and i > j. (Mimic the tubing construction in Section 5d of [7].) The pseudo-gradient flow can then be used to isotope this T' into Wj. With the preceding understood, the lemma follows with the realization that the intersection number of T, as above, with Bb-, as above, is a linear functional of the entries of the matrix (Si,j)i>j. And, this is, by assumption, the zero matrix. 0
c) Z and W x W. This subsection describes Z C W x W, a submanifold with boundary. For the most part, the discussion here mirrors the discussion in Section 3g, h of [7] where an analogous Z is defined. The stage is set with the following Definition: DEFINITION 2.2. Let M o, Ml be compact, oriented 3-manifolds with the rational homology of 8 3 •
. A simple type cobordism: A cobordism W between M o and Ml is of simple type if the following criteria are met: 1) W is oriented and connected. 2) W has a good Morse function with only two critical points, both of index 2. 3) H 2 (W; Z) ~ Z2, and the intersection form of W is conjugate over GL(2; Z) to H(m) for some integer m "I- O• . A simple type Morse function: Let W be a cobordism of simple type. Let ~ [0,1] be a function and let v be a pseudo-gradient for I. Then (I, v) are of simple type if the following criteria are met: 1) 1_1(0) = M o and 1-1(1) = MI. 2) dl "I- 0 near 8W. 3) I has only two critical points, (bl'~)' both with index 2. 4) 15/16 < l(b 2 ) < 1/2 < I(bt} < 17/16. 5) There are integers ml > 0, m2 > 0 and ml,2; and there are orientations of the descending disks from b1 and ~ such that
I: W
(2.9)
0"1
== ml . [Bbl-J + ml,2 . [Bb2-J and
0"2
== m2 • [Bb2-J
generate the image in H2(W, Mo; Z) of H2 (W; Z). 6) The pseudo-gradient v is good in the sense of Definition 3.1 in [7J. With the stage set, assume below that W is a cobordism of simple type, and that (I,v) are a pair of Morse function and pseudo-gradient on W which are also of simple type.
424
CLIFFORD HENRY TAUBES
As in Section 3g of [7], introduce
Z
(2.10)
=((x,y) E W x W: F(x,y) =fey) - f(x) = O}.
Define Z C Z by intersecting the latter with the compliment in W x W of (open) small radius balls about (bl' bl ) and (b 2 , b2 ). That is, mimick the constructions in Sections 3i and 3h of [7]. Some properties of Z are listed below: A manifold: Z is a manifold with boundary,
(2.11)
az = (Mo x Mo) U (Ml
X
Mt) U (S3
X
S 3ht U (S3
X
S3)b2 j
here (S3 x S 3h is the link around Z's singularity at (b, b). (See Section 3h of
[7].) Orientation: The manifold int(Z) is naturally oriented using the orientation from W x W along with dF to trivialize the normal bundle to int(Z) in W x W. Orient the various components of (2.11) as described in Section 3h of [7]. Homology: The rational homology of Z is described by
2.3. Let W be as described above. Then the following hold: Ho(Z) ~ R. Hl(Z) ~ o. The inclusion Z C W x W induces H 2 (Z) ~ H 2 (W x W) ~]R4. There is a surjection
LEMMA
1) 2) 9) 4)
(2.12) Here, L± ~ ]R are freely genemted by embedded 9-spheres in Z as described in Equations (3.92) and (3.33) of {7J. Proof. Mimic the proof of Lemma 3.7 in [7]. D 3 Constructing Tl and T 2 • The constructions in [7] aside, the proof of Theorem 1.3 is mostly occupied with constructions on W 2 x W 2 , where W 2 is described in (1.4). The previous subsection introduced a factorization of such a W2 as a sequence of cobordisms of simple type, each with a Morse function f and pseudo-gradient v of simple type. (See Proposition 2.1 and Definition 2.2.) The required constructions for W2 in (1.4) can be reduced to a series of identical constructions, one on each simple type cobordism factor in (2.5). With the preceding as motivation, this section will restrict attention to a cobordism W of simple type with a Morse function f and pseudo-gradient v which are of simple type also. The purpose of this section is to describe a very useful pair of 2-dimensional sub manifolds of W, Tl and T 2 , whose fundamental classes generate H 2 (W) and give the intersection form H(m). Thus, this section serves as a second digression before the construction of E±.
METABOLIC COBORDISMS
425
a) Reconstructing T2 • The submanifold T2 is obtained by smoothing the corners of a CO embedding of a smooth surface into W. This embedding can be obtained as follows: Step 1: Let V C W denote the set {x E W : I(x) ~ 1/4}. To construct T 2 , first introduce the number m2 from (2.9) and take m2 push-off copies of Bb2 _ n V, all with the same orientation. Make these push-offs so that f restricts to each copy with only one critical point, a maximum. And require that said maximum be close to b2 in the following sense: Use the Morse coordinates of (3.2) in [7] and the Euclidean metric on lR" to measure distance. With this understood, the distance from each such minimum to b2 should be much less than the number r which is used in (3.29) of [7] to define the boundary of Z. To be precise, work in the Morse coordinates of (3.2) in [7] near b2 • Choose m2 distinct unit vectors {ncr} in the (X3, X4) plane. Then, choose f > 0 but with f «r. Define the o:'th push-off of Bb2- to be the set (3.1)
Step 2: Use B~2_ to denote the resulting m2 push-offs of Bb2; this is an oriented, submanifold with boundary in V. It is important to realize that 8B~2_ C M 1/ 4 == 1- 1(1/4) is a disjoint union of m2 embedded, oriented circles. These circles bound an oriented, embedded surface with boundary R2 C M 1 / 4 which intersects 8B~2_ as 8R2. Take such an R2 which is connected and which has no compact components. Set (3.2) This is a (tame) CO-embedding of a smooth surface; the embedding is smooth save for the corners along 8R2 • However, these corners are right angle. corners in a suitable coordinate system and can be smoothed without difficulty. The resulting smooth submanifold of W is T 2 • Step 3: The push-offs B~2- can be constructed so that T~ has the following properties: 1) No pseudo-gradient flow line intersects T~ more than once. 2) No pseudo-gradient flow line is anywhere tangent to B~2-' 2) T~ has empty intersection with Bb2-' (3.3) 3) The restriction of 1 to B~2- has only index 2 critical points, and precisely one on each component. 4) Each component of B~2- intersects B~+ transverally in exactly one point. To satisfy (3.3), first note that the explicit description in (3.2) for B~2- obeys (3.3). (This is because the vectors {ncr} in (3.1) are assumed to be distinct.)
426
CLIFFORD HENRY TAUBES
Second, note that B" _ can be made so that: (3.3) holds, 8B"2_ lies on the boundary of an embedded solid torus N C M 1/4, and Past(B~l-) n M1/4 lies in the interior of N. Note that the core circle of N is Bb2- n M1/4' (Recall from Section 5a in [7] the definition of the past and future of a set U (written past(U) and fut(U), respectively). For example, past(U) C W is the set of points which can be obtained from U by traveling along pseudo-gradient flow lines to decrease I.) The Morse coordinates in (3.1) extend over a neighborhood of Bb2- in W, and with this understood, the tubular neighborhood N is described by
N ~
(3.4)
(X1,X2, X3,X4) :
x~
+ x~
::; E
and x~
+ x~ = x~ + x~ + c,
Here c > 0 is an appropriate constant. Equation (3.3) follows by showing that 8B~2_ bounds an embedded surface with boundary in the compliment ofint(N). And, such a surface exists because the class T2 E H2 (W; Z) has zero self intersection number. With the coordinates of (3.1) and (3.4) understood, the submanifold R2 can be assumed to intersect a neighborhood of N as the set of (Xl, X2, X3, X4) which obey:
1) 2)
(3.5)
(X3, X4)
x~
= t . no
3/8. (3.11) 2) The restriction of f to the image of B~l- has only index 2 critical points, and there is precisely one on each component. The embedded image of C u B~l _ gives a piecewise smooth embedding in W of a union of disks. Indeed, the embedded image of C U B~l _ has a corner where the images of C and of B~l- intersect, that is, along f-l(ht}. Choose
429
METABOLIC COBORDISMS
in advance a neighborhood of this corner, and the image of C U B~l_ can be smoothed in the chosen neighborhood so that the result, 1 - , has the following properties:
Br
1)
2) 3)
4) 5) (3.12)
6) 7)
B~' _ = B~l_ where f < h2 • B~~ _ agrees with the image of C where f 2: hl.
