Surveys in Differential Geometry vol.3 (Surveys in Differential Geometry)

Volume III Surveys in Differential Geometry Lectures on Geometry and Topology held at Harvard University, May 3 -5, 19...

Author: Hsiung C.C. | Yau S.-T. (eds.)

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Volume III

Surveys in Differential Geometry Lectures on Geometry and Topology held at Harvard University, May 3 -5, 1996, sponsored by Lehigh University's Journal of Differential Geometry

Edited by C.C. Hsiung S.-T. Yau

International Press

JOURNAL OF DIFFERENTIAL GEOMETRY Editors-in-Chief C.C.HSIUNG

S.T.YAU

Lehigh University Bethlehem, PA 18015

Harvard University Cambridge, MA 02138

Editors

JEFF CHEEGER

H. BLAINE LAWSON, JR. State University of New York Stony Brook, NY 11794

New York University New York, NY 10012

SIMON K.DONALDSON University of Oxford Oxford OX1 3LB, ENGLAND

RICHARD M. SCHOEN Stanford University Stanford, CA 94305

Associate Editors MICHAEL H. FREEDMAN

University of California La Jolla, CA 92093 NIGEL HITCHIN

Mathematics Institute University of Warwick Coventry CV4 7AL, ENGLAND

SHIGEEUb1I MORI

Faculty of Sciences Nagoya University Nagoya 464, JAPAN ALAN WEINSTEIN

University of California Berkeley, CA 94720

Surveys in Differential Geometry: Lectures on Geometry and Topology given at Harvard University, May 3-5, 1996, sponsored by the Journal of Differential Geometry. C. C. Hsiung and S: T. Yau, Editors-in-Chief. ISBN 1-57146-067-5 International Press Incorporated, Boston PO Box 38-2872 Cambridge, MA 02238 All rights are reserved. No part of this work can be reproduced in any form, electronic or mechancial, recording, or by any information storage and data retrieval system, without specific authorization from the publisher. Reproduction for classroom or personal use will, in most cases, be granted without charge. Copyright ® 1998 International Press Printed in the United States of America The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.

Surveys in Differential Geomety Vol. 1: Lectures given in 1990, edited by S.-T. Yau and H. Blaine Lawson

Vol. 2: Lectures given in 1993, edited by C.C. Hsiung and S.-T. Yau

Vol. 3: Lectures given in 1996, edited by C.C. Hsiung and S.-T. Yau

Vol. 4: Integrable Systems, edited by Chuu Lian Terng and Karen Uhlenbeck

Preface This year is the 30'" anniversary of the Journal of Differential Geometry and the 80th birthday of the founder

of the journal , Chuan-Chih Hsiung of Lehigh University, who also initiated the idea of expanding the classical (or traditional) coverage of the subject of "differential

geometry" to be as broad as today's. So this conference was held together with a joint celebration of these two big

events, and these proceedings are published in honor of the celebration.

C. C. Hsiung Lehigh University

S: T. Yau Harvard University

Table of Contents Local Index Theory, Eta Invariants and Holomorphic Torsion: A Survey Jean-Michel Bismut ....................................................................1

Thurston's Hyperbolization of Haken Manifolds Jean-Pierre Otal ........................................................................77

Quasi-Minimal Semi-Euclidean Laminations in 3-Manifolds David Gabai ............................................................................195

Embedded Surfaces and Guage Theory in Three And Four Dimensions P. B. Kronheimer ....................................................................243

The Geometry of the Seiberg-Witten Invariants Clifford H. Taubes ..................................................................299

Local index theory, eta invariants and holomorphic torsion : a survey by

JEAN-MICHEL BISMUT

Departement de Mathematique Universit6 Paris-Sud URA D1169 du CNRS Batiment 425 91405 ORSAY Cedex FRANCE [email protected]

Abstract. The purpose of this paper is to review various results related to the local families index theorem, eta invariants, and the Ray-Singer holomorphic torsion.

Acknowledgments. The support of Institut Universitaire de France (IUF) is gratefully acknowledged.

Also the author is indebted to Jeff Cheeger and Paolo Piazza for very helpful comments.

2

Local index theory, eta invariants, and holomorphic torsion

The purpose of this paper is to survey various recent developments in local index theory, including applications to eta invariants, Quillen metrics on determinant bundles, and analytic torsion forms. In this refined index theory, the construction of secondary objects plays an important role. On the analytic side, these will be the forms (which are extensions of eta invariants), analytic torsion forms (which extend the Ray-Singer torsion). These objects refine on the construction of the index bundle [A], [AS2]. On the geometric side, they include Chern-Simons forms [ChSi], Bott-Chern forms [BoCh] and Bott-Chern currents [BGS4,5], which refine on the classical cohomological objects of index theory like A, Td, ch. The ultimate purpose of the new theory is to relate the analytic secondary invariants to corresponding geometric secondary invariants. This line of thought is most clearly illustrated by the Riemann-Roch formula in Arakelov geometry of Gillet-Soule [GS3,4], which applies to arithmetic varieties. In its simplest form, it evaluates the artihmetic degree of a determinant bundle (whose evaluation involves Quillen metrics at places at infinity) in terms of integrals of arithmetic characteristic classes [GS1,2) (whose construction involves Bott-Chern classes and Bott-Chern currents at places at infinity).

Needless to say, we will always work here in the context of real or complex geometry. Still the idea that the above objects fit "naturally" in an algebraic context has been a powerful motivation for their development. Let us now briefly discuss in more detail the content of this survey. As we said before, this paper is organized around the local index theorem [P1,2], [Gi1,2], [ABoP]. Let Z be a compact even dimensional Riemannian oriented spin manifold, let Dz be a Dirac operator acting on smooth sections of the twisted spinors STZ 0 . The Atiyah-Singer index theorem [AS1]

asserts that the index Ind (Di) E Z of D+ (which is Dz restricted to twisted positive spinors) is given by (0.1)

Ind (D.ZF) =

Jz

A(TZ) ch(g)

where the right-hand side is an integral of a characteristic class, which is a cohomology class.

Let Pt (x, y) be the heat kernel for exp(-tDZ,2). The Mc-Kean Singer formula [McKS] says that for T > 0, (0.2)

Ind (DMZ,) = Trs [exp(-tDZ,2)]

= f ns

[I't(x, z)] doz(es)

(in (0.2), Mrs is our notation for a supertrace, which is a graded trace). The local index theorem, conjectured in [McKS] and proved in [P], [Gil,2], [ABoP], asserts that as t -; 0, we have "fantastic" cancellations in Tr6 [Pt (x, x)] (which means that as t -+ 0, Tre [Pt (x, x] is non singular),

and that (0.3)

Trs [Pt(x,x)]dvx(x)-+{A(TZ,VTZ)Ch (C,V)}

Jean-Michel Bismut

3

where in (0.3), the corresponding characteristic forms in Chern-Weil theory are calculated using the Levi-Civita connection OTZ on TZ, and the given connection V on . Of course, from (0.2), (0.3), we recover (0.1). In the above form, the local index theorem was used in [ABoP] to give a new proof of the Atiyah-Singer index theorem. In [APS1], Atiyah-Patodi-Singer developed an index theory for manifolds with boundary. If Z is a compact even dimensional oriented spin manifold with boundary, the index problem on the Dirac operator Dz on Z imposes global boundary conditions on 8Z. The index formula of [APS1] takes the form

Ind(D+)= f

(0.4)

7D6a (s)

is a meromorphic function of S, which is calculated in In (0.4), terms of the spectrum of a Dirac operator DSZ on the boundary 8Z . The quantity 3IDe2 (0) is called a reduced eta invariant. To establish (0.4), the

local index theorem plays an essential role. In fact, the first term in the right-hand side of (0.4) is the integral of a closed differential form.

Formula (0.4) is quite important. In effect, f A(TZ, VTZ) Ch (f' VI) is 2

_D&Z (0)

a Chern-Simons invariant. On the other hand, is a global spectral invariant of 8Z, which is a prototype of the analytic secondary invariants VDez (0) which will be considered later. Then (0.4) implies that mod(Z), is equal to a Chern-Simons invariant. This is a simple prototype of a refined index theorem. Such a theorem was formulated first in the context of differential characters by Cheeger and Simons [CSi]. Section 1 is devoted to a short exposition of local index theory and eta invariants.

Let now it : X -+ B be a fibration with compact fibres as before. Let (Dn )bES be the corresponding family of Dirac operators. In [AS2], Atiyah and Singer have shown how to associate to this family an (analytic) index bundle Ind (D.Z}) E K°(B). They also defined a topological index, and they proved a corresponding families index formula. When mapping K°(B) in H(B, Q) by the Chern character ch, the formula of [AS2] takes the form (0.5)

ch(Ind (D+)) = ir. [A(TZ) ch(e)] in H(B, Q).

The local index theorem of [B2] refines on the right hand-side of (0.5), by replacing it by an explicit geometrically constructed differential form ir.[A(TZ,VTZ) Ch (4, Vi)]. When Ind(DII) is a honest vector bundle, the theory of [B2] replaces the analytic index by an analytically constructed differential form ch(kerDZ, VkerD5,u). Then an essential by-product of [B2], [BeV], [BeGeV], [BC1] is the construction of an explicit differential form i on B such that (0.6)

d = 1r. [A(TZ,VTZ) ch(e,Vt)] - ch(kerDz, V1--DZ,u)

4


Quillen's superconnections [Q1] are an important tool to obtain the above results. Superconnections provide a useful extension of Chern-Weil theory to Z2-graded (and possibly infinite dimensional) vector bundles. In [B2], VC)] is produced by refining the local index the form ir.[A(TZ,VTZ) theorem (0.3) in a relative context, hence its name of a local families index theorem.

Equation (0.5) has been extended in [BC2,3], [MeP1,2] to a families index theorem for Dirac operators on manifolds with boundary, by using either the techniques of Cheeger [C1,2,3] on manifolds with conical singularities, or the b-calculus of Melrose [Me]. The important concept of a spectral section [MeP1,2] has emerged in this context. If (E, gE) is a holomorphic Hermitian vector bundle on a complex manifold, E is naturally equipped with the holomorphic Hermitian connection DE. We will denote by Td(E, gE) the form Td(E, DE). It is a sum of forms of type (p,p). If it : X -+ S is a holomorphic submersion with compact fibre Z, if T Z is a holomorphic is equipped with a Hermitian metric gTZ, and, if

Hermitian vector bundle on X such that R1r.e is locally free, a natural equation related to (0.6) is (0.7)

2i1-T =

-v. [Td(TZ,gTZ) ch(e,g£)]

is the metric on R1r.e obtained from gTZ,g{ by Hodge theory where along the fibres. Assume that X is Kahler and let wX be the corresponding Kahler from.