The restriction of f to B~'l- has only index 2 critical points where f > 3/8; and there is precisely one on each component. Each component of B~'l- is either a push-off copy of Bbl -, or else one of Bb2 -' A component of B b1 - which is a copy of Bbl- intersects Bbl + transversely in a single point, Such a component also intersects fut(T2 ) in a finite set of halfopen arcs with their boundaries on M 3 / 8 • The closures of each half-open arc is an embedded arc whose other end-point is the intersection point with Bbl+' Furthermore, f restricts to each half-open arc with no critical points where f < 3/8. A component of Bbl _ which is a copy of Bb2- has empty intersection with fut(T2 ). No pseudo-gradient flow line is anywhere tangent to
B"bl-' Let (3.13)
Rr ==
R~
- C and define
T1il -= R"1 U B"bl-'
This submanifold obeys (3.10). (Where f 2: hl,T{' is obtained from R~ by flowing the latter along pseudo-gradient flow lines.) d) Intersection of fut(T{') with T2 and T{' with fut(T2 ). The intersection between fut(T{') n T2 is the union of a finite set of half-open arcs each of which has its endpoint at one of the points of T{' n T 2 , and viceversa. (Note that T{' n T2 = R~ n T 2 .) The closure of each half open arc is an embedded arc with its other end point where B~2- intersects B b2 +. There are at least m such arcs. The intersection ofT:' with fut(T2 ) is more complicated. After perturbing T{' slightly, this intersection can be assumed to have the following form: It consists of a finite, disjoint set of closed arcs, half-open arcs, and open arcs in T{'; and disjoint from these arcs, there is a finite set of disjoint, embedded circles. Each point of T{' n T2(= R~ n B:'2-) will be a boundary component for some arc, either half open or closed. (But, there may be more or less arcs than boundary components of arcs.) The closure of a half-open arc will be a smooth arc whose other endpoint lies on B:': _ n Bbl + (and thus in a push-off copy of Bb. _ in B:". _). The closure
430
CLIFFORD HENRY TAUBES
of an open arc will also be a smooth arc, but with both of its endpoints in B~l_nBbl+. To see that such is the case, introduce R2 == fut(R2)n/- 1 (7/16). This will intersect TI' in the m2 push-off copies of Bbl-. (In fact, its intersection number with the union of said m2 copies is equal to m. See (5) and (6) of (3.12).) Each intersection point of & with TI' has one half-open arc component or one open arc component of T{'n fut(T2) passing through it. FUrthermore, each half-open arc component intersects precisely one point of TI' n &, while each open arc component intersects precisely two such points. Each half-open arc component intersects /-1([7/16, 1]) as a push-off copy of a pseudo-gradient flow line for / in Bbl- which ends in ~j and each open arc component intersects /-1 ([7/16, 1]) in a pair of such push-offs. The circles in T{'n fut(T2) can be assumed to lie in the interior of Rr. (See (5) and (6) of (3.12).) It is important to note that there are at least m half-open arcs components of TI/n fut(T2)j any less would be incompatible with the assumed value of m for TI' . T2. H a pair of points in TI' n R2 are points on the same open arc, then these points will have opposite local intersection numbers for T{' n T 2 • A similar argument shows that for at least m of these arcs, both the intersection point in B~/l _ n R2 and the endpoint in Rr n B~2 _ are points of positive local intersection number for T{' n R2 and for Rr n T2, respectively. With the preceding understood, fix one half-open are, (3.14) which intersects B~/l _ n R2 at a point of positive local intersection number, and which ends in Rr n B~2- at a point with positive local intersection number. e) HOInology of TI and T2 and the linking matrix.
There is one additional constraint that must be imposed on TI' j and this one also requires advanced knowledge of T2. Suppose that TI' and T2 have already been constructed. The surface T2 has some genus 92 ~ O. As such, its first homology has a basis which is represented by the fundamental class of a set, {112fj}~~1 C int(R2 ), of 2·92 embedded, oriented circles. Take nl (from above) like oriented, push-off (in R 2 ) copies of each "72a • Together, these form a set {p~J, where (3 runs from 1 to 292, and where i runs from 1 to nl. The pseudo-gradient flow pushes R2 isotopically into M 3 / 8 as the submanifold fut(R2) n M 3 / 8 , and thus the circles {p~J are pushed isotopically into M 3 / 8 as a set, {Pfj;} C M 3 / 8 , of 2 . nl • 92 circles. Fix the set of circles {Pfji} once and for all. These circles will be used to constrain Rr j but a short digression is needed to define these new constraints. Start the digression by observing that the surface T{' has some genus 91 ~ 0 and so its first homology is represented by the fundamental class of a set of 2·91 embedded, oriented circles, {111a}!~1 C int(Rn C M 3 / S • These generators should be chosen to be disjoint from the arc vO which is described in (3.14). (This is possible because VO is an arc with one endpoint on aR~ and the other in the interior of R~'.)
METABOLIC COBORDISMS
431
The manifold M 3 / s , being diffeomorphic to Mo, has the rational homology of Sa. This means, in particular, that some number nl ~ 1 of like oriented pushoff copies (in of each "110 bounds an embedded surface with boundary, Sa C M 3 / s. No generality is lost by assuming that R~ intersect each of the circles {pP.} transversally. Likewise, there is no generality lost here by requiring that the {"Ila} which generate H 1 (Tt) be disjoint from the set {ppJ. Push-off, in R~, the nl copies of each "I1a. Make these close to "Ila to insure that the push-off isotopy is disjoint from {pP.}. Find the submanifold with boundary Sa C M 3 / s which intersects the nl push-off copies of "Ila as its boundary. In general position, each such Sa will intersect each of the circles PP. transversally. So, there is a 291 x 292 matrix A == (Aa,p) where Aa,p is the sum of the intersection numbers between the sudace Sa and the nl circles {pP.} ~';1. (Here, the index {3 is fixed.) The matrix A will be called the linking matrix between the set {"Ila} and the set {pp.}. Note that the entries Aa,p are divisible by the integer nl, and that the definition of Aa,p requires the apriori choice of push-offs {p~J of {"I2P}· With the preceding understood, the point of this subsection is to remark that there is an isotopy of R~ in Ma/ s (reI aR~, the arc va, and R~ nT2 ) to a surface R*1 C M a/ s so that the linking matrix A* between the isotoped circles, {"I*la}, and {pp.} has all entries zero. In fact, this can be accomplished using finger moves to isotope "I1a to change its linking number with each PPI but leave unchanged the linking number with each PP.>I. (Note that the linking number with PPI can be changed only by multiples of an integer which divides nl, while the entries of the matrix Aa,p are divisible by nd Each such finger move changes R~ by an ambient isotopy which fixes the compliment of a small ball in R~ and which stretches the interior of this ball over a regular neighborhood of some arc in M 3 / s . The ability to simultaneously change all entries of A to zero is based on the fact that the finger move isotopy moves R~ only in tubular neighborhoods of arcs. Because each finger move changes R~ only in the neighborhood of a point, these finger can be made away from aR~, the path vo. For the same reason, the finger moves can be done so as to leave R~ n T2 unchanged. With the preceding understood, it will be assumed in the sequel that there exist nl ~ 0 and a set of:
Rn
1) (3.15)
2) 3)
circles {"I2P} C T2 which generate H 1 (n) for the homology of T 2 , nl push-off copies, {{PPi} ~';1}' of {'f/2P} , circles {711a} which generate HI (T{'),
with the property that the resulting linking matrix A = (A a ,/1) has all entries zero. Furthermore, {111a} will be assumed disjoint from Vo of (2.14) and from fut( {p~.,}).
432
CLIFFORD HENRY TAUBES
f) Definition of T1 • With T{' understood, the surface Tl C W can now be constructed by isotoping T{' into the future a small amount along pseudo-gradient flow lines. This construction of Tl is accomplished by the following steps:
Step 1: Find an embedding (3.16) with the following properties:
(3.17)
1) cP is the end of an isotopy which moves points along pseudo-gradient flow lines. 2) cp is the identity where I ~ 3/8 + 1/100. 3) Let M == cp(M3 /s). Then 11M> 3/8. 4) I restricts to cp(VO) with out critical points. 5) inf(Jlcp({1]la})) > sup(Jlcp({pfji nT{'})). 6) I restricts to CP(B~'l_) with only index 2 critical points, one on each component.
To find such a cp, use the pseudo-gradient flow to construct a diffeomorphism (3.18)
1-1 ([3/8,7/16]) ~ M3/s
x [3/8,7/16],
where the pseudo-gradient flow lines are mapped to the lines p x [3/8,7/16], and where I is given by projection onto the second factor. With respect to (3.18), the embedding cp sends (p, t) to (p, g(p, t)), where g is a smooth function. It is left to the reader to find g which makes (3.17) true. (Remark here that {1]la} are disjoint from VO and from {Pfj.}.}
Step 2: With r.p understood, define (3.19) Also, introduce Rl == c,o(Rl'). Here are some important properties of T 1 :
METABOLIC COBORDISMS
433
1) No pseudo-gradient flow line intersects Tl more than once where f ~ sup(fIT2 ). 2) No pseudo-gradient is anywhere tangent to TI where
(3.20)
f ~ sup(fIT2 ). 3) Tl n T2 = int(Rt} n B b2 -·
4) Where f ~ 3/8 + 1/100, the restriction of f to TI has only index 2 critical points.
4 A start at Ez. This section begins the construction of the subvariety as in (1.6). The plan is to factor the cobordism W 2 from (1.4) as a sequence of cobordisms of simple type (Definition 2.2), and to define a E. for each component, simple type cobordism in this factorization. Then, EZ2 in (1.6) is defined to be the union of these E. for the constituent simple type cobordisms which comprise W2 • With the preceding understood, assume in this section and in Section 5 that W, and the Morse function f and the pseudo-gradient v are of simple type, as defined in Definition 2.2. Use the definitions in Section 2c to define Z C W x W. Sections 4 and 5 will construct a particular oriented, dimension-4 subvariety with boundary Ez C Z. The boundary of Ez will sit in 8Z. Furthermore, Ez will contain a class az E H4 (E z ,8E) which obeys the conclusions of Lemma 4.1 in [7]. As in Section 4c and (4.10) of [7], the variety Ez will be given as a union EZ2
(4.1) Here, 6. z is as described in Section 4d of [7], and EL,R are as described in Section 4e of [7]. (Remember: 6. z is the intersection of Z with the diagonal in W x W. Meanwhile, ER,EL are the respective intersections of Z with 7 x W and W x 7; here 7 C W is the pseudo-gradient flow line which starts at Po E Mo and ends at PI E Md a) A first pass at E_. Recall that the future of a set U C W (written fut(U)) is the set of points in W which can be reached from U by traveling along a gradient flow line in the direction of increasing f. Introduce (4.2) Equations (3.6) and (3.20) ensure that TI x fut(T2) and fut(Tt} x T2 intersect Z transversally, each as a smooth submanifold with boundary. These assertions are proved with the following fact: Let U C W be a submanifold which intersects no pseudo-gradient flow line more than once, and which is nowhere tangent to a pseudo-gradient flow line. Then fut(U) C W is a smooth submanifold with boundary, and that boundary is U.
434
CLIFFORD HENRY TAUBES
b) E{_ as a cycle. To consider Ef_ as a cycle, it is necessary to understand first the boundaries of (T1 x fut(T2)) n Z and (fut(Td x T 2) n Z. One finds (4.3)
8[(T1
x fut(T2)) n Zl
= [(T1
x T 2) n Zl u [(T1 x fut(T2)) n 8Zl,
and, likewise,
(The conditions in (3.6) and (3.20) are used here.) It follows from (4.3), (4.4) that orientations exist for both (fut(Td x T 2 ) n Z and (T1 x fut(T2)) nz such that 8[Ef_l has support (as a cycle) in (S3 X S3hl U (S3 x S3h~. With the preceding understood, write
(4.5) where, Sbl+ C (S3 X S3hl while Sb~- C (S3 X S3h2' It is left as an exercise to prove that Sbl + can be identified as being some number of push-off copies of the right-hand sphere, (S3hl+ c (S3 x S3)bl; while Sb2- consists of some number of disjoint, push-off copies of (S3)b~_ C (S3 X S3)b2' (See the proof of Lemma 4.1, below.) The next task is to determine the homology classes of the cycles that Sbl + and Sb2- define.
»n Z and (fut(Td x T 2) n Z of
LEMMA 4.1. The components (Tl x fut(T2 E{_ can be oriented so that as a cycle,
(4.6) Proof. Orient E{_ as follows: To begin, orient T1 and T2 to make their intersection number [Til' [T2l equal to m. Let 01,2 E A 2T(T1,2) denote the respective orientations. Next, orient fut(Td and fut(T2) by using -v A 01,2, where v is the pseudo-gradient for f. (Note that v is tangent to fut(T1,2) and is inward pointing along Tl or T2') Orient T1 x fut(T2) as 7rL * 01 A 7rR * (-v A 02) and orient fut(Td x T2 as 7rL * (v A 01) A 7rR * 02. Notice that the former is oriented using the product orientation, but the latter is oriented in reverse. This insures that the respective orientations which are induced on T1 X T2 are, in fact, opposite. Near b1 , T1 is identified with m1 like oriented, push-off copies of the descending disk Bbl-' Using the Morse coordinates of (3.2) in [7], this descending disk is given by setting X3 = X4 = O. And, one can assume, without loss of generality, that 01 = 8"'1 " 8"'2' Here, the orientation for W can be assumed to be
o
=Near 8"'1 " 8"'2 " 8"'3 " 8"'4 . b ,fut(T is a union of some number of disjoint components.
These 2) 1 components can be described as follows: The pseudo-gradient flow isotopes T2
435
METABOLIC COBORDISMS
to where f ~ 7/16 in W. This isotopic image, '£..2' intersects TI transversally; in fact, '£..2 intersects TI in the ml push-off copies of Bb1 _. Each intersection of '£..2 with Bb1- defines a component fut(T2 ) near bl , and likewise each intersection point of '£..2 with one of the ml push-off copies of B b1 - defines a component of TI x fut(T2) near (b l , bl)' Thus, the intersection points of '£..2 with the ml push-off copies of Bbl- are in 1-1 correspondence with the components of Sbl +. Using Morse coordinates of (3.2) in [7] near bl , a typical component offut(T2) near bl is given by {x : Xl = O,X2 > O}. IT this component corresponds to a positive intersection point of '£..2 with TI, then this component can be assumed oriented by -OX2 "OX8 "OX4; here OX2 is equal to v where Xl, X3 and X4 all vanish and X2 > O. Thus, the corresponding component of TI x fut(T2) is oriented by
(4.7) where
(4.8)
Xl
= Xs = X4 = YI = Y3 = Y4 = 0
and Y2
> O.
Here, the orientation for the intersection of TI x fut(T2) with Z is given by contracting (4.7) with -dY2 + dx 2. The resulting orientation is OX1 " (OX2'+ ( 112 ) " OilS" 0114 , The induced boundary orientation is given by contracting this with -dX2 - dY2; and the result is OX1 "OY8 "0114 , Meanwhile, (SSh1+ = {(x,y) : YI = Xs = X4 = O,X2 = r}. At the point in (4.8), (SShl+ is oriented by OX1 "0113 " 0114 also. Notice that this orientation is the same as that of the boundary of the given component of TI x fut(T2), and this component, by assumption, corresponds to a positive intersection point between TI and L. To summarize the preceding, a component of Sb1 + is oriented the same as (S3hl+ if the corresponding intersection point between TI and L is positive; while it is oriented in reverse if the corresponding intersection point between TI and T 2 is negative. This observation justifies the factor of m in the first term on the right side of (4.6) because the algebraic intersection number between Tl and '£..2 is equal to that between TI and T 2, which is m. Consider now the analogous calculation near (b 2 , b2 ). Here, the roles of TI and T2 are interchanged. The intersection of TI and T2 occur along B~2 _; the m2 push-off copies of Bb2-' Thus, the components of fut(Tt} x T2 near (b2, b2) are in 1-1 correspondence with the intersection points of TI and B~2- as are the components of S62-' Use the Morse coordinates of (3.2) in [7] near b2 • A typical component of fut(Tt} x T2 near (b 2, b2 ) is given as (4.9)
{(X,y) : Xl
= Y3 = Y4 = O,X2 > O}.
IT the component above corresponds to an intersection point of TI with B~2which has positive intersection number, then the orientation of (4.9) is given by OX2 " OX8 " OX4 " 0Y1 "0112 at points where (4.10)
Xl
= Xs = X4 = YI = Ys = Y4 = 0
and X2 >
o.
CLIFFORD HENRY TAUBES
436
The orientation for the intersection of (4.9) with Z is given by contracting its orientation with -dY2 + dx 2 • The resulting orientation at (4.10) is OZ3 "OZ4 " 0111 " (0112 + OZ2). The boundary orientation is obtained by contracting again with -dX2 - dY2; the result is OX8 "OZ4 "0111 . Note that this orientation equals the given orientation on (8 3 )62_' The preceding is summarized as follows: A component of Sb2 _ is oriented as (8 3 )62_ if the corresponding intersection point of Tl and T2 is positive; and the component is oriented negatively if the corresponding intersection point is negative. Thus, the factor of m in the second term on the right in (4.6) also follows from the fact that Tl . T2 = m. 0 c) E 1 - as a smoothing of E~_. As defined by (4.2), E~_ is the union of a pair of 4-dimensional submanifolds with boundary in Z which meet along a common boundary component which is (Tl x T2 ) n Z. There are no obstructions to smoothing the crease along (Tl x T 2 ) x Z to obtain a smoothly embedded, oriented submanifold with boundary, El- C Z. The next few subsections will describe some additional properties of E 1 _.
d) E1+' Introduce the switch map
(4.11)
9:WxW---+WxW,
which interchanges the coordinates. This map preserves Z. Define 9(E~_) and E1+ 9(El-). Thus,
=
E~+
_
(4.12) e) The intersection with
~z.
Make the standard identification of ~w C W x W with W (project on either right of left factor). This identifies ~z with the compliment in W of the union of an open ball about b1 and an open ball about b2 • And this identifies E~~ n~z with the intersection of
(4.13) with the compliment in W of said balls. To begin the analysis of (4.13), note that fut(Td nT2 is the union of a finite set of half-open arcs which start at the points of Tl n T2 (this is the same as Rl n B~2_). The closure of each of these half-open arcs is an embedded arc whose other endpoint is in B~2- n Bb2+. Remark that there are at least m such arcs. The intersection of Tl with fut(T2 ) is the image under the embedding r.p in (3.16) of T{' n fut(T2 ). The latter is described in Section 3d.
METABOLIC COBORDISMS
437
It follows from the description in Section 3d of T{' n fut(T2 ) that the intersection of E l - with i:::.. z is the disjoint union of some number of arcs and some number of circles. The end-points of the arcs lie 8El _ n i:::.. z , that is, on (S3 x S3hl U (S3 X S3h2. It is important to note that there are at least m such arcs which join m points of Sbl+ n (i:::..sahl with m points of Sb2- n (i:::.. s ah2. Furthermore, the proof of Lemma 4.1 shows that for at least m of these arcs, the one end point in Sbl+ and the other in Sb2- lie in components which are oriented positively with respect to the given orientations of (S3)bl + and (S3)b2_' respectively. In fact, there is an arc, v C E l - n i:::.. z , which connects a positively oriented component of Sbl + with a positively oriented component of Sb2-' and which is characterized as follows: Before smoothing E~_ to E l -, this v was an arc in E~_ which intersected Tl n fut(T2) as l;?(vO) n i:::.. z , where VO is the half-open arc in (3.14).
f) Intersections with
EL,R'
The submanifold E l - can be assumed to have empty intersection with EL,R. Indeed, the flow line 'Y between Po E Mo and Pl E Ml misses a small ball around bl and b2 ; and a small perturbation of Rl and R2 will insure that 'Y misses these surfaces ·also.
g) Normal framings. The claim here is that E l - has trivial normal bundle in Z, and that there is a trivialization of said normal bundle which restricts to each component of Sbl + and Sb2- as the constant normal framing. (Recall from [7] that the constant framing of S3 x point in S3 x S3 is the normal framing which is given by 7rR * f, where 7rR maps S3 x S3 onto the right factor of S3, and f is a normal framing of the point.) The establishment of this claim requires the following six steps. Step 1: This first step identifies E 1 _: LEMMA
Tl
4.2.
The submanifold E 1 - is diffeomorphic to the compliment in
x T2 of a finite number of disjoint, open balls. Proof. The identification of E 1 _ starts with the identification
(4.14) where U is a finite set of disjoint, open balls. Meanwhile, (4.15)
[Tl x fut(T2 )] n Z ~ [(Rl x R 2 ) U (B~l_ x R 2 ) U (B~lj- x B~2-)]- U'.
Here, Rl x R2 and B~l _ x R2 are attached along their common boundary component, 8B~1_ X RI. Meanwhile, (B~l- x R 2 ) U (B~I-" x B~2-) are attached along their common boundary component, B~l_ X 8B~2_. Finally, U' C int(B~l_ x R 2 ) is a finite, disjoint collection of open balls.