When gTZ is obtained from wX by restriction to TZ, the forms T(wX,g{) were constructed in [BGS3], [BK] and were called analytic torsion forms, because in degree 0, Ti°>(wX,g£) coincides with the Ray-Singer analytic torsion [RS] of the corresponding Dolbeault complex. In fact of special interest are (1) and TO). In [BF1,2], (BGS3], i-7(1) appears as a connection form on the determinant line bundle (det ker DZ)-1, and TO) (wX, gC) is the natural correction to the obvious Hodge metric on the line bundle (detR7r.g)'1 introduced by Quillen [Q2] to construct the

Quillen metric on (detRlrg)-1. When suitably interpreted, in degree 2, equations (0.6) and (0.7) appear as curvature theorems for natural connections on the line bundles (detker Dg)-1 and (det Ri.t)-1. Of course, one of the points of [BF1,2], [BGS3] is that such curvature theorems still hold without any assumption on kerDZ or Rn.g. Objects like j and T(wX,g{) are secondary invariants which refine on the family index theorem of Atiyah-Singer [AS2] or on Riemann-RochGrothendieck. These last theorems are naturally functorial, in the sense they are compatible to the composition of maps. It is natural to ask whether or T (wX , g9) have related functorial properties. A first simple question is to ask how T (wx, 9e) depends on wX, g£. It was shown in [BK] that T (wX, g£) depend on wX, g{ via obvious Bott-Chern

classes [BoC], [BGS1]. Another question is related to the behaviour on

or T (wX, g{) by the composition of two submersions. This question was solved in [BC1], [BeB],

Jean-Michel Bismut

s

[D], [Ma], using the idea of adiabatic limits. Let us give an elementary application of this idea. In fact, if

0->L--3M-+N-+0 is an exact sequence of holomorphic vector bundles, and if 9M, 9N are Hermitian metrics on M, N for e > 0, set

Then if Q is any characteristic polynomial, one can easily show that if gL is the metric on L induced by 9M, (0.10)

Q(M,9M)-

Q(L,gL)Q(N,gN).

Now (0.10) can be applied to the exact sequence

0-*TZ-*TX -*7r*TS-->0. Following a terminology introduced by physicists [W], studying geometric or spectral objects depending on a metric gTx = gTx + ir*gTS as a -i 0 E is called passing to the adiabatic limits. As was observed in [BF2], there is an analogue of (0.10) for the LeviCivita connection of a fibered manifold. On the other hand, the Leray spectral sequence for the de Rham or Dolbeault complexes of a fibered manifold makes the left-hand side of RiemannRoch-Grothendieck compatible with the composition of submersions. The behaviour of i and T (wx, g{) under composition of submersions

was obtained by adiabatic limit techniques. In [BC1], [D], [BerB], the case where the last submersion maps to a point was considered, and the results were expressed as results on the adiabatic limit of eta invariants, or on Quillen metrics. In [Ma], corresponding results were obtained for the composition of arbitrary holomorphic submersions. The above results also rely on an observation of Mazzeo-Melrose [MazMe] relating the adiabatic limit of the spectrum of the Dirac operator to the Leray spectral sequence.

Similar questions can be asked in the case of embeddings. We will explain the problem in the context of complex geometry. Let i : Y - X be an embedding of complex manifolds. Let 71 be a vector bundle on Y. If X v) of holomorphic is projective, there is a resolution of i*i7 by a complex vector bundles on X. Then by definition, the direct image fit) E K(X) is given by

i!7 _ [C] in K(X),

(0.11)

the point being to show that [£] does not depend on C. Now by RiemannRoch-Grothendieck, (0.12)

ch(i!17) = i* (Td-'(Nylx) ch(n)) in Hm'(X, Q),

6


so that (0.13)

ch(C) = i. (Td-1(Ny/x) ch(rl)) in Heven(X, Q)

Let gf, gn, gNs-lx be Hermitian metrics on g, 77, Ny/x. By analogy with (0.7) it is natural to ask whether one can refine (0.13). Namely one can ask for the existence of a current T(g, g£) on X such that (0.14)

30T(0,9{) = Td-1(Ny/x,g"7') th(11,9°)Sy - ch(C,9£)

The current T(e, gC) has been constructed in [B3], [BGS4]. Again it is natural to study the compatibility of the currents T (C, g£) to the composition of embeddings. This has been done in detail in [BGS5]. Having now constructed objects T(wx,gC) and associated to a

submersion or an embedding, which are compatible to the composition of submersions or of embeddings, the last obvious final step is to study the compatibility of these objects to the composition of an embedding and a submersion. This has been done in [BL] when the submersion maps to a point, and in [B5,6] in the general case. In [BL], the main result is formulated naturally in terms of Quillen metrics. In the proof of [BL], [B5,6], a mysterious secondary invariant associated to a short exact sequence of holomorphic Hermitian vector bundles appeared, whose construction was somewhat puzzling. A preliminary step for the proof of [BL), [B5,6] was the explicit evaluation of this class in [B4]. The most elaborate formula in [B5,6] expresses a combination of analytic torsion forms as a sum of integrals along the fibre of analytic torsion currents and of Bott-Chern classes. This indicates that the refined objects introduced above fit in a refined Riemann-Roch algebra. As explained before, Gillet and Soule [GS3] have explained how, in the case of arithmetic varieties, these results can be used to prove a Riemann-Roch-Grothendieck formula in Arakelov geometry. They have proved such a formula in (GS4] for the first Chern class, and their proof for higher Chern classes is pending. This paper is organized as follows. Section 1 is devoted to the local index theorem and the eta invariant. In Section 2, we review various results on the local families index theorem and the j forms. Finally, in Section 3, we consider analytic torsion forms and analytic torsion currents. Part of the material contained in this survey already has been reviewed in [B7].

Jean-Michel Bismut

7

1. The local index theorem and the eta invariant. In this Section, we review a few well-known results on the local index theorem for Dirac operators closed manifolds, on the index theorem for manifolds with boundary, and we also give related results on eta invariant. This Section is organized as follows. In a), we state the local AtiyahSinger index theorem for closed manifolds of Patodi [P1,2], Gilkey [Gil,2], Atiyah-Bott-Patodi [ABoP]. In b), we state the Atiyah-Patodi-Singer in-

dex theorem on manifolds with boundary [APS1] and we introduce the associated eta invariant. In c), we give the formula for the signature of a manifold with boundary obtained by Atiyah-Patodi-Singer [APS1]. In d), we describe result by Cheeger [C1,2] and Chou [Ch] on the index theorem on manifolds with conical singularities. In e), we recall the result by Cheeger [C3] on the L2 signature of such manifolds. Finally in f), we review briefly the approach by Melrose [Me] to the Atiyah-Patodi-Singer index theorem, using the b-calculus. For a detailed approach to the local index theorem, we refer to BerlineGetzler-Vergne [BeGeV].

a) The local Atiyah-Singer index theorem for Dirac operators on closed manifolds. Let Z be an even dimensional compact oriented spin manifold. Let gTZ be a Riemannian metric on TZ. Let STZ = S+Z ® S_TZ be the Hermitian Z2-graded vector bundle of (TZ,9TZ) spinors. VSTZ be the Let VTZ be the Levi-Civita connection on (TZ,gTZ). Let VC) be a Hermitian vector connection induced by VTZ on STZ. Let VSTZ®f be the obvious bundle equipped with a unitary connection. Let connection on STZ ®g. Let c(TZ) be the bundle of Clifford algebras of (TZ, gTZ). It is generated over R by 1, X E TZ, and the commutation relations (1.1)

XY+YX = -2(X,Y) .

Then STZ is a c(TZ)-Clifford module. If X E TZ, we denote by c(X) the action of X E c(TZ) on STZ. Then c(X) acts like c(X) ®1 on STZ ®t. Let DZ be the Dirac operator associated to (9TZ, Va). If e1,. .. , en is an orthonormal basis of TZ, n

(1.2)

DZ =

c(e;)De{TZ®f

Then Dz is an odd operator, i.e. it exchanges C°°(Z, S.TFZ0) and C- (Z, S_TZO

g). Also Dz it is a first-order self-adjoint elliptic operator. Let Dt be the restriction of Dz to C°O (Z, Sf Z 0 ), so that (1.3)

DZ

=[D+ z

DZ1 0

1.

Since D.Z. is elliptic, it is a Fredholm operator. By definition, the index Ind (D.Z.) E Z is given by (1.4)

Ind (DZ) = dimker(D.ZF) - dim ker(D?).


8

Put x/2

A(x) =

sinh(x/2)

(1.5)

Td(x) =

1 - e_y We identify A and Td with the corresponding multiplicative genera, the Hirzebruch genus and the Todd genus. Similarly the Chern character ch is the additive genus associated with the function exp(x). If 7r : F -+ Z is a complex vector bundle, and if P is a real invariant polynomial, let P(F) E Heen(Z,R) be the corresponding characteristic class. If OF is a connection on F, and RF = OF,2 is its curvature, we denote

by P(F, VI) the closed even form P in cohomology.

(__) on Z, which represents P(F)

Then, by the Atiyah-Singer index theorem [AS1],

Ind (D+) =

(1.6)

If E = E+ ® E_ is a Z2-graded vector space, let r = ±1 on Et define the Z2-grading. If A E End (E), we define its supertrace Trg [A] by (1.7) Tr8 [A] = Tr[rA]. The algebra End (E) is naturally Z2-graded, the even (resp. odd) elements commuting (resp. anticommuting) with r. If A, B E End (E), we define the supercommutator [A, B] by the formula

[A,B] = AB - (-1)degAdesaBA. By [Qi], if A, B E End (E), (1.8)

Trg [[A, B]] = 0.

(1.9)

Observe that since Dz is elliptic, for t > 0, exp(-tDZ.2) is trace class. We then have the formula of Mc Kean-Singer [McKS].