CLIFFORD HENRY TAUBES
438
Remember that (fut(Ttl x T2) n Z and (Tl x fut(n)) n Z are attached along their common boundary to obtain E l _. This common boundary is (4.16) where (Sl)m2 ~ (B~2_ n M). With (4.16) understood, one can see (4.14) and (4.15) as a decomposition of Tl x T2 less some number of open balls by writing TI ~ B~I_ URI and T2 ~ B~2_ U R2. 0 Step 2: The normal bundle to E I - in Z is an oriented three-plane bundle, and since E I - is not closed, this 3-plane bundle is classified by its 2nd StieffelWhitney class, W2' This class is zero for the following reasons: First, W2 (TW) = 0 since W is assumed to be a spin manifold. Thus, w2(T(W x W)) = O. Second, remark that T(W x W) IZ ~ T Z EEl~, where ~ is the trivial, real line bundle. Thus, w2(TZ) = O. Restricted to EI_,TZ ~ TEl _ EEl vE1 _, where vE1 _ is the normal bundle in question. Now, Tl x T2 is a spin manifold, and therefore w2(E l -) = OJ so w2(vE l -) = 0 as claimed. Step 3: Having established that E I - has trivial normal bundle in Z, it remains yet to establish that this normal bundle has a trivialization which restricts to each component of BEI - as the constant normal framing. Here is an outline of the argument: a) Remember that E I - is the image of an embedding of the compliment in TI X T2 of some number of open balls. With this understood, the proof establishes that this embeddin~ extends as an embedding of TI x T2 into W x W. This extension will be called E I _. b) The proof establishes that the normal bundle in W x W to EI - splits as N EEl~, where N is a trivial 3-plane bundle, and where ~ restricts to E I - C EI as the normal bundle to Z in W x W. c) The proof establishes that N is a trivial 3-plane bundle over EI _, d) Thus, N restricts to E I - as vEI-j and the restriction of a framing of N to E I _ gives a framing of vE1 _ which is homotopic to the constant framing over each component of BEl _. Step 4: To esteblish Step 3a, above, remark that a component, C of BEl on (83 X 8 3 )61 has a neighborhood in E I - which can be assumed to have the following form in coordinates from Lemma 3.6 in [7]: (4.17)
{(x, y) : X3
= X4 = Y2 = 0
and x~
+ x~ + y~ + y~ = yn,
where YI ~ (r/2)l/2. Here, C is given by (4.17) with YI = (r/2)1/2. Note that C is the intersection with (83 x 8 3 )61 of a push-off of the ascending 4-ball from the critical point (b l , bl ) for the function F on W x W which is given in (3.20) of [7]. Thus C bounds an embedded 4-ball in W x W, for example, the ball B C (W x W - Z) which is given by (4.18)
{(x,y) : X3
= X4 = Y2 = O,Yl = (r/2h/2 and x~
+ x~ + y~ + y~
$ r/2}.
METABOLIC COBORDISMS
439
Each boundary component of E 1 _ has its analogous Bj and these can be taken to be mutually disjoint, being all push-off copies of a descending 4-ball for F from (b l , bl ) or from (b 2 , ~). Glue these 4-balls to E 1 - along their common boundaries and smooth the corner along aE1 - to obtain EI -, an embedding of TI x T2 into W x W which extends E I -. Step 5: To establish Step 3b, note that the normal bundle to E1- in W x W splits as vEI - EEl ~ where ~ is spanned by a section of T(W x W) along EIwhich has positive pairing with dF. With this understood, consider the vector field -a/aYI in the coordinates of (4.17), (4.18). This vector field is nowhere tangent to E1 - and restricts to a neighborhood of B in EI - to have positive pairing with the -I-form dF. Thus, -a/aYI extends the preceding splittin~ of the normal bundle of E 1 - in W x W to a splitting of the normal bundle of E I in W x W as N EEl~, where N = vEI _ over E I _. Step 6: The fact that vEI _ is trivial implies that w2(N) = O. Thus, N is the trivial bundle if N's first Pontrjagin class vanishes. This class is computed as follows: Since PI (T(TI x T2)) = 0, it follows that PI (N) is the same as PI (T(W x W»IE 1 _. Thus, N is trivial if PI (T(W x W) is trivial as a rational class. The latter is trivial because PI (T(WxW» ~ lI'L *PI (TW) +lI'R*PI (TW)), and both these classes vanish because W is has non-trivial boundary. h) A fiducial homotopy class of normal framing. The previous subsection establishes that there are homotopy classes of normal framings for E 1 - in Z which restrict to each component of aEI _ as the class of the constant normal framing. The purpose of this subsection is to describe a subset of such classes which behave nicely when restricted to a specific set of generators for H1(EI -). To make this all precise, it proves useful to first digress to describe a set of generators of HI (E1 -). To begin the digression, take the generators {7]la} for HdT{') and {7]2,B} for HI (T2 ) as desctibed in (3.15). Choose a point Xl E RI and a point X2 E R 2 • Then, generators for H1(EI - ) are given by (4.19)
1)
{Sla
2)
{S2,8
== (CP(7]la) X fut(x2)) n Z}, == (Xl x fut(7]2,8)) n Z}.
Fix generators {Sla, S2,B} as above. End the digression. Ideally, a normal frame for E I - should restrict to these circles as a product normal frame, e = (el,e2,e3), for RI x fut(R 2 ) in W x W with the following properties: (4.20)
1) 2) 3)
el is normal to cp(M3 / S ) in W and (dJ, el) < O. e2 is normal to RI in cp(M3 / S )' e3 is normal to fut(R2} in W and (dJ, e3) = O.
LEMMA 4.3. Given generators {Sla, S2,8} for H.(EI_) as described in (4.19), there is a nonnal /rame for E 1 _ in Z whose restriction to each component 0/
440
CLIFFORD HENRY TAUBES
8E1_ is a constant normal frame, and whose restriction to each
8
E {Slo,S2/3}
is described by (4.EO). Relllark: A normal frame for E 1 - which is described by Lemma 4.3 will be called a fiducial normal frame. Proof. The restriction of a given normal frame of E 1- to S E {Slo, S2/3} can be written as 9 . e, where 9 : S ~ SO(3). H 9 is null-homotopic, then., and only then can be homotoped to a frame whose restriction to s is equal to e. With the preceding understood, note that a map 9 : Sl ~ SO(3) is classified by the class in H1(Sl; Z/2) ofthe pull-back of the generator, u , of the module HI (SO(3); Z/2). Therefore, a normal frame for E 1- (which is homotopic to a constant frame on each component of 8E1 -) defines an element >.(e) E (EBoH1(SlO; Z/2)) EB (EB/3H 1(S2,B; Z/2)) which is the obstruction to deforming to a fiducial frame. By the way, note that when h : E 1 - ~ SO(3), then >'(h . e) = >.(e) + i* h*u, where i is the inclusion map of (UaS10) U CU/3S2,B) into E 1 _. To prove the lemma, take a normal frame for E 1 - and define a map q : (UoS1a) U (U/3S2/3) ~ Sl as follows: H s E {Slo,S2,B} and >.ce) has trivial summand in HI (s; Z /2), then make ql s the constant map. Otherwise, make qls a diffeomorphism to 8 1 (a degree one map.) Because {S10, S2,B} generate H 1(E 1-), this map q extends as a map q : E 1- ~ Sl which is trivial near 8E1_. Let j : Sl ~ SO(3) generate H1CSO(3» and set h == j 09: Then >'(h 0 e) = 0 because of the equalities i*h*u = (j 0 q)*u = >.(e). 0
e
e
e
e
e
i) H2(E1_) and H 2(E 1_). Lemma 4.2 implies that (4.21) Of course, H 2 (E1 -; 1R) is isomorphic to (4.21), but the proof of the results in the introductory section requires a set of generators for H 2 (E 1-i 1R). To give such generators, it is necessary to first choose orientations for Tl and T2 so that their intersection number equals m. Choose a point PI E B~l _ with f(Pt> < f1T2. Also, choose a point P2 E R2 which is on a gradient flow line which ends on MI. With these choices understood, then (4.22)
== (T1 x fut~» n Z,
1)
T1-
2)
T2- == PI x (fut(T2) n f- 1(P1»
are embedded sub manifolds of E 1 - each of whose fundamental class is a generator of H 2 (E 1-). To obtain the remaining generators, it is necessary to first choose embedded circles, {1/10} c R~ and {1/2,B} C R2 which generate the respective first homology of T;' and T 2 • Equation (3.15) introduces an integer n1 ~ 1 and then, for each /3, a set {P~J~l of n1 like oriented, push-off copies (in R 2) of 1/213. Let 1/~13 == UiP~ •. Orient this submanifold of M 1 /4 by taking the given orientation of
METABOLIC COBORDISMS
441
Pp;.
each For £Uture applications, it should be assumed, as in Section 3e, that {171a} is disjoint from fut({71~.a})· For each a, fix a set, 71~a C R~, of n1, like oriented, push-off copies of 711 a . Do not make a big push off: The push-off isotopy must not intersect £Ut( {71;.a} ) nor should (3.17.5) fail with {71~a} replacing {711a}. The remaining generators of H 2 (E1 _) can be taken to be the fundamental classes of (4.23)
j) Pushing off H2(E1 -). The second homology of EL with real coefficients is generated by (4.24) The second homology of ER with real coefficients is generated by the corresponding [TIRJ == 8.[T1 L] and [T2RJ = 9.[T2LJ. The inclusion map from EL U ER into Z identifies these four classes as generators of H 2 (Z). (Use real coefficients here and through out this subsection.) The inclusion map of E 1 - U ER U EL into Z induces a homomorphism (4.25) with the property that
1) (4.26)
2)
t· t .
3)
t·
([TI-J- [TlL])
= 0,
([T2-J - [T2 R]) = 0, [Ta •.a-J = 0.