Proposition 1.1. For t > 0, Ind (D+) = Tr.

(1.10) PROOF :

[exp(-tDZ,2)]

By spectral theory,

lim Tr$ [exp(-tDZ,2)I = Ind

(1.11)

t ++00

Also we have the "Bianchi" identity [DZ, DZ,2] = 0.

(1.12)

Using (1.9), (1.12), we get (1.13)

at ']

[exp(-tDZ°2)]

Trs [DZ°2 exp(-tDZ 2)] _

- z Tr. [[DZ,DZexp(-tDZ,2)]] = 0. Note that in the last steps of (1.13), one should express the various quantities in terms of smooth heat kernels to justify the use of (1.9).

Jean-Michel Bismut

9

Let Pt(x, y) be the smooth kernel of exp(-tDz 2) with respect to the Riemannian volume dy . Then by (1.10), (1.14)

Ind (D+) = fz Tr8 [Pt(x, x)] dx.

Put n = dim Z. By general results on elliptic differential operators, we know that for x E Z, as t -+ 0, P

(1.15)

Tr$ [Pt(x, x)] = E

a-n/2+k(x)t-n12+k + 0.(t-n/2+P+i)

,

k=O

and Ox(t-n/2+P+1) is uniform in x E X. Moreover the aj(x) only depend on the complete symbol of Dz near x. By (1.14), (1.15),

fa1(x)dx=0 (1. 16)

forj960,

Lao= Ind (D) .

Let VTZ be the Levi-Civita connection on (TZ, 9TZ). In [McKS], Mc Kean and Singer conjectured that "fantastic cancellations" should occur in (1.16) so that for the considered operator Dz,

of = 0, j < 0, (1.17)

ao(x) = {A(TZ,VT-) ch(C, V{)}mom

Needless to say, (1.16), (1.17) would imply the index formula (1.6).

Theorem 1.2. Equation (1.17) holds. There are two kinds of proofs of (1.17). The algebraic proofs of Patodi [P1,2], Gilkey [Gil,2], Atiyah-Bott-Patodi

PROOF :

[ABoP] describe explicitly the aj (j < 0) as polynomial functions of the metric gTZ, the connection V and their derivatives. For j < 0, arguments of Gilkey show that there are no such polynomials other than 0. Also ao is shown to be a universal combination of certain Chern-Weil forms. One then only needs to verify the identity (1.17) for ao on sufficiently many examples,

given by the Pn(C). . The direct proofs of Getzler [Gel,2], Bismut [Bi], Berlin-Vergne [BeV]. These proofs were stimulated by arguments by Alvarez-Gaume [Al], using functional integration, which suggested that there should be some explicit algebraic mechanism forcing the above local cancellations. The proof of Getzler [Ge2] uses a powerful restating technique on the

Clifford algebra. It is explained in detail in [BeGeV, Chapter 4]. Of particular importance is the Getzler operator [Gel,2], [BeGeV, Proposition 4.19], which appears when doing the Getzler rescaling on the Clifford algebra. This operator is an harmonic oscillator, and its heat kernel produces the genus A in (1.17). Theorem 1.2 is often called the local index theorem for Dirac operators.

0

Local index theory, eta invariants, and holoiuorphic torsion

10

b) The Atiyah-Patodi-Singer index theorem.

Let Z be a compact manifold of dimension n with boundary Y = 8Z.

Let OZ x [0,1] be a tubular neighborhood of 8Z in Z, u being the inward normal coordinate. Let gTZ be a Riemannian metric on TZ, which is product

near 8Z, i.e. gTZ = 9TOZ + IduI2

(1.18)

Let E, F be complex Hermitian vectors bundles on Z, which, near Y,

are pull-backs of vector bundles on Y. Let D : C°°(Z,E) -+ C°°(Z,F) be an elliptic first order differential operator. Let a : Ely -4 Fly be a be an elliptic first bundle isometry. Let A : C°O(Y,Ely.) -+ order differential operator, which is self-adjoint with respect to the obvious L2 Hermitian product. We assume that on 8Z x [0,1],

D=aI L+A) .

(1.19)

Let P>o (resp. P 0, let At be the superconnection

At =VE+v`V.

(2.14)

We extend 'p to a mapA(T*B)®R,C--A(T*B)ORC. DEFINITION 2.5. Put

at ='p Trs [e%p(-A')] (2.15)

Qt =

rBAt

1

(2i,) 1/210

21

Trs L at exp(-At )J

One verifies easily that, the forms at, Rt are real. Also by Theorem 2.4,

at is closed, and [at] = ch(E). By an obvious extension of Chern-Simons theory to superconnections, 8t at

= -d,8t .

ao = ch(E, VE). Assume that ker V has locally constant dimension, i.e. ker V is a Z2graded smooth vector subbundle of E. Then [E] = [ker V] in Ko(B) .

(2.18)

Let V1 rv be the orthogonal projection of VE on ker V. Now we state a result by Berlin-Vergne [BeV, Theorem 1.9], [BeGeV, Theorems 9.2 and 9.7].

Theorem 2.6. As t - +oo, at = ch (ker V, Vk't') + O (2.19)

Ot=0\ t3/2 1 / DEFINITION 2.7. Put (2.20)

n1=

f 0

f3tdt.

f 1


22

Theorem 2.8. The form

dij = ch(E, DE) - ch(ker V, Vker V )

(2.21) PROOF :

is real and odd, and moreover

This follows from (2.16), (2.17), (2.19).

O

DEFINITION 2.9. We will say that (E, gE, DE, V) splits if E = ker V ®Im V,

if the connection V E preserves ker D and Im V, and V is a unitary odd section of End (Im V) preserving VIm V

Now we state a result characterizing the form i uniquely. This is an obvious analogue of corresponding results for Bott-Chern classes in [BGS1].

Theorem 2.10. There exists a unique way to associate to (E, gE, VE, V ) an odd form in C°°(B,Aodd(T*B))/dCoo(B,Aeven(T*B)) having the following three properties

a) i is functorial. b) If (E, 9E, yE, V) splits, then i = 0. c)

The following identity holds dr1= ch(E, DE) - ch(ker V, Vker V )

(2.22) PROOF :

If (E, gE, DE, V) splits, then At = DE,2 + tPIm V

(2.23)

and so (2.24)

It follows that if i is taken as in (2.20),

q= 0.

(2.25)

So the form it of Definition 2.9 has properties a), b), c). To establish uniqueness, one verifies easily that over S x [0,1], one can deform (E, 9E, V E, V) at

s = 0 into a split object at s = 1. Let (E, gE, yE, V) be the corresponding object on S x [0,1]. Then if ij is taken as in our Theorem,

(2.26) q = f (ch E, DE) - ch(ker V, ykerV )) modulo coboundaries, 0,11

which characterizes the class of it uniquely.

J

0

Jean-Michel Bismut

23

d) Superconnections : the odd case. Let now ir : E -> B be a Hermitian vector bundle on B. 1. Then E 0 C(o) is a Z2Let v be an odd variable such that graded vector bundle, and End (E) ® C(o) is a Z2-graded algebra. Let Tr° be the functional from A(T*B)®(End(E) ® C(o)) into A(T*B) such that if w E A(T*B), A E End (E), Tr°(wA) = 0. (2 27) Tr°(wAv) = w Tr(A). Again Tr° vanishes on supercommutators. DEFINITION 2.11. A superconnection is an odd differential operator acting on C°°(B, A(T*B))®(E ® C(a)) such that (2.6) still holds Let A2 be the curvature of A. Then A2 E COD(B, (A(T*B)®(End (E)(& C(Q))evea).

Now we have the result of Quillen [Q1, Section 5].

Proposition 2.12. The odd form (2.28) a = Tr° [exp(-A2)] is closed and exact.

By proceeding as in the proof of Theorem 2.4, we see that a is closed. Moreover, we find that [a] does not depend on A. Also if A = VE, a = 0. The proof of our Proposition is complete. PROOF :

Let 7r : (E, gE) -3 B be a complex Hermitian vector bundle. Let VE be a unitary connection on (E,gE).

Let V E C°°(B, End (E)) be self-adjoint. For t > 0, let At be the superconnection

At=VE+V Vo.

(2.29)

DEFINITION 2.13. Put

at = (2i)1/2 '1 ° [exp(-A2)] (2.30)

,6t =

1p Tr V, 7r

21

exp(-At) J ° Lt &

As before, the forms at,13t axe real and by Proposition 2.12, the form at is closed and exact. Again as in (2.16),

(2.31)t = -d/3t. Clearly (2.32)

ao = 0.

Assume that ker(V) is of locally constant dimension, so that ker V is a vector subbundle of E. Let Vker(v) be the orthogonal projection of VE on ker V.

By [BeV, Theorem 1.9], [BeGeV, Theorems 9.2 and 9.7], we have


24

Theorem 2.14. As t --* +oo,

at 0\/ (2.33)

I

Qt=C(i). DEFINITION 2.15. Put

rl =

(2.34)

r+o,Otdt. 0

Theorem 2.16. The even form it is real and closed. PROOF :

By (2.32), (2.33)

ao=0,a00=0.

(2.35)

Using (2.31), we find that

dvl=0.

(2.36)

0 Let E>0 (EoeEo) - ch(Eo, -1 on Eo a VEo®E 0, let At be the superconnection on H,

At = pH," + ADZ - c(T)

(2.55)

4Vt_

Observe that At is fibrewise elliptic, so that exp(-Ai) is fibrewise trace class.

DEFINITION 2.21. Put

at = W Tr. [exp(-A?)] 1 (8A 1 At = (2i7r)1/29 Trs [ 8t exp(-At)J

(2.56)

Now we have the local families index theorem of [B2, Theorems 3.4, 4.12 and 4.16].

Theorem 2.22. The forms at and Qt are real and the form at is closed. The cohomology class [at] of at is constant and (2.57) [at] = ch(Ind DZ) in Heven(B, Q) Also (2.58)

Finally as t -+ 0,

at = ir.. (A(TZ,VTZ) (2.59)

o£)) + 0(t),

at=0(1) REMARK 2.23. The fact that at is closed follows from the arguments in Theorem 2.4. The most difficult result is (2.59). In fact the main point of [B2] was to produce the "right" superconnection At such that a result like (2.59) would hold.