As discussed in Section 4h, the submanifold E 1 - has a trivial normal bundle in Z with a fiducial homotopy class of framing which restricts to each component of 8E1 _ as the class of the constant normal framing. Choose a framing from such a homotopy class and use one of the frame vectors to push each of the submanifolds T1 -,T2 -, and {Ta •.a-} into Z - E1-. IT P2 is chosen so that fut(P2) is disjoint from T 1 , then the submanifold T 1 is disjoint from EH U 6.z. IT PI is chosen to be disjoint from fut(T2 ), then the submanifold T 2- is likewise disjoint from EH U6. z . As {71la} and £Ut( {71~.a}) are assumed to be disjoint, {Ta •.a-} is disjoint from 6.z. And, because of (3.17.5), {Ta ..a-} is disjoint from E H • Thus, T1 -, T2 - and all {Tal.a-} can be pushed off of E 1 - into Z - EI where (4.27) in an essentially canonical way. Both EL and ER have a canonical homotopy class of normal bundle framing. The canonical homotopy class of normal framing for EL is the class of the normal framing which is obtained by pulling back via the projection 7rR a normal bundle
CLIFFORD HENRY TAUBES
442
framing for the arc 'Y in W. Similarly, the canonical homotopy class of normal framing for ER is obtained by pulling back via the projection 7rL the same normal bundle framing for 'Y C W. Fix a framing in the canonical homotopy class for EL'S normal bundle and use one of the framing basis vectors to push TlL and T2L off of EL into :E1- Then, push TIR and T2R off of ER into :El by the analogous method. These push-offs define a homomorphism (4.28) and the purpose of this subsection is to prove LEMMA 4.4. The classes ([T1 -] erate the kernel of ,'. Thus, ker(t')
-
[TIL]), ([T2-] - [T2R]) and {[Ta,p-]} gen-
= ker(t).
Proof. The proof considers each of the three kinds of classes in turn. Case 1: The class [T1 -] - [TlL]. To begin, remark that there is a natural push-off, Tf_, ofll_ into 1;1 which is obtained by using (4.22.1) withP2 replaced by a point p~ E M I / 4 - R2 which is a push-off of P2. This sort of push-off can be defined by a normal framing, (el' e2, e3), for E I - which has the following property: Along T 1 - C (Tl X fut(T2)) 11 Z, the frame is the restriction from Tl x fut(T2) of a product frame, where
(4.29)
1) 2)
is normal to fut(T2) in Wand (df,ea) = 0, (el. e2) is a normal frame for Tl in W.
ea
The push-off TL as described above is then obtained by pushing off T 1 - along the normal vector e3. Now Tl has trivial normal bundle (its self intersection number is zero), so there is a normal frame as in (4.29) for E 1 - along T 1 _. Furthermore, LEMMA 4.5. There is a fiducial normal frame from Section 4h whose restriction to T 1 - is described by (4.29).
This lemma is proved below; accept it for the time being to continue with the proof of Lemma 4.4 for [T1-] - [TIL]. An acceptable push-off of TlL is defined as follows: Take a point PofMl/4 which is near too, but not equal to 'Y n M 1 / 4 • A push-off of TlL into Z - :El is (Tl X fut(Po) n Z. Since R2 and Rl both are connected, and both have non-trivial boundaries, one can find a path p, in M 1 / 4 with one endpoint P~ and the other Po and whose future is disjoint from Tl, T2 and 'Y. With this understood, then (Tl x fut (p,)) n Z is an isotopy in Z - :El between the push-offs of T 1 - and TlL.
e
Proof. Let denote a normal frame from Section 4h. There is a map 9 : T 1 _ --+ 80(3) such that g. <elTl - ) is described by (4.29). With this under-
stood, the lemma follows if such a map 9 can be found which is null homotopic. Now, a map 9 : T 1 - --+ 80(3) is null homotopic if and only if the map lifts to a
METABOLIC COBORDISMS
44S
map into 8 s . The obstruction to such a lift is an element (J(9) E HI (T1-; 7l. /2) which is the pull-back by 9 of the generator of H1(80(3); 7l./2). Note that (J(91 . 92) = (J(91) + (J(92). Store this information. Consider now the homotopy classes of normal frames which have the form of (4.29). Given that es is constrained to lie on a fixed side of R 2 , these are in 1-1 correspondence with the homotopy classes of normal frames of T 1 • The latter set is isomorphic (though not canonically) to the set of homotopy classes of maps from T1- to 80(2) ~ 8 1 • Meanwhile, a map h : T1 - ---+ 8 1 is distinguished up to homotopy by an invariant (J1 (h) E HI (T1-; 7l.) which is the class of the pull-back by h of the generator of H1(8 1). Furthermore, let j : 80(2) ---+ 80(3) denote the usual inclusion. Then j 0 h : Tl- ---+ 80(3) and one has (J(j 0 h) = (J1 (h)mod(2). The lemma now follows from this last comment because H1(T1 _;7l./2) ~ HI (T1 -; 7l.) ® 7l./2. Case 2: The class [T2-J- [T2RJ. The argument for [T2-J- [T2RJ is essentially the same as the preceding one and will not be given. Case 3: The classes {[Ta,pl}. The first step is to consider the trivialization of the normal bundle of E 1- along a given Ta,p. LEMMA 4.6. There are fiducial normal frames for E 1 - (as d"efined in Section 4h) that restrict to Ta,p as the restriction of a normal frame e = (e1, e2, es) which is a product normal frame for the submanifold R1 x fut(R2) c W x W as described by (4.20).
This lemma is proved below; accept it momentarily to continue the proof of Lemma 4.5. Describe the push-off of Ta,p_ into Z - ~1 as follows: Let M s/ s == f- 1(3/8). Push l1~a into M 3 / s along the pseudo-gradient flow. Use !l~a to denote the resulting set of circles. Then (4.30)
is a acceptable push-off of Ta,p- into Z - ~1' Now, remark that l1~a bounds a smooth surface 8 a c cp(Ms/ s ), ana this means that rl' bounds in M s/ s , the bounding surface, Sa, is obtained by flowing 8 a alongthe pseudo-gradient flow lines into M s/ s . With the preceding understood, (4.31)
bounds L,p- in Z. Note that (4.31) is disjoint from E1- and from E1+' It is also disjoint from E L , and it is disjoint from ER if Sa is chosen to miss the point of intersection of'Y with cp(Ms / s ). The intersection of (4.31) with az is (4.32)
CLIFFORD HENRY TAUBES
444
This may be non-empty. However, by assumption, the linking matrix of Section 3e has all entries zero, which means that the intersection points in (4.32) can be paired so that the local intersection numbers (±1) of the points in each pair cancel. The cancelling of these local signs in pairs implies that an ambient surgery in Z of the interior of (4.32) (remove (SO x B3) which intersect !:::.Z and replace with (B1 x S2)'S which do not) will result in a submanifold with boundary in Z which is completely disjoint from ~1 and which bounds L,{3-' Proof. Upon restriction to TO/,/3, a fiducial normal frame ~ for E 1- (as described in Section 4h) has the form 9 . e for some 9 : TO/,{3 ~ SO(3). IT g is null-homotopic, then ~ can be homotoped in a neighborhood of T ,{3 to restrict to TO/,{3 as the restriction of e. With the preceding understood, remark that a map 9 as above is null homotopic if and only if 9 lifts to a map into S3. The obstruction to finding such a lift is g*a E H 1(TO/,{3; Z/2), where a generates H1(SO(3); Z/2). Now, TO/,{3 is the disjoint union of push-off copies of an embedded torus, (c,o(7110/) x fut(712/3)) n Z. The first homology of this embedded torus is generated by the circles (c,o(711a,) X fut(x2)) n Z and (Xl x fut(712{3)) x Z; here Xl E c,o(7110/) while X2 E 712{3. This fact with Lemma 4.3 insures that g*a is zero. 0 Q
k) Intersections with
L.±"
Reintroduce the function F on W x W which assigns fey) - f(x) to a point (x, y). As remarked in Section 3 of [7], the critical points of F are the points (p, q) where p and q are critical points of f. The descending 4-ball from the point (b 2 , b1 ) intersects Z as an embedded 3-sphere which will be denoted by S(2,1)' (In (3.32) of [7], this 3-sphere is denoted by S(b2,bd-; but such notation is not necessary here.) Likewise, the ascending 4-ball from (b 1 , b2 ) intersects Z as an embedded 3-sphere which will be denoted by S(1,2) (rather than S(bl,b 2)+ as in (3.33) of [7]). The purpose of this subsection is to prove LEMMA 4.7. The intersection numbers of E 1- and of E1+ with S(2,1) add 'Up to zero. The intersection numbers of E 1- and of E1+ with S(1,2) also add 'Up to zero.
Proof. To consider the case of S(2,1). note that the descending ball from (b 2, bt) is Bb 2 + X B b1 -. The intersection numbers of E 1- and of E1+ with S(2,1) (in Z) are minus the respective intersection numbers (in W x W) of E l - and of E1+ with Bb2+ x Bbl _. Consider first the intersection number of E 1- with Bb2+ x Bbt-' There are no intersection points in (fut(Tt) x T 2 ) n Z because the intersection between Bbl- and T2 occurs near f- 1(1/4), while on futeTI),f ~ 3/8. As for (Tl x fut(T2 n Z, note that Bb 2 + has intersection number ml,2 with T l ; one intersection point is in each of the Iml,21 copies of Bb2- which sit in B~l _. Each of these intersection points can be assumed to have a different value of I, but all such values occur near l(b 2 ). Meanwhile, Bbl~ has intersection
»
METABOLIC COBORDISMS
445
number m/ml with fut(T2) n ,-I(f(b2 )). This number is computed using the following facts:
(4.33)
1) The intersection number of Tl with fut(T2) n ,-I(f(b2) is the same as that of Tl with T2· 2) Tl intersects fut(T2) n ,-1 (f(b 2 )) only in the push-off copies of Bbl- in cp(B~/l_).