Observe that, in general, (2.59) does not hold for the "simpler" superconnection OH," +,JDZ. Needless to say, from [B2, Theorem 2.22], we recover the cohomological form of the families index theorem of Atiyah-Singer [AS2] given in (2.2). Still our proof [B2] is completely local on the basis B, hence the fact Theorem 2.22 is called a local families index theorem. Still to prove (2.59), we use the fibrewise heat kernel for exp(-At ), and prove a corresponding convergence result which is local on the fibre Z.

Let gTB be a metric on TB. For e > 0, let gTX be the metric on TX =THX ®TZ, (2.60)

gTX =

7r

s9TB ®9TZ

Assume that X and B are compact and that B is even dimensional, oriented and spin. Then X is also even dimensional, oriented and spin. Let STB be

the vector bundle of (TB,gTB) spinors. For e > 0, let De be the Dirac operator acting on C°D(X, STX (& l;), associated to (gE X, Vi).

Let Pt (x, x') (x, x' E X) be the smooth heat kernel for exp(-tDx,2) associated to the volume dv(x'). In view of Theorems 1.2 and 2.22, it is natural to ask whether, given t > 0, as e -+ 0, Zr$ [Pt (x, x)] has a limit.


28

Let Qt(x, x') be the smooth fibrewise kernel of exp(-At ). Following a terminology introduced by physicists [W], the idea of studying the limit of certain quantities ass -+ 0 is called passing to the adiabatic limit. A first result in that direction is as follows [B2, Theorem 5.3].

Theorem 2.24. For t > 0, (2.61)

lim

[Qt(x,x)]A(TB,VTB)}II,ax(B)

[Pt (x,x)] _ {w Tr8

In [B2], the idea is to view (2.61) as a consequence of the local index theorem over B with coefficients in the infinite dimensional Z2-graded vector bundle H. PROOF :

Put (2.62)

DH =

EC(fa) (f.)

(Vf.0

+ 21 div(fQH,) )

.

Then a simple computation [BC, eq. (4.26)] shows that (2.63)

DE = DZ + eDH - eXc(f«)c(fi)c(T(f , f)).

At least formally, (2.61) follows easily from local index theoretic techniques over B. 0

Let V'E X be the Levi-Civita connection on (TX, gTX ). Put SE

(2.64)

= v'TX - VTX

Then by (2.50), we find that

PTZSE = PTZS,

(2.65)

PT"XSE=EPTHXS.

From (2.64), (2.65), we find that as c -> 0, the connection VTX has a limit. More precisely, (2.66)

VITX -VTX +PTZS.

From (2.66), we find [BF2, eq. (3.196)] that as c -+ 0, (2.67)

A(TX, V'E X) -+ A(TZ,VTZ)xr*A(TB,

VTB).

Now while A(TX, V'E X) ch V{) appears naturally when applying the local index Theorem 1.2 to the operator DX over X, A(TZ, VTZ) ch(4, V) appears naturally in the local families index theorem stated in Theorem 2.22.

Jean-Michel Bismut

29

Given Theorem 2.24, the "local in the fibre" version of the local families index Theorem of [B2] stated in Theorem 2.22 is just the assertion that the following diagram commutes (2.68)

. {A(TX, V'9 X)

Trs [Pt (x, X)]

t-*O

ch(,, V£}max(X)

{{A(TZ,VTZ)

{p Trs [Qt(r, x)}

A(TB, VTB)}max(B) t,o

Vt)}max(Z) A(TB, VTB)}max(B)

Needless to say, the explicit form for At in (2.55) was found by trying to make the above diagram commute by brute force. The comparison of formulas (2.55) and (2.63) for At and DX provides overwhelming evidence that At is the "right" superconnection. Finally observe that given Theorems 1.2 and 2.24, and also (2.66), a proof of Theorem 2.22 can be given, which makes the commutativity of the above diagram a tautology, by showing that the convergence as t --* 0 in the upper row is uniform in e E]0,1].

Assume that kerDZ is of locally constant dimension. Then kerDZ) is a smooth subbundle of H. Let V'' Dz ,1L be the orthogonal projection of VH° on kerDZ. The we have the following result of Berline-Getzler-Vergne [BeGeV, Theorems 9.19 and 9.23], which extends Theorem 2.6 to an infinite dimensional situation.

Theorem 2.25. As t --3 +oo, c,t = ch(ker DZ, Vker Dz,'°) + ® (2.69)

at=o (i). DEFINITION 2.26. Let

f 1

be the odd form on B r+oo

(2.70)

= J0

f3tdt.

By (2.58), (2.59), (2.69) we get the following result.

Theorem 2.27. The odd smooth form i on B is such that (2.71)

dri =7. [A(TZ,VTZ)

ch(kerDZ, VkerDz,u)


30

REMARK 2.28. From equation (2.71), one deduces easily how, modulo coboundaries, depends on (THX,gTZ,g£). This is because any two sets of such data can be deformed into each other.

f) The determinant bundle. We make the same assumptions as in Section 2 e). Complex lines form a group under the 0 operation. In particular, if A is ®a-1 = C, the canonical a complex line, let A-1 be the dual line, so that A complex line.

If E is a complex vector space, put det E = Ama"E .

(2.72)

If E = E+ ® E_ is a Z2-graded vector space, set (2.73)

det E = det E+ ® (det E-)-'.

DEFINITION 2.29. For b E B, set (2.74)

ab = (detkerDb)

Then in [Q2], Quillen has shown how to glue the \b'S into a honest line bundle A, even though, in general, the dimension of ker Df,b is not locally constant. The idea is as follows. For a > 0, let U. be the open set (2.75)

U. = {b E B,a V Sp(D62)}

.

Let H10,6i be the direct sum of the eigenspaces of Db'2 for eigenvalues µ < a.

Put (2.76)

\10,6i = (det Hio'6i) -I

Then a10,61 is a smooth line bundle on Ua.

Given 0 < a < a', let Hla,a i be the direct sum of eigenspace of Db '2 for eigenvalues p E [a, a']. Set (2.77)

\ia,4 i

= (detH(a 61)'1 .

Then \[a,"] has a canonical nonzero section det D.Z}'i6'6'i which is smooth on v, fl Ua . Also on U6 fl U°', (2.78)

\[O,a') - ANA

®aia,6 ]

.

Since \[6,a'] is canonically trivialized, on U6 fl U6' (2.79)

\10,a'l ^,'\10'61.

Jean-Michel Bismut

31

DEFINITION 2.30. The inverse determinant bundle A is the line bundle which restricts to x10,61 on U°.

By (2.79), we find that for any b E B,

Ab ^ (detkerDb )

(2.80)

By the Atiyah-Singer family index theorem [AS2] (see eq. (2.2)),

c1(A) = -a. [A(TZ) ch(C)J (2)

(2.81)

.

In [Q2], [BF1,2], Quillen and Bismut and Freed have shown how to equip A with a smooth metric 11 j and a unitary connection Da such that (2.82)

c1(,\,0,\) = -7r. [A(TZ, VTZ) ch(e, Ve)] (2)

.

Here we will concentrate on the construction of the imaginary part of the connection VA [BF1,2]. Assume first that ker DZ is of locally constant dimension.

A=

Then (detkerDZ)-1 is a smooth line bundle and

(detkerDZ)-1.

The connection VkerD2,u induces a connection 10' on A. Put 2V = IV + (2.83) +2i,670).

Clearly (2.84)

c1(A,10a) = - [ch(kerDz,

Vk.rDz,a.)] (2)

From (2.84), we deduce that (2.85)

c1(A,2Vx)=-[ch(kerD2,VkerDz,u)1(2)-d

By (2.71), (2.85), we find that (2.86)

1).

JJ

c1(A,2V") = _jr.

The connection V' in [BF1,2] of differs from 1V-\ by an exact real form, so that (2.82) follows from (2.86). The remarkable fact is that even if ker Dt is not of locally constant dimension, in [BF1,2], it is possible to define the connection 20a by formulas similar to (2.83). The idea is to construct over U6 a connection V'10'`] by a suitable modification of (2.83), and to establish that the connections V1(0'°', suitably modified, define a connection VI on A, for which (2.82) still holds.

g) The local families index theorem : the odd case. Now we assume that the fibres Z are odd dimensional. Put Hb = C°° (Zb, (sTZ ® e)IZ.)

(2.87)

Then (D2) is a family of self-adjoint operators acting on H. Take o as in Section 2 d).


32

DEFINITION 2.31. For t > 0, put (2.88)

At = VH,u + \DZa - c(T )a

4f

Again, exp(-At) is fibrewise trace class. DEFINITION 2.32. Put

at = (2i)l/ZcpTrU [exp(-At)] (2.89)

Qt =

WTr°

[

tt exp(-At)]

Now we state a result of [BF2, Theorem 2.10].

Theorem 2.33. The forms at and fat are real, and the form at is closed. The cohomology class [at] of at is constant, and (2.90)

[at] = ch(DZ) in Hodd(B, Q) .

Also

T

8a = -df3t.

(2.91)

Finally as t -> 0 (2.92)

at = 7r, [A(TZ,VTZ)

0£)] + 0(t),

Qt = 0(1) .

Assume now that ker Dz is of locally constant dimension. Then ker Dz is a vector bundle on B. We have the obvious analogue of Theorems 2.14 and 1.25.

Theorem 2.34. As t -* +oo,

(2.93)

DEFINITION 2.35. Let

be the even form f+oo/3tdt.

(2.94)

n=

0

Jean-Michel Bismut

Theorem 2.36. The even form

PROOF :

is such that

d i j = 7r. [A(TZ,VTZ) Ch g, V )]

(2.95)

33

.

O

This follows from (2.91)-(2.93).

h) The odd local families index theorem and the eta invariant. Observe that 1

at°)

(2.96)

2

TT

[D

Z

exp(-tDZ,2)]

By (2.92), ast-+0, (2.97)

$O) = 0(1). 71DZ

Now by (1.25), (2.97) guarantees that the eta function (s) is holomorphic at 0. Note that this result extends Theorem 1.4 to the case where Z does not necessarily bound. This result is also a consequence of [APS3, Theorem 4.5]. By (2.94), (2.96), +00

1

(2.98)

(0) =

2ir

J

t-1/2 Tr [Dz exp(-tDZ,2)] dt.

0

Using (1.25), (2.97), (2.98), we get (2.99)

0l =

Moreover in degree 1, (2.95) specializes to (2.100)

d,-?(O) = Jr, [A(TZ VTZ)

V4)]

In view of (2.99), formula (2.100) gives a local expression for d2rl(0), which,

when the fibres Z bound, can also be derived from the index theorem of Atiyah-Patodi-Singer [APS1]. When B is a point (i.e. in the case of a single fibre), the condition that ker(DZ) is of locally constant dimension is empty. However in general, this

condition is non empty, since it implies that the family (DZ) is trivial in K1(B). Set (2.101)

ADZ (s)

= 2 (7IDZ (s) + dim ker DZ)

Then i7DZ (0) is called the reduced eta invariant of DZ. In [APS3, Section 2], r1DZ (0) is shown to define a smooth function with values in R/Z, and the general form of (2.100) is (2.102)

d,-7D Z

(0) = 7r. [A(TZ, VT-)

V ), (1) .


34

i) The holonomy Theorem. Now we use the assumptions and notation of Sections 2 e) and 2 f).

Let s E S1 -- c8 E B be an oriented smooth curve to B. In [W], Witten raised the question of calculating the holonomy of a connection Da on A in terms of the eta invariant of the odd dimensional oriented compact spin manifold M = 7r-1(C). Let gTX be a Riemannian metric on TX, let 9TB be a metric on TB.

Put

9T X = 9Tx +

(2.103)

1 r*9Ts

Let gTM be the metric on TM induced by gTx on TM. We equip Sl with the non trivial spin structure. Then since TZ is spin, TM inherits an obvious spin structure. Let DM be the Dirac operator on M associated to gT M, VC.

Let if E Sl be the parallel transport with respect to the connection Va along s E Sl --+ c8.

Then we have the following result by Bismut-Freed [BF2, Theorem 3.16] and Cheeger [C4].

Theorem 2.37. The limit as c -+ 0 of CDs' (0) E R/Z exists, and moreover (2.104)

ro=exp(-2iir m17 DM(0))

PaooF : Assume first that c bounds A in B. Then by (2.82) (2.105)

r1 = exp

-2i7r

r

A(TZ, VTZ) ch(¢, 0£)

On the other hand, by the index Theorem of Atiyah-Patodi-Singer [APS1] (see Theorem 1.3), (2.106)

jID"' (0) =

A(TM, gE M) ch(t;', V{) in R/Z. x -,(A)

As in (2.67), one verifies easily that as e -+ 0, (2.107)

A(TM, gT M) -a A(TZ, VTZ).

From (2.106), (2.107), we find that as e -+ 0, (2.108)

i?DM (0) _+

Jir-'(o) A(TZ, OT Z) ch(e, Vf) in R/Z.

By (2.105)-(2.108), we get (2.104) in this case. In general, using (2.102) and (2.107), one finds easily that lira 1IDM (0) E R/Z exists. The main point of e-i0

Jean-Michel Bismut

35

[BF2], [C4] is to extend (2.104) when c does not bound in B. Then if for b E C, D6 is invertible, a direct study of the formula (2.98) for IID (0) by the methods used in the proof of (2.68) shows that as a -r 0, D'

(2.109)

+l

(0)

_+f

from which (2.104) follows easily. When ker DZ is nonzero and not even a vector bundle over c, a non trivial perturbation argument shows that (2.104) O still holds.

REMARK 2.38. Theorem 2.37 is one of the motivations for studying adiabatic limits of eta invariants, when instead of the circle c, the base of the fibration is arbitrary.

j) Adiabatic limits of eta invariants. Assume first that B is an odd dimensional compact oriented spin Riemannian manifold. Let E = E+ ® E_ be a Z2-graded Hermitian vector bundle as in Section 2 c), and let V E End (E) be a self-adjoint section of End odd (E), such that ker V is of locally constant dimension. Let V+ be the

restriction of V to Et. In the sequel, the assumptions of Section 2 c) will be in force. In particular QE = DE+ ® VE- is a split unitary connection on E = E+ ® E_, and Qker v is the orthogonal projection of DE on ker V. Also the odd form q was defined in Definition 2.7.

Let DB,Ef, DB,kerv* be the Dirac operators associated to the above data, acting on smooth actions over B of STB ® Ef, ST" ® ker V. Let ffDB,E* (0),

DB,k.r v* (0)

be the corresponding reduced eta invariants. Theorem 2.39. The following identity holds (2.110)

j7DB B+ (0) _

j1DB,E_ (0)

= 11DB,ker V.1. (0) _ 7DB,ker V_ (0) + f A(TB, OTB)vl in R/Z. B

In view of (2.21), (2.102), it is clear that both sides of (2.110) vary in the same way. By a simple deformation argument, we may as well PROOF :

assume that E+ = E0,+ ® F, E_ = E0,_ ® F, V is the identity on F and vanishes on Eo,+ ® E0,_, VE+ = VEO,+ ® VF DE_ = VE-,_ ® VF. In this situation, by Theorem 2.10, q vanishes, and (2.110) is a trivial identity. 0

Assume now that B is instead even dimensional, that E is a Hermitian vector bundle, that V is a self-adjoint section of End (E) such that ker V is a vector bundle, and yE is a unitary connection on E. We use the notation of Section 2 d).

Let DB,E, DB,kerv be the Dirac operators acting on smooth sections of Coo(B, STB (9 E), COO (B, ST B ® ker V). Clearly E

VDB

'

(21111

(0) = 1 Ind (DB°E) in R/Z

eB,ker V (0)

= 1 Ind

(DB,ke''v)

in R/Z.

36


Theorem 2.40. The following identities holds

A(TB,VTB)il in R/Z.

i7D(0) = j7Da.ke.v(0)+

(2.112)

B

By Theorem 2.17 and the Atiyah-Singer index theorem [AS1],

PROOF :

(2.113)

JB

A(TB,VTB) = (Ind (DB,E'o) - Ind (DB'E 0, we replace gTB by 4. Let D'B be the corresponding Dirac operator. Then by (2.102), 7D'B (0) remains constant in R/Z. Now, we give a refinement of Theorem 2.39, established in [BC1, Theorem 2.28]. Theorem 2.41. If kerV = {0}, then the limit as c -4 0 of yID'B (0) exists in R, and moreover

m j7D'! (0) = LA(TB,VTB).

(2.116)

PROOF :

(2.117)

E

The main point in the proof of [BC1] is to show that lim

2

1 Tr [D'B exp(-tD'B'2)] =

JB A(TB,VTB)Qt,

which in turn follows from local index theory techniques.

Jean-Michel Bismut

37

If B is even dimensional, an obvious analogue of Theorem 2.41 holds. Now we make the same assumptions as in Sections 2 a) and 2 e). Sup-

pose that X and B are compact, that X is odd dimensional and that B is oriented and spin. Then X is also oriented and spin. Let 9TZ, 9TB be metrics on TZ, TB. Put (2.118)

91 TX = 1

.*9TB ®9TZ

Let DE be the Dirac operator on (X, g' x) as in Section 2 e). Using the variation formula for eta invariants, one verifies easily that as a -+ 0, s (0) converges in R/Z. Now we state the main result of [BC1, Theorems 4.35 and 4.95].

Theorem 2.42. If kerDZ = {0}, as c -+ 0, ijD; (0) converges in R, and moreover (2.119) PROOF :

lim o'D` (0) = fBA(TB,TB Formally, the proof of Theorem 2.42 is closely related to the

proof of Theorem 2.41. In fact, (2.119) is an infinite dimensional version of (2.116), as should be clear from formula (2.63). The proof has three main steps:

One proves that as c -> 0, 11

(2.120)

Tr

exp(-tDx'2)J

r -+ L A(TB,VTB)at.

One controls the lowest eigenvalue of De as a -* 0. One uses a version of finite propagation speed to control the integrand 0 in (2.98) uniformly in a as t -+ +oo. REMARK 2.43. In [D], Dai has given a very interesting extension of Theorem 2.42 to the case where kerDZ is not necessarily zero. Dai's result apply in particular to the case where B is odd dimensional, and DE is the signature operator of [APS1] associated to the metric g6 x In this case, there is no spectral flow, so that lim ijDt (0) exists in R. Us6-40 ing results of Mazzeo and Melrose [MazMe] relating small eigenvalues of DE '2 to the Leray spectral of the fibration, Dai obtains a formula extending (2.119) by adding to the right-hand side of (2.119) the reduced eta invariant of a signature operator on B (twisted by the cohomology of the fibres) and

a sum of half integers. These integers are the "signatures" of the Leray spectral sequence (E,., d,.), for r > 3. Adiabatic limits of eta invariants appeared naturally in the context of the solution by Atiyah-Donnelly-Singer [ADS] of the Hirzebruch conjecture

[Hir] on the signature of Hilbert modular varieties. The fibrations which appear in this context are fibrations by tori over a torus basis. The calculation of [ADS] was recovered in the context of j forms in [BC5]. For a L2 approach to the same problem, we refer to [Miil].


38

k) The families index theorem for manifolds with boundary. lr

Let now X be a manifold with boundary, let B be a manifold, and let : X -+ B be a fibration, whose fibres Z are smooth compact manifolds

with boundary. We assume the fibres Z to be even dimensional, oriented and spin. Let 9TZ be a metric on TZ, which is fibrewise product near BZ. Let THX

be a horizontal vector subbundle of TX, such that THXjax C TOX. Put THBX =THXIax. Then THBX is a horizontal subbundle of TBX. Let g{, Ve) be a Hermitian vector bundle on X with unitary connection.

Assume first that Z is even dimensional. For b E B, let Db be the Dirac operator with the Atiyah-Patodi Singer boundary conditions on BZ. In order that the index bundle Ind (D+) to be well-defined, it is crucial that the family of boundary Dirac operators DOZ does not have spectral flow. So we first assume that kerDOZ = 0.

(2.121)

In this case, (D+) is a family of Fredholm operators and its index Ind (D+) E K°(B) is well-defined. Let q be the even form on B constructed in Definition 2.35, which is attached to the family The following result is proved in Bismut-Cheeger [BC3, Theorem 6.11]. DOZ.