Thus, Bb2+ x Bbl- has intersection number m· m1,2/m1 with E 1- (so 8(2,1) has intersection number equal to -m· ml,2/ml with El-). Now turn to the intersection number of Bb2+ x Bbl- with E1+. Here, there are no intersections in (fut(T2) x T1 ) n Z because Bb 2 + intersects fut(T2) where , > 7/16, while Bbl- intersects Tl where, is approximately 3/8. On the other hand, Bb2+ has intersection number m2 with T 2 , once in each copy of Bb 2 - that makes up B~2_. Each such intersection takes place near b2 • Meanwhile, the intersection number between Bbl- and fut(T1) n- 1 (b 2 ) is equal to -m· ml,2/(ml . m2). This number is computed using the following facts:
1) Tl has zero intersection number with itself. 2) A push-off copy of Tl can be constructed which intersects T1 (4.34) as a push-off of B~l- intersects fut(Rt) n ,-1(7/16). 3) T2 has m intersections with Tl, one in each of the push-off copies of B b2 - that comprise B~2-.
Thus, B b2 + x Bbl- has intersection number -m ·ml,2/m1 with E1+ (so 8(2,1) has intersection number m· ml,2/ml with E1+). The case for 8(1,2) follows from the preceding computation because 8(1,2) = 9(8(2,1») while 9 interchanges E 1- with E1+. 0 5 The Construction of E±. The previous section began the construction of Ez in the case where W is a cobordism of simple type as described in Definition 2.2. (See (4.1).) This section will finish the construction of Ez for such a cobordism. Indeed, (4.1) is missing only definitions of E±; and this section will construct E± from El± via ambient surgery in Z. The surgical techniques here are those from Sections 7-10 of [7]. a) Constructing E2-: Push-offs and tUbings. Begin with E 1 - of the preceding section. Using the fiducial normal framing at the end of Section 4h, make 2m disjoint, push-9ff copies of E 1 - in addition to the original. Orient the first m copies as the original, and orient the last m copies in reverse. Use E~_ to denote the resulting disjoint union. The boundary of E 1 - is described in (4.5) and (4.6). That is, it is a union of 3-spheres which are push-off copies of (8 3 hl+ or of (8 3 h2_ in (8 3 x 8 3 )bl or in (8 3 x 8 3 )h, respectively. As described at the end of Section 4e, there is an
446
CLIFFORD HENRY TAUBES
arc component v of E 1 _
n D:.z that connects a positively oriented component
8 1 C 8E1 - n (83 x 8 3 h1_ with a positively oriented component 8 2 C 8E1 _ n (83 x 8 3 h2' Each of the first m - 1 push-off copies of E 1 - contains a push-off copy of
8 1. Let {81a}::~01 denote this set of push-offs. Here, 810 is the original 8 1 in
the original copy of E 1 _. Use {82a}::~l to denote the corresponding copies of 8 2 (with 8 20 denoting the original), and let {va,}::~l denote the corresponding copies of v. Note that the components of (8E~_ n (8 3 x 8 3 h1) - {81a } can be paired up so that each pair contains one positively oriented sphere and one negatively oriented sphere. The spheres in each pair should be tubed to each other as described in Section 7b of [7] (see (7.3) in [7]). Note: As m ~ 1, there is at least one pair to tube here. Likewise, the components of (8E~_ n (8 3 x 8 3 h2) - {82a } can be paired so that each pair contains one positively oriented sphere and one negatively oriented sphere. The spheres in each pair should be tubed to each other as described in the same Section 7b of [7]. There is at least one pair to tube here too. Use E 2 - to describe the submanifold (with boundary) of Z that results. By construction, (5.1) Note as well (see Section 7e of [7]) that after a small perturbation, the intersection of E 2 - with D:.z will be transversal, and given by (5.2) where C C int(D:.z) is a disjoint union of embedded circles. One can argue as in Section 7f of [7] that E 2 - has trivial normal bundle in Z with a framing which restricts to each component of 8~_ as the constant normal framing. Meyer-Vietoris (as used in the proof of Assertion 6 of Proposition 7.4 in [7]) shows that H2(E2_) ::::: H2(E~_). Define E2+ 8(E2_) and define E2 as in (4.27) with E2± instead of El±. Define the homorphism , : H 2 (E2- U EL U ER) - t H 2 (Z) from the inclusion into Z, and define " : H 2 (E2 - U Et U ER) - t H 2 (Z - E 2 ) by analogy with (4.28) using the homotopy class of normal frame for E 2 - which is inherited (as in Section 7f of [7]) from the canonical homotopy class of normal frame for E 1 _. Then
=
(5.3)
ker(L)
= ker(,'),
just as in Lemma 4.4. To prove (5.3), note first that (5.3) holds for E~_ since E~_ is the disjoint union of some number of push-off copies of E1_0 Next, remark that E 2 - = EL except near az. Finally, note that the homologies which prove Lemma 4.4 for E 1 - are made away from az.
METABOLIC COBORDISMS
447
As a final comment about E 2 -, remark that the tubing can be done in such a way that E 2- has empty intersection with EL,Ri and it can be done so that the tubing avoids the spheres 8(2,1) and 8(1,2) of Lemma 4.7. In any event, the fundamental class [~_] in H 4 (Z, OZ) will equal [E1 -].
b) Constructing E 3 _: Removing circles. The goal here is to take some number N1 of like oriented, push-off copies of E 2 - and do surgery on the circles in its intersection with t1 z . The goal is to obtain a manifold E 3 - with the following properties: PROPOSITION 5.1. There is an oriented submanifold (with boundary) E 3 - C Z and an integer N ~ 1 with the following properties: 1) The fundamental class [E3-] in H 4 (Z, oZ) is equal to m- 1 • N . [E1-], and, in particular, obeys
2)
The boundary of E 3 - is a submanifold of oZ, given by
oE3-
3)
4) 5)
= (U~=1S1a) U (U~=1S2a),
where each S1a is a push-off copy of (S3 x point) C (S3 x S3hl' while each S2a is a push-off copy of (point xS 3 ) C (S3 x S3h2. E 3 - has empty intersection with Mo x Mo and with M1 x M1. E 3 - has empty intersection with EL and with ER. E 3 - has transversal intersection with t1z and E3 -
n t1z = U~=1Va,
where {va} are all push-off copies of an arc. Furthermore, for each a, Va has one end point on S1a n t1z and the other on S2a n t1z. 6) E 3 - has trivial normal bundle in Z, and this normal bundle has a fiducial frame ( which restricts to each S1a and each S2a as the constant normal frame. 7) E 3 - is obtained from the disjoint union, E~_, of some number N1 of like oriented, pwh-off copies of E 2 - by ambient surgery in Z on embedded circle~ in E~_ n t1z. This surgery naturally identifies H2(E3_i Q) ~ ffiN1H2 (E~_ i Q). 8) Define E3+ 9(E3_) and define E3 as in (4.27) with E 3 ± instead of El±. Define the homorphism t : H 2(E3- U EL U ER) --t H2(Z) from the inclusion into Z, and define t' : H 2(E3- U EL U ER) --t H 2(Z - E 3 ) by analogy with (4.28) using the homotopy class of normal frame for E3from Assertion 6, above. Then ker(t) = ker(t'). 9) The intersection numbers of E 3 - and E 3 + with the sphere 8(1,2) (of Lemma 4.7) sum to zero; and the same is true for the intersection numbers of E 3 - and E3+ with 8(2,1).
=
CLIFFORD HENRY TAUBES
448
Proof. The submanifold E 3 - is constructed by mimicking the proof of Proposition B.1 in [7]. To be brief, the first step is to invoke Propositions 8.3 and 8.7 in [7]. Copy the arguments in Sections Be, Bd and Be of [7] to verify that the assumptions of Propositions B.3 and B.7 can be met with the following choices of A,B,X and 0:
1)
(5.4)
2)
3) 4)
A is the interior of some number NI of like oriented, push-off copies ofE2 _. B = int(~z). X = int(Z). 0 is the compliment in int(Z) of the closure of a regular neighborhood of az u v U EL U ER.
Here, v C ~z n E 1 - is described in Section 5a, above; and it is assumed that 0 does not contain the NI . m push copies of v which are the arc components of An~z. (Note that there is a basis (Le., [T1 ] and [T2D for B's second homology in which the B's intersection form is a 2 x 2 matrix with zero's on the diagonal. A symmetric, bilinear form with this property is even.) In proving Assertion 7, note that the union of the circles in E~_ n ~z is homologically non-trivial because the construction of E 2 _ required at least one pair of tubings near each of (83 x S3hl,2' The prooffor Assertion B of Proposition 5.1 is as follows: The assertion holds with EL replacing E 3 - everywhere since E~_ is a union of push-off copies of ~_. Meanwhile, the surgery which changes E~_ to E 3 - takes place in a regular neighborhood of ~z, and the homologies which prove that ker(t) = ker(t') can be made with support away from ~z. (See the proof of Lemma 4.4.) Assertion 9 of Proposition 5.1 follows from Assertion 1 and Lemrna 4.7. D c) Constructing E4±; straightening EH n E 3 -
•
The intersection of EH with E 3 - can be something of a mess. After small perturbations of E 3 ±, this intersection has the form (5.5)
EH
n E 3 - = rue,
=
where r c Z is the union of r 1 U~=l va with some N - 1 like oriented, push-offs of r 1 into Z - ~z. These push-offs can be assumed as close to ~z as desired. Meanwhile, C C int(Z) - ~z is a disjoint union of embedded circles. By the way, (5.5) can be established using (3.17.4). Argue as in Section 9 in [7] to prove that ambient surgery on a pair, E~_, of like oriented push-offs of E 3 - , with ambient surgery on a pair, E~+, of like oriented push-offs of E3+ will result in submanifolds E4± with the following properties: PROPOSITION 5.2. There are connected, oriented submanifolds (with boundary) E 4 - C Z and EH C Z and an integer N ~ 1 with the following properties: 1) The fundamental classes [E4±J in H4(Z, aZ) equal m- 1 • N . [El±] and furthermore obey
METABOLIC COBORDISMS
8[E4+]
2) Let e : Z
3)
449
= N· [S3]bl_ + N· [S3]b2+.