Theorem 2.44. The following identity holds (2.122) ch(Ind D+) = 7r..[A(TZ,VTZ) ch(e, Vf)]

- it in

Heven(B, @)

The basic idea in [BC2,3] is to replace the manifolds with boundary Z by the manifolds with conical singularity ZZ = ZUOZC[°'1(BZ). Then we equip the fibres Z with a family of metric of conical type. By proceeding as in Proposition 1.7, for P large enough, the Atiyah-Patodi-Singer family PROOF :

(D+) and the family of L2 Dirac operators (D") have the same index. To the family 21, one attaches a natural Levi-Civita superconnection At, to which the techniques of [B2] are formally applied. Note here that the advantage of using Z1 is that the Atiyah-Patodi-Singer boundary conditions only appear in implicit form. Observe that equation (2.95) explains why the right-hand side of (2.122) is closed.

In [MeP1], Melrose and Piazza have extended Theorem 2.44 in a fundamental way. In fact, they observe that even if ker DOZ is non trivial, by the

family index Theorem of Atiyah-Singer [AS2], the family (DaZ) E K'(B) is trivial. They show that if B is compact, the triviality of the family (DOZ) is equivalent to the existence of a spectral section P, i.e. a smooth family of

Jean-Michel Bismut

39

self-adjoint projections Pb : C°°(BZb, (STaZ ®l;)IZ,) - C°O(BZb, (ST"Z f)IZ,), such that there is R > 0 for which for any b E B,

DOZU = Au, A > R, then Pu = u (2.123)

A < -R, then Pu = 0.

Then Melrose and Piazza [MePl] prove that if for every b E B, the Atiyah-Patodi-Singer projection P>O,b is replaced by Pb, the family of Dirac

operators D+,n associated to the boundary conditions attached to P has a honest index bundle Ind (D+'p). They construct an even form ilp on B, formally similar to the form i in (2.94). However in Melrose-Piazza's construction, the term /DaZ is replaced by a more complicate expression J Di°Z, where D'eZ interpolates between DOZ for t « 1 and a suitable perturbation DaZ + Ap (with Ap depending on P) for t >> 1. The family Ap is smoothing and such that DeZ + Ap is invertible. It can be seen as providing an explicit trivialization of the zero class (DeZ) E K'(B). Modulo exact forms, the form i ' only depends on P and not on the particular choice of Ap. Then Melrose and Piazza [MePI] prove :

Theorem 2.45. The following identity holds (2.124)

ch(Ind(D+'p))=i.[A(TZ,vTZ)

(f,of),-f'

in

Heven(B,Q)

Besides in [MePI], Melrose and Piazza compare the forms ip for different choices of P. When the fibres Z are odd dimensional and the family DeZ is invertible, Bismut-Cheeger [BC4, Section 6] conjectured a formula like (2.124)

for a family of self-adjoint Dirac operators Dz. This conjecture has been proved and extended by Melrose-Piazza [MeP2]. They adapted the idea of a spectral section in this new context, produced a superconnection whose Chern character forms are shown to represent the index, and established the corresponding index formula.


40

III. Analytic torsion forms and analytic torsion currents. The purpose of this Section is to review the properties of the analytic torsion forms of [BGS2] and [BK] , and of the analytic torsion currents of [B2], [BGS4,5]. As explained in the introduction, analytic torsion forms are naturally associated to a family of Hermitian Dolbeault complexes. Analytic torsion currents are associated to an embedding i : Y -+ X and a resolution of a holomorphic Hermitian vector bundle p on Y by a holomorphic complex of Hermitian vector bundles on X. Analytic torsion forms and analytic torsion currents are secondary objects which refine the Riemann-Roch-Grothendieck theorem for submersions and immersions at the level of differential forms or currents. This Section in organized as follows. In a), we construct the torsion forms associated to a holomorphic Hermitian complex of vector bundles [BGS1], and we relate them to the secondary classes of Bott-Chern [BoCh]. In b), we consider a holomorphic submersion it : X -* S, and a holomorphic Hermitian vector bundle on X. When this fibration is Kahler (in a sense to be described), we show that the Levi-Civita supercounection of Definition 2.20 "respects" the holomorphic structure of the problem. When Rir e is locally free, we construct analytic torsion forms on S, which refine on the 7 forms of Definition 2.26. In c), we introduce the Quillen metrics on the inverse of the determinant The construction of the Quillen of the cohomology metric only involves the component of degree 0 of the above analytic torsion

forms. Then we state the curvature theorem of [BGS1,3] for the Quillen metric on a(C). In d), we describe the results of [BerB] and [Ma] on the compatibility of the analytic torsion forms to the composition of submersions.

In e), we construct the analytic torsion currents of [B2], [BGS4,5]. In f), we show that these currents are compatible to the composition of immersions.

In g) and h), we describe the results of [BL] and [B5,6] on the compatibility of the analytic torsion forms and analytic torsion currents to the composition of an immersion and a submersion. In i), we give a short introduction to the proof of the main result in [BL]. In j), we develop a simple but crucial technical tool in [BL], the Hodge theory of the resolution of a point. In k), we explain the construction in [B4] of the analytic torsion forms associated to a short exact sequence of holomorphic vector bundles, which plays an important role in the proof of the main result in [BL] and [B5]. In particular the evaluation of [B4] produces the genus R of Gillet and Soule [GS3] in the final formula.

a) The torsion forms of a holomorphic Hermitian complex. Let S be a complex manifold. Let (3.1)

(E, v) : 0 -+ E,,,, -+ ... -+ Ee -+ 0 V

v

be a holomorphic complex of vector bundles on S . Put (3.2)

E+= ED E, i even

,

E_=®E2. i odd

Jean-Michel Bismut

41

Then E = E+ ® E_ is Z2-graded.

Let gE _ ® 9E, be a Hermitian metric on E. Let VE = ® VE' be i=o

M

i=o

the holomorphic Hermitian connection on E _ ® E. Let v' be the adjoint i=0

of v. Put V = v + v`.

(3.3)

Then V is a self-adjoint section of End dd(E). For t > 0, put

At = QE + fv, A't = 0E + fv`,

(3.4)

At =VE+fv. Clearly

At = At + A' .

(3.5)

Also At is a superconnection of the kind we already met in (2.14). Let N be the number operator on E, i.e. N acts by multiplication by k on Ek. The following result is established in [BGS1, Proposition 1.6]. Proposition 3.1. The following identities hold 112

=0,

A'2t

= 0,

At = [At , Alt] [At , At ] = 0 , [_4;, Afl = 0, aA't' _ 1 [All, NJ at

as A' tt

PROOF :

t'

2t

-1 - 2t [At, N]

Since (E, v) is a holomorphic complex,

A"t = 0.

(3.7)

Also

aA't (3.8)

_

v

1

at - 2 f=2t [At , N]

The other identities in (3.6) follow easily from analogues of (3.7), (3.8).


42

DEFINITION 3.2. For t > 0, put at = W Trs [eXP(-A,')] (3.9)

at =

1

(2iir)1/2 `°

1 as `r Lt & exp(-Aa )J

7t = W Tr$ [N exp(-At )]

Observe that in our context, the forms at and fit were already introduced in Definition 2.5. Let PS be the set of smooth real forms on S which are sums of forms of type (p, p). Let PS,0 be the subspace of the a E PS such that a = 8f3 + &y, with f3 and y smooth. Now we have the result of [BGS1, Theorem 1.15].

Theorem 3.3. The forms at and ryt lie in PS. Moreover

aat

W - -df3t,

(3. 10)

_ 1 (6a)-Y, 2i7r

2t

In particular (3.11)

aat _ as ryt 2iir t at

We only prove part of Theorem 3.3. The first identity in (3.10) was already established in (2.16). Using (3.6), we obtain the second identity in (3.10). PROOF :

Assume now that the homology H(E,v) is of locally constant dimension. Then H(E, v) is a holomorphic Z-graded holomorphic vector bundle on S. Clearly (3.12)

H(E, v) c ker V.

Let 9H(E,v) be the metric on H(E, v) induces by gE via (3.12). One verifies easily that Vim'v = pker V VE is the holomorphic Hermitian connection on (H(E, v), gX (E Ul ). Put (3.13)

ch'(E,gE)=E(-1)'ich(E.,gE`). i=o

Then by an analogue of Theorem 2.6, as t -> +oo, (3.14)

'Yt=ch'(H(E,v),gI(E, ))+0

\f/

Jean-Michel Bismut

43

DEFINITION 3.4. For s E C, 0 < Re(s) < 2, put (3.15)

R(E,9E)(s) =

1

Jotoo

r(s)

t8-1(7t -'Yoo)dt.

By (3.14), R(E,gE)(s) is a holomorphic function of s, which extends holomorphically to s = 0. DEFINITION 3.5. Set

(3.16)

T(E,gE) _ sR(E,9E)(s)Ie=o

Recall that the odd form n was defined in Definition 2.7. Now we have the result of [BGS1, Theorem 1.17].

Theorem 3.6. The form T(E,9E) lies in PS. Moreover

2 T(E,gE) = ch(H(E,v),9H(E,v)) - ch(E,9E), (3.17)

2

PROOF :

These identities follow easily from Theorems 2.6 and 3.3.

DEFINITION 3.7. We will say that ((E,v),gE) is split, if E_ = F; ® Ft_1 e Hi,vjE; vanishes on F= ® H; and is the identity on F;_1, and the above splitting is orthogonal with respect to gE;.

Now we state a result of [BGS1, Corollary 1.30].

Theorem 3.7. There is a unique way to associate to ((E,v),9E), with H(E, v) of locally constant dimension, a class T (E, v) E PS/PS,' such that

a) T(E,gE) is functorial. b) If (E, gE) is split, T (B, gE) = 0. c) The following identity holds,

(3.18)

2 T(E,9E) = ch(H(E,v),9H(E.1)) - ch(E,gE).


44

PROOF :

Existence is almost obvious by the above construction. As to

uniqueness, observe that over S x P1, one constructs easily a complex (E, v)

and a metric 9E such that ((E,v),9E)sx{o} _ ((E,v),9E),

(3.19)

((E,v),9E)sx{o} is split. Using the obvious equation 8a

(3.20)

tar

log I

2 = Z12

6{o} -61.),

we get (3.21)

T(E,gE) = fpi logIzI 2 (ch (H(E,v), 9H(E'"))

- ch(E,9E))

in

ps/ps,o 0

which guarantees uniqueness. REMARK 3.8. Observe that (3.22)

R(E,9E)(o)(s) = fts [N[V2)-y]

so that (3.23)

T(E,gR)(o) = - Trs [Nlog(V2)] .