~ Z denote the switch map. Near 8Z U tl. z ,
The boundary of E4- is a submanifold of 8Z, given by 8E4_ = (U~=ISla) U (U~=lS2a), where each Sla is a push-off copy of (S3 X point) C (S3 X S3)bl' while each S2a is a push-off copy of (point xS 3) C (S3 x S3h2.
4) E4± have empty intersection with Mo x Mo and with Ml x MI. 5) E4± have empty intersection with EL and with ER. 6) E4± have tmnsversal intersection with tl.z, and E4-
n tl.z
= E 4+ n tl.z = U~=l Va,
where {va} are all push-off copies of an arc. Furthermore, for each cr, Va one end point on Sla n tl.z and the other on S2a n tl.z . E4- has tmnsversal intersections with E4+. Furthermore, h~s
7)
8)
9)
10)
11)
where r is the union of r l == Ua=INva and some N - 1 like oriented, push-off copies of r 1 in Z - tl. z . E4± have trivial normal bundles in Z, and these normal bundles have frames (± with properties which include: The frames (± restrict to each boundary component as the constant frame. Furthermore, where Assertion 2 holds, (+ = e * ((_). E4± are obtained from the union, E~±, of one or possibly two like oriented, push-off copies of E 3± by ambient surgery in int(Z - tl. z ) on the circles in E~_ n E~+. These surgeries natumlly identify H2 (E4±; Q) ::::: ffiH2(E~±; Q). Define E4 as in (4.27) with E4± instead of E1±. Define the homorphism. t : H2(E4_UE4+UELUER) ~ H 2(Z) from the inclusion into Z, and define £': H2(E4-UEHUELUER) ~ H 2(Z-E 4) by analogy with (4.28) using the homotopy class of normal frame for E 4± from Assertion 7, above. Then ker(t) = ker(t' ). The intersection numbers of E 4- and E4+ with the sphere S(1,2) (of Lemma 4.7) sum to zero; and the same is true for the intersection numbers of E4- and EH with 8(2,1).
The fact that Z is path connected implies that E4± can be constructed to be path connected. See Lemma 8.10 in [7] and its proof. Remark that the last assertion of Proposition 5.2 follows from Assertion 1 and Lemma 4.7. The argument for Assertion 9 proceeds as follows: Since E~± are
450
CLIFFORD HENRY TAUBES
disjoint unions of push-off copies of E3±, Assertion 9 holds if E~± everywhere replace E4±' Now, E4± is constructed by surgery on E~±i and these surgeries can be performed away from the generators of H 2 • Furthermore, the surgeries take place in a regular neighborhood of a surface with boundary or 3-ball in Z, and so can be performed away from the homologies which establish Assertion 9 for E~±. (See the proof of Lemma 404.) d) The meld construction and E±. This section constructs E± from E4± using the meld operation of Section 10 in [7}. In this regard, note that the behavior of E 4 - near 8Z U t1z is described by (10.2-5) in [7] modulo notation. To be precise, there is a regular neighborhood U C Z of 8Z n t1z such that E4- n U is a set {Ya}:=l (with N from Proposition 5.2), where {Ya~2} are disjoint, like oriented push-off copies of Y1 • Meanwhile, Yi is the image of a proper embedding into U of the space in (10.3) of [7]; this being the compliment in the open unit 4-ball of the open balls B± of radius 1/8 and centers (±1/4, 0, 0, 0). Note here that the boundary of B+ is mapped diffeomorphically onto (8 3 )b1+ C (83 x 8 3 h1' and the boundary of B_ is likewise mapped onto (8 3 h2_ C (83 X 8 3 h2' Meanwhile, the arc along the x-axis between (±1/8, 0, 0, 0) is mapped to the arc v C E 4 - n t1 z . The {Ya >2} are described by (lOA) and (10.5) in [7]. The melded space, E_, is then described by (10.8) in [7]. (See also (10.9) and (10.10) in [7].) As for E+, the neighborhood U can be chosen to be invariant under the switch map (4.11) and such that E4+nU = 9(E4_ nUl. With this understood, define E+ n (Z - U) E4+ n (Z - U), and define E+ n U 9(E_ n U). Note that
=
1)
(5.6)
2)
3)
=
[E±] = m- 1 • N . [El±] in H4(Z, 8Zi Z). 8[E_] = N· [8 3 ]b1+ + N· [8 3 ]62_ and 8[E+] = N· [8 3 ]b 1_ + [8 3 ]b2+' H2(E±iCij):::::: H2(E4±iCij).
6 Completing the proof The purpose of this last section is to complete the proof of Theorem 1.3 along the lines that were outlined in Section Ie. Thus, suppose that Mo and MI are compact, oriented 3-manifolds with the rational homology of 8 3 • Assume that Mo and MI are spin cobordant by a cobordism whose intersection form is equivalent to a sum of metabolics (see (1.3)). As descibed in (1.4), one can find such a cobordism which factors as WI UW2 UW3 , where WI and W3 have the rational homology of 8 3 , and where W 2 has a good Morse function with only index 2 critical points. As in Proposition 2.1, the cobordism W 2 can be factored as Uj'=l W 2"j, where each W 2,j is a cobordism of simple type (Definition 2.2) between a pair, F j - l and Fj , of rational homology spheres. Here, Fo = M~ and Fn = M{. Define Z2 == Uj Z 2,i, where each Z2,J C W2,j X W2,j is defined as in Section 2c. The id~ntifi~ation of Fj x Fj C Z2,j with Fj x Fj C Z2,j+1 is left implicit here. Use thIS Z2 ln (1.5)
METABOLIC COBORDISMS
451
Fix base points in each Pj. Then define {Ez N ·} as in (4.1). With this understood, set EZ2 == UjEzN. after making the implicit boundary identifications. Use this EZ2 in (1.6). Step 3 of the outline in Section 1c constructs a closed 2-form Wz on the compliment of Ez which obeys Wz I\wz I\wz = 0 near Ez. The construction of Wz proceeds by first constructing a closed 2-form ~z on the compliment of Ez in a regular neighborhood Nz of E z in Z. The form ~z will be built so that it satisfies Condition 1 of Lemma 4.2 in [7]. Also, f!lz 1\ f!lz = O. The form will then be extended over the compliment of Ez of a neighborhood of az U Ez so that its pull-back to any boundary component M x M - EM is a form which computes 12(M). Here M x M is any of Mo x Mo, Ml X Ml or any (8 3 x 8 3)bli' (83 X 83h2i' The next question is whether the form f!lz so constructed can be extended over Z - Ez. The author does not know when such is the case. However, it is shown below that there is a closed 2-form JJ on N z which obeys JJ 1\ JJ = 0, which vanishes near az, and is such that Wz == f!lz - I' extends over Z - Ez as a closed form. Note that such a form will satisfy the third condition in (1.7). The form I' will vanish near EZI and near Ezs' Furthermore, JJ will be written as JJ = Ej =lnJJ2,j, where 1'2,j has compact support in the interior of Z2,j' With this understood, the construction of 1'2,j can be made independently for each factor Z2,j which comprises Z2.
a) f!lz near Ez and
az.
The construction of the closed 2-form f!lz on the compliment of Ez in a regular neighborhood Nz C Z of Ez U az proceeds by mimicking the constructions in Sections 11a - 11i of [7] which construct Wz near Ez when the cobordism between Mo and Ml has the rational homology of 8 3 • The conditions in Theorem 1.3 that W be spin and that the canonical frame be represented by c in the kernel of the homomorphism tw arise here. The verification of Condition 1 of Lemma 4.2 for Wz proceeds as in Section 11k of [7], and the reader is referred there. (But note Assertion 10 of Proposition 5.2.) b) The obstruction from cobordisms of simple type.
At this point, the proof of Theorem 1.3 must diverge from the proof of Theorem 2.9 in [7] because the restriction homomorphism H2(Z) -+ H2(Ez) will not generally be surjective. (Use real coefficients here and throughout this section.) Thus, the second part of Lemma 4.2 in [7] can not be invoked. This failure of surjectivity obstructs the extension of f!lz to Z - Ez. This extension obstruction will be studied by using the fact that restriction to the Z2,j defines isomorphisms H2(Z) ~ ffijH2(Z2,j) and H2(Ez) ~ ffi j H 2(Ez2 .;). (Meyer-Vietoris proves these assertions.) These direct sum decompositions imply that the obstruction to extending Wz over Z - Ez can be understood by restricting attention to Z2 - EZ2 and even further, by restricting attention to
Z2,j'
More precisely, the obstructions to extending I",lz can be
CLIFFORD HENRY TAUBES
452
understood by restricting attention to the very special case of a cobordism of simple type (as in Definition 2.2). With the preceding understood, agree, for the remainder of Section 11, to restrict attention to a particular cobordism of simple type. Simplify notation by using W now to denote this simple type cobordism. Then, Z c W x W and Ez C Z are defined accordingly. With Z as just redefined, note that the extension obstruction may well exist because rank(H2(Ez» ~ 10 while rank(H2(Z» = 4. Indeed, Lemma 2.3 describes H2(Z)(~ JR4), while Meyer-Vietoris with Proposition 5.2 find
(6.1) H 2(E z ) ~ H
=
H2(~Z) E9
H2(EL) E9 H2(ER) E9 H2(E_) E9 H2(E+).
In fact, the restriction map from Z to Ez maps H2(Z) injectively into H2(EL)E9
H2(ER)' c) Analyzing the obstruction.
Let W be a cobordism of simple type and let Z C W x W and let I: z C Z be defined accordingly. Let N z C Z be a regular neighborhood of I:z. Introduce (6.2)
ii : H2(Z - Ez) ~ H2(Nz - I:z) ii : H2(Nz) ~ H2(Nz - I:z)
and
to denote the pull-back homomorphisms. One can conclude from the MeyerVietoris exact sequence that (6.3) and the purpose of the subsequent arguments is to prove PROPOSITION 6.1. Equation (6.9) can be solved with a closed 2-/orm f3 on N z which obeys f3 A f3 = 0 and which vanishes near az .
=
Remark that the lemma implies that Wz !!lz - iif3 extends over Z - Ez (as a in (6.3)) and it obeys Wz A Wz A Wz = 0 near Ez as required. d) Strategy for the proof of Proposition 6.1.
The proof of Proposition 6.1 starts with the remark that the various framings that were introduced in the construction of Wz can be used to construct a homomorphism (6.4)
with the property that the composition of j with i2 (the dual of ii in (6.2)) gives the identity. Indeed, each of ~z, E L,R and E4± have natural trivializations of their normal bundles. And, these trivializations can be used to push-off the generating cycles for the homology groups in question. (For EL,R, see the proof of Lemma 4.4, and see Assertion 9 of Proposition 5.2 for E4± .) In this regard, note that an application of Meyer-Vietoris shows that the dimension 2 homology
453
METABOLIC COBORDISMS
of Ez can be represented by submanifolds in dz, EL,R and in the smooth parts of E±; and these submanifolds can be assumed to be disjoint from E± n dz and from EL,R n dz. The homomorphism j has the property that (6.5)
(wz,j(·)}
= o.
(This is because j is defined by the same homotopy class of normal framing which is used to define wz.) Put (6.5) aside for the moment to consider the composition (6.6) which will be denoted by t'. (The arrow i 1 in (6.6) is induced by the inclusion.) Define Q C H 2 (Ez) by the exact sequence (6.7)
0 ~ Q ~ H2(Ez) ~ ker(t')* ~ O.
Note that the restriction induced monomorphism H2(Z) ~ H 2(E z ) factors through Q. H the quotient Q/H2(Z) is zero, then it follows from (6.7) that (6.3) can be solved with f3 == o. H the quotient Q/ H2 (Z) is one dimensional, and if a generator can be represented by a form f3 with {3" (3 0, then Proposition 6.1 follows. Thus, the proof of Proposition 6.1 will proceed with a proof that the dimension of Q/ H2 (Z) is one or less. The proof will end by finding a generator (when dim(Q/H 2(Z)) = 1) which is represented in Q by a form with square zero (see (6.10), below). By the way, the following lemma will be the principle tool for finding closed forms with square zero:
=
LEMMA 6.2. Let X be an oriented ..I-manifold, and let ReX be a compact, oriented, embedded sur/ace. Suppose that R has zero self-intersection number. Given an open neighborhood 0 C X of R, there is a closed 2-form J.l. with J.l. " J.l. = 0 which is supported in 0 and which represents the Poincare dual to R in H~omp(X).
Proof. The surface R has trivial normal bundle. Use this fact as in (6.12) of [7] to define a fibration from a neighborhood of R in X to the unit disk in IR2 which sends R to the origin. Use such a map to pull-back from said unit disk a 2-form with compact support in the interior and with total mass equal to one. Set J.l. equal to this pull-back. 0 e) The dilllension of Q / H2 (Z).
Here is the answer to the dimension question: LEMMA
6.3. The dimension of Q / H2 (Z)) is one or zero.
Proof. The inclusion of I:z into Z induces the homomorphism t : H 2 (I:z) --+ Then, the dimension of Q/H2(Z) is equal to the dimension of ker(t)/ ker(t'). H2(Z).
454
CLIFFORD HENRY TAUBES
To prove that ker(t)/ ker(,-/) has dimension 1 or less, consider an integral class u E H 2 (Ez) with t·u = 0, but with t'·u '" O. Since '·U = 0, there is a bounding 3-cycle T C Z. The cycle T is a sum of singular simplices; and these simplices can be chosen to have the following property: Each is a smooth map from the standard 3-simplex into Z which is transversal to each of t:J. z , E L , E R , E4± on the interior of every codimension p = 0,··· ,3 face of the standard simplex. (Thus, the boundary of the standard simplex is mapped into the complimep.t of Ez.) With this understood, it makes sense to speak of the intersection number of T with each of t:J.z, EL, ER, E4±' Note that the intersection number between T and E4± can be assumed to be divisible by the integer N of Proposition 5.2. This can be achieved by replacing u with N . u. Observe now that intersections of T with any of EL, ER, E 4± can be removed by changing T to T', where T' has extra intersections with t:J.z. For example, one can add to T some multiple of [Po x Mo] to remove the intersection points with EL at the expense of adding such points to t:J.z. Likewise, adding to T multiples of [S3]bl_ will remove intersections with E 4- and add intersections with t:J.z. Note that all of T'S intersections with E4- can be transferred to t:J.z because E4- is connected, and because T'S intersection number with t:J.z is assumed'divisible by the integer N from Proposition 5.2. The cases for ER and E4± are analogous. (See, e.g., (9.9a,b) in [7].) It follows from the preceding that ker(t)/ ker(t' ) is at most one-dimensional. This is because any element in this quotient can be represented by a closed 2-cycle which bounds T as above, whose intersections with E z lie in t:J.z only. Given two such elements, a non-trivial linear combination would be represented by a closed 2-cycle which bounds T as above with absolutely no intersections with Ez. Such a linear combination would be zero in ker(t)/ker(t' ). 0
f) ker(t). The final step in the proof of Proposition 6.1 is to consider the possible generators of Q/H2(Z) in the case where this group has dimension l. A generator of this group is represented by a class l E H2(Ez) which annihilates the kernel of t / , but which is non-zero on a class u E H 2 (E z ) which is annihilated by '- but not by ,-'. The analysis of l proceeds by considering various possibilities for ker(£)/ ker(£I). Remark that if this group has dimension 1, then it can be represented in ker(£) by some generator. In H 2(t:J. Z ) ffi H 2(EL) ffi H2(ER) sits a two-dimensional subspace of ker(t). An element in ker(£) n (H2(t:J. Z ) ffi H 2(EL ) ffi H 2(E R )) has the form (6.8)
where u. sits in the summand with the corresponding label. Here, each u. pushes forward to W as the same class Uo E H2(W). Then, two generators of the kernel of tin H2(Llz) ffi H 2(EL) Ell H 2(ER) are given by u as above with Uo = [Tll and with 0"0 = [T2 ].
METABOLIC COBORDISMS
455
The remaining generators of the kernel of " can be taken to have the form 1)
0'+1 - O'R1 and 0'+2 - O'L21 0'-1 - O'L1 and 0"-2 - 0'R2, {A±c}.
2)
(6.9)
3)
Here, 0'±l,2 E H2(E±), while O'L1,2 E H 2(EL) and O"R1,2 E H 2(ER) project to H 2 (W) as multiples of [T1,2], respectively. Meanwhile, {A±c} E H 2(E±) is a finite set of classes, and each is represented by a push-off of some Ta,p:I: as described in (4.23). LEMMA
6.4.
The classes in (6.9) are annihilated by,'.
Proof. This follows from Assertion 9 of Proposition 5.2. With the preceding lemma understood, it follows that a generator of ker(t)/ ker(,') is described by (6.8).
g) If 0'
= 0'lJ. -
O'L - O"R is not annihilated by
t'.
In this case, there exists 0" as above with either 0"0 = [T1] or 0"0 = [T2]' For arguments sake, assume 0'0 = [T1]' Let f3lJ.l E H2(ElJ.) be the pull-back by the map 7rL to W of the Poincare dual to [T2]. Then f3lJ.l pairs non-trivially with O"L and so with 0". Let f3R2 E H2(ER) be the pull-back by 7rR of the Poincare dual to [T1]. Note that f3R2 pairs trivially with 0". It follows that there is c E lR. such that f3' f3lJ. 1 + C • f3R2 annihilates the ker(,,) in H 2 (6.z) E9 H 2(E L ) E9 H 2(ER). This f3' will have trivial pairing with the classes in (6.9.1), and it will have trivial pairing with 0"-1 - O"L1 in (6.9.2), but unless c = 0, it will pair non-trivially with 0"-2 - O"R2 in (6.9.2). However, note that the Poincare dual, f3-2 E H 2(E_), to 0"-1 pairs trivially with 0"-1 and non-trivially with 0"-2. And so, there is a real number c' such that
=
(6.10)
f3
=f3lJ.l + c . f3R2 + c' . f3-2
annihilates all of the classes in (6.9.2). Note that f3 will also annihilate the classes in (6.9.3). By appeal to Lemma 6.2, each of f3lJ.1I f3R2, and f3-2 can be represented by a closed form with square zero and with support away from az. (This is because [Td and [T2] are classes with square zero in W.) Furthermore, Lemma 6.2 insures that these forms can be constructed to have disjoint supports. Thus, f3 will vanish near az and have square zero as required. REFERENCES
[1]
[2]
S. Axelrod & I. M. Singer, Chern-Simons perturbation theory, Proc. XXth DGM Conference (New York, 1991, S. Catto and A. Tocha, eds.), World Scientific, 1992, 3-45. _ _ , Chern-Simones perturbation theory. II, J. Differential Geometry 39 (1994) 173-213.
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[8]
CLIFFORD HENRY TAUBES
F. Hirzebruch, W. D. Neumann & S. S. Koh, Differentiable manifolds and quadratic forms, Marcell Decker, New York, 1971. M. Kontsevich, Feynman diagrams and low dimensional topology, MaxPlanck-Institute, Bonn, Preprint, 1993. C. C. MacDuffe, The theory of matrices, Springer, Berlin, 1933. J. Milnor, Lectures on the h-cobomism theorem, Notes by L. Siebenmann and J. Sondau, Princeton University Press, Princeton, 1965. C. H. Taubes, Homology cobordisms and the simplest perlurbative ChernSimons 9-manifold invariant, Geometry, Topology and Physics for Raovl Bott, ed. S.T. Yau, International Press, 1995. E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 117 (1988) 351-399.