By (3.18), we get (3.24)

cl(detE,gE)=cl(detH(E,v),gdetH(E,v))-

0aT(E'9E)(o).

2iI

The interpretation of (3.24) is easy. In fact there is a canonical holomorphic isomorphism [KMu]

det E = det H(E, v). Let us briefly describe this isomorphism. First assume that H(E, v) _ {0), i.e. (E, v) is acyclic. Then (3.25) says that det E has a canonical non zero section T(E, v). To construct r(E, v), we choose w,,, E det E,,,, w,,, 54 0, wm-1 E AdimE,,,-i-dimE,,,Em-1 such that vw m Awm-1 E det E,,,_1 is non (3.25)

Zero, Wm_2 E AdimEm-2-dimB -1+dimEmE76_2 such that vwm-1 Awm-2 E

det E,,,_2 is non zero... . These choices are possible because (E, v) is acyclic. Then (3.26) r(E, v) = (wm ® (vw,,, A wm_1)-1 0 (vwmm-1 A wm-2) ®... )(-1)m

does not depend on the above choices. When H(E, v) is non zero, the construction of the canonical isomorphism (3.25) is similar. Then one verifies easily that 9detE = 9detH(E,v) exp{T(E,gE)(o)}, (3.27) from which (3.24) follows immediately.

Jean-Michel Bismut

45

The class of forms T (E, 9E) E Ps / pS O appears as a prototype of BottChern classes [BoCh], [BGS1]. Let us give a construction of these classes in the simplest case.

Let p : F -a S be a holomorphic vector bundle. Let gF,g'F be two Hermitian metrics on F. Let Q be a characteristic polynomial. The following result in established in [BGS1, Theorem 1.29].

Theorem S.S. There exists a unique way to assign to (F, gF, g'F) a class Q(F,gF,g'F) E PS/PS,O such that a) Q(F,gF,g'F) is functorial. b) If gF = 9'F, Q(F',9F,9'F) = 0. c) The following identity holds

2nQ(F',9F,9'F) = Q(F,9'F) - Q(F,9F)

(3.28)

PROOF :

We just outline a construction of gF_ Q(F,g5,g'F) [BGS1]. Extend

F to a vector bundle F on S x P1. Let

be a metric on F such that

9Sx{o} = 9F, 9sx{oo} = 9'F Put (3.29)

Q(F, 9F, 9'F) = - fP I log(1zI2)Q(F, 9F)

Then by (3.20), (3.28) holds.

m M Let gE = ED g', g'E = ® g'E; be two set of Hermiti i=O

i=O

Let gH(E,v), 9IH(E,v) be the corresponding metrics on H(E, v).


T(E,g'E) -T(E,gE) =

ch(H(E,v),gH(E,v),9tH(E,v))

- ch(E,9E,9'E) in Ps/PS,O. PROOF :

Using Theorem 3.7, our Theorem is a straightforward conse-

quence of the uniqueness of Bott-Chern classes stated in Theorem 3.8.

b) The Levi-Civita superconnection of a Kahler fibration and the analytic torsion forms. Let ir : X - S be a holomorphic submersion with compact fibre Z. Let be a holomorphic vector bundle on X. Let Ri.g be the direct image oft C. Let wX be a real closed (1, 1)-form on X, such that the restriction of wX to TZ is the Kahler form wT2 of a Hermitian metric gTZ on TZ = TX/S.

If JTRZ is the complex structure of TfZ, if U, V E TR,Z, WTZ(U,V) =


46

(U, JTRZV). Let gf be a Hermitian metric on £, let Ve be the holomorphic Hermitian connection on gf ). Let THX be the orthogonal bundle to TZ with respect to wX. Let (sl(Z,C1Z),8Z) be the family of relative Dolbeault complex along the fibres Z. We equip &1(Z, a Z) with the L2 metric attached to gTZ, g£,

(s, s') = L (s, s')

(3.31)

(2

)amZ

Let 8*Z be the formal adjoint of 8Z with respect to (3.31). Set

DZ=BZ+aZ*.

(3.32)

Then by [Hi], vtDZ is a family of standard Dirac operators along the fibre Z. The only minor difference is that the fibres Z only have spin` structure. To the data (9TZ,THX) we can associate the objects constructed in Section 2 e). The following result is proved in [BGS2, Theorem 1.7]. Theorem 3.10. The connection VTRZ onTRZ preserves the complex structure of TR,Z. It induces the holomorphic Hermitian connection on (TZ, gTZ). As a 2 -form, T is of complex type (1,1). Let VA(T*(0")Z)Ot be the connection induced by VTZ, V on A(T*(0,1)Z)®

If U E TZ, let UH E TX X be the horizontal lift of U. DEFINITION 3.11. If U E TRS, ifs is a smooth section of SZ(Z, CIZ) over S,

put Vn(Z,41Z)S

(3.33)

= DU(X

The following result is established in [BGS2, Theorem 1.14].

Theorem 3.12. The connection Vsz(Z,fiz) on SZ(Z,eiz) preserves the Hermitian product (3.91) on SZ(Z,t;IZ). Its curvature is of complex type (1, 1). Also (3.34)

IVO(Z,f1Z)",3Z] =o

,

[on(Z,t1Z)',eZ*]

=0.

Amazingly enough, (SZ(Z, eIZ),8Z) appears to be a "holomorphic" Hermitian vector bundle compact over S. By (3.34), we find that (3.35)

(V

Z 2= 0 , + 8)

z. 2

(Vn(Z,fiz)' + 8 ) = 0.

The explanation for (3.35) given in [BGS2,

Theorem 2.8) is that us-

ing the smooth identification A(T*(o.1)X) = A(T*(0,1)Z)®1r*A(T*(0,1)S), pn(Z,fIz)" +aZ is exactly the full Dolbeault operator 8X acting on SZ(X,C).

47

Jean-Michel Bismut

Recall that A(T*(0'I)X) ®g is a c(TRZ) Clifford module. Namely if X E TZ, let .X* E T*(0,1)Z correspond to X by the metric gTZ. Then if X E TZ, Y E TZ, put (3.36)

c(X) _'X*A , c(Y) _ -v iy.

Extend c to a linear map TRZ OR C -+ End ((A(T*(°°1)Z) ®C'). Then if X, X' E TRZ OR C, (3.37)

c(X)c(X')+c(.X')c(X)=-2(_X,X')9TZ

Let (fa) be a basis of TRS, let (f a) be the dual basis of TRS. Put (3.38)

c(T(I'0)) = 1faf"c(T(1,0)(f« ,fe )), c(T°'1)) = 1fafRc/lT(°'1)(fa ,f ))

With the notation in (2.54), c(T) = c(T(1,o))+c(T(°,1))

(3.39)

DEFINITION 3.13. For t > 0, put

Bt = (3.40)

Bt =

aZ + ,/t-

0n(Z,tlz),,

z. +DO(Z,E1z)'

_ c(T

_

(I,0))

2 2t c(T(0,1)) 2-,,/2-t

'

Bt = Bt + Bt. Then by (2.55) and Theorem 3.12, for t > 0, Bt is exactly the superconnection At in the sense of [Bi], i.e. Bt is a Levi-Civita superconnection.

Put (3.41)

wX =wTZ+wH.

In particular wH E 7r*A2(TRS) is the restriction of wX to TRHX = 7r*TRS.

Let Nv be the number operator of S1(Z,eiz), i.e. Nv acts by multiplication by k on 12k(Z,e1Z).

DEFINITION 3.14. For t > 0, put (3.42)

Nt=Nv+ -

H .

Then Nt E (A(TRS)® End (12(Z,C'jZ)))ee°

48


The following result is proved in [BGS2, Theorem 2.6]. Theorem 3.15. The following identities hold,

B"i = 0

(3.43)

B'E =0,

,

Bt = [B't', Bt]

,

[B,", Bt ] = 0

,

[Bt, Bt ] = 0 ,

8Bt _-1[B"Nt] of 8Bt at

2t

=

1 2t

t

'

[Bt, Nt]

REMARK 3.16. The identities in (3.43) are remarkable. They guarantee that the Levi-Civita superconnection Bt also has natural holomorphic properties,

i.e. it splits as Bt = B,' + Bt. Besides, by comparing (3.43) with (3.6), Nt appears as the right "number operator" associated to Bt. DEFINITION 3.17. For t > 0, set

(3.44)

at = cp as [exp(-Bt )] r8B 1

I3 = ( 2i7r) I"

Trs

L att

1

eXP(-B,2)]

it = p Trs [Nt exp(-Bt)] . Now we state a result taken from [BGS3, Theorems 2.9 and 2.16]. Theorem 3.18. The forms at, /3t, ryt are real. The forms at and ryt lie in PS. The cohomology class of at is constant, and (3.45)

[at] = ch(R7r t;) in Hevea(S,R).

Also,

(3.46)

oat = -dat, at

at=-2g7r(a-a)2t In particular, (3.47)

Bat

asyt

8t

2iir t

Finally as t -+ 0, at =7r. [Td(TZ,gTZ) (3.48)

0(t),

C-1 7t= t +Co+®(t) , C-1,CoEPs.

Jean-Michel Bismut PROOF :

49

We just sketch the proof of part of Theorem 3.18. Equation

(3.45) follows from Theorem 2.22. Equation (3.46) follows from (3.43) as in (3.10). The first equation in (3.48) follows from Theorem 2.22. The second equation in (3.48) is proved in [BGS2] by local index theoretic techniques.

Now we assume that Ra.l: is locally free. So Ra.l; is a holomorphic Z-graded vector bundle on S, and moreover (Rir.i:)s = H(Z., elz,). Since H(Z,CIZ) ^- kerDZ,Rir.g inherits a smooth metric gR"4.

Theorem 3.19. As t - +oo, R",C

1

(3.49) ch'(R7r.4,9T,,.E)

'Yt =

PROOF :

+ 0 (;)1

With the notation of Theorem 2.25, using [BGS3, Theorem

3.11] (which relies on (3.35)), one shows easily that VkerD2,a is just the Theorem 3.19 is then holomorphic Hermitian connection on an obvious modification of Theorem 2.25. DEFINITION 3.20. For s E C, 0 < Re(s) < 2, put (3.50)

R(.',9£) = - 1

r(s) JO

+oots-'(7t

- y o)dt.

By (3.48), (3.49), R(wX, gf) is a holomorphic function of s, which extend holomorphically near s = 0. DEFINITION 3.21. Set (3.51)

T(wX,9{) =

asR(wX,9{)(0)

Recall that the form was defined in Definition 2.26. Then we have the result of [BGS2, Theorem 2.20], [BK, Theorem 3.9].

Theorem 3.22. The form T (w", gf) lies in PS. Moreover

.52) (3.52)

PROOF :

Equation (3.52) follows from Theorems 3.18 and 3.19.


50

In Remark 2.28, we observed that the dependence of (modulo coboundaries) on the various geometric data is quite explicit. Also in Theorem 3.9, we found that the dependence of T(E, 9E) E ps/ ps,o on the metric gE can be explicitly given in terms of Bott-Chern classes. It is then natural to ask how T(wx, gf) depends on (wx, gf ). In fact

if only gf is made to vary, equation (3.52) and the methods of [BGS1] used in the proof of Theorem 3.7 provide the answer immediately. However

if WX also varies, the answer certainly does not rely on the methods of Theorem 3.7. In fact, for Theorem 3.22 to hold, it is crucial for wX to be closed. So in order to calculate T (w'x, 91) - T (wx, gf) using (3.52), a necessary condition would be, for example, that the fibres Z have the same volume for wX and w'X. Let (w'X, g'f) be taken as before. The following "anomaly formulas" were established in [BK, Theorem 3.10], extending earlier work in degree 0 [BGS3, Theorem 1.23].


T(w'x,g'f) -T(wX,9f) = ch

gIR'.E)

-1. [Td(TZ,9TZ,9'TZ) ch(t,91) + In particular, T(wx 9f) E ps/ps,o only depends on

91f)] in PS/P-9,0.

(gTZ,gf).

The last statement in Theorem 3.23 is if particular importance. It says that, as should be the case, the class of T(wx, gf) in Ps/PS,' only depends on the geometric data which appear in the right-hand side of the first equation in (3.52).

c) Quillen metrics. Assume first that S is a point. Let gT Z, gf be the Hermitian metrics on TZ, £. DEFINITION 3.24. Put (3.54)

9(s)

Tr8 [Nv [DZ,2] -8]

Then 0(s) is a linear combination of the zeta functions of the Laplacian DZ,2 acting on forms in f(Z, I:1Z) of degree 0,1... , dim Z.

Put (3.55)

(detH(Z,1,z))-1

Then .1(1') is a complex line. The metric gH(Z,fjz) induces a metric I on .\(£). In [Q2], Quillen introduced the following metrics on A(t).

Ialfl

Jean-Michel Bismut DEFINITION 3.25. The Quillen metric I) (3.56)

II

I

Ia(n) exp

51

on A (C) is given by

{-(o)}

The underlying motivation for formula (3.56) is equation (3.27). In fact (3.56) is a way of making sense of the metric 9(detta(Z4iz))-" which (detQ(Z,e1Z))-1 does not exist. The quantity 86(0) is called the Ray-Singer analytic torsion [RS]. Let now n : X -+ S be a holomorphic sumersion with compact fibre Z. Let t be a holomorphic vector bundle on X. By a construction due to Grothendieck-Knudsen-Munford [KMu], there is a canonically defined holomorphic line bundle A(t;) on S, called the inverse

of the determinant of the direct image Ring. In particular ifs E S, there is a canonical isomorphism (3.57)

A(g)8 = (det H(Z8, 1z, ))

Needless to say, if R7r.g is locally free,

A(g) = (detRir.g)-'

(3.58)

In the general case, we will still use the notation A(g) = (det Rir4) Let 9TZ, g{ be arbitrary Hermitian metrics on TZ, C. Then by the construction in Definition 3.25 and using (3.57), the fibre A(g)8 can be equipped with the Quillen metric II II,,(£), A first result on Quillen metrics is as follows [BGS3, Theorem 3.14].

Theorem 3.26. The Quillen metric is a smooth metric on A(C). If Rn.g is locally free, Ia(,) and II II,,(,) are smooth. The remarkable fact is that in the general case,II IIA(,) is still smooth. However formula (3.27) partly explains the smoothness of II II,,({) PROOF :

I

DEFINITION 3.27. We will say that 7r: X -T S is locally Kahler if there is a covering of S by open sets U such that 7r-'(U) is Ki hler.

We now state the result of [BGS3, Theorem 1.27].

Theorem 3.28. Assume that it : X -+ S is locally Kdhler and that 9TZ is fibrewise KBhler. Then (3.59)

ci(A(g),II

II.,(e)) _ -7r. [Td(TZ,gTZ)ch(6,g{)](2)

Local index theory, eta invariants, and holoinorphic torsion

52

Clearly, we can assume that X is Kahler. Let gTx be a Kahler metric on TX, with Kahler form wx, and assume first that gTZ is the metric on TZ induced by gTx. By (3.50), (3.54), PROOF :

R(wx,g£)(01(s) = 0(s),

(3.60)

and so

T(wx,g{)(0) _ g(0) as

(3.61)

Suppose that R7r.C is locally free. By (3.52), we get (3.62)

aaT(wx,9f)(°) _

IA(e)) - ir. [Td(TZ, yTZ)

From (3.62), we get (3.59). If Rir.i< is not locally free, more work is needed to establish (3.59) [BGSS]. Suppose now that g'T Z is a metric on TZ which is only fibrewise Kahler. Then by (3.53), (3.63)

log (II 11

Ih(E)/= f

From (3.59) (established for gTZ) and (3.63), we get (3.59) for 9'TZ.

d) Adiabatic limits of Quillen metrics, analytic torsion forms, and composition of submersions. Let 7r : X -+ S be a submersion of compact complex manifolds. Let { be holomorphic vector bundle on X. We assume that Rir.i; is locally free. Let gTx be a Kahler metric on X, let wx be the corresponding Kahler form. Let gTS be a Kahler metric on S. Let gC be a Hermitian metric on e.

Put A = (det H(X,

fix))-1

(3.64)

a' = ®(det H(S,

,

(-1)4+,

Let (3.65)

C (X,e) =F°52(X,ia) D F1f(X,e) D ... D FdimS+1S2(X,o = 0

be the obvious filtration by the partial degree in A(T"(°,1)S) of the Dolbeault complex 52(X, g). Let (E,., d,.) be the associated spectral sequence. Then (3.66)

EZP,e)

= HP(S, R'

By (3.66), it follows that the lines A and A' are canonically isomorphic.

Jean-Michel Bismut

53

We can equip the line \ with the Quillen metric associated to gTX,g{, and the line A' with the Quillen metric associated to gTS,gR".f Consider the exact sequence

0 -i TZ -+ TX -> ir`TS -+ 0.

(3.67)

Let Td(TX,TS,gTx,9TS) E pX/pX,O be the Bott-Chern class [BoCh], [BGS1] such that (3.68)

'9'9 Td(TX,TS,gTx gTS)=Td(TX,9TX)-Td(TZ,gTZ)lr*Td(TS,gTS) The following result is established in [BerB, Theorem 3.1]. Theorem 3.29. The following identity holds (3.69)

log

11

Ik

Ia /2

- f Td(TS,9TS)T(wx,9£) +J Td(TX,TS,gTx,gTS)ch(t;,g£)

x

We identify Ta, (x) to the corresponding additive genus. By definition, the genus Td' is the product of the additive genus Td' genus Td. In [BerB, Theorem 3.2], it is shown that (3.69) is essentially equivalent to the following statement. For e > 0, put (3.70) Let II

TX ge = 9TX

+ 1 .9TS

Ila, be the Quillen metric on ,\ associated to (g, x, ge)

Theorem 3.30. As a -+ 0, (3.71)

log VIII IIIL) - f 1r` Td'(TS) Td(TZ) ch(l;) log(e) 2

-+ -

Td (TS, gTS)T (wx, 9£) + log

\

. III

Illa)

REMARK 3.31. The proof of Theorems 3.29 and 3.30 is a combination of the adiabatic limit techniques of Bismut-Cheeger [BC1], and of the Leray spectral arguments of Mazzeo-Melrose [MazMe] and Dal [D]. Recently, Ma [Ma] has established an extension of Theorem 3.29 for the higher analytic torsion forms T(wX, g4). Namely let (3.72)

Z -W itz/YI

irw/vI

Y - b-V

rw/s AV/s-S


54

be a commutative diagram of holomorphic submersions, with compact fibres

Z and Y. Let C be a holomorphic vector bundle on W. Assume that Rirw/s.e, Rnw/v.f and Rirv/s.Rirw/v.g are locally free. Let g{ be a Hermitian metric on t;. Let ww, wV be (1,1) closed forms on

W, V having the properties described in Section 3 b). Then by proceeding as in Section 3 b), the three direct images vector bundles described above inherit Hermitian metrics. Let be the analytic torsion forms on V and S associated to 7rw/v,7rw/s,7rv/s, and the given metrics. A problem which arises in [Ma] is the adequate definition of (3.73)

a = ch

gR7r%v/s.R1rw/v.4)

so that (3.74) 2

;8

2iir

9RAv/s.Ruw/v.4)

a = ch

- ch

In fact there is a spectral sequence E, of sheaves on S such that E2 = which abuts to Under an adequate assumption of ampleness, this spectral sequence is trivial, so that the definition of a is easy. If the Er are locally free, there

is also a natural definition of a. In general, if W and V are projective, a definition of a is given in [Ma]. Then Ma's result is as follows.

Theorem 3.32. The following identity holds (3.75) Tw/s (ww, 91) = Tv/s

(wv,

9R"w/ v., )

+7rw/s. [Td(TY,gT3')Tw/v(wW,9£)] + a - 7rw/s. [Td (TZ,TY, gTZ, gTy) ch(1;, 9')]in Ps/Ps,o

e) Analytic torsion currents.

Let i : Y -+ X be an embedding of complex manifolds. Let 77 be a holomorphic vector bundle on Y. Let (3.76)

v) : 0

v

v

o-0

be a holomorphic complex of vector bundles on X, which, together with a holomorphic restriction map r : l;'olY -+ 77, provides a resolution of the sheaf i.Oy(7l), i.e. we have an exact sequence of sheaves (3.77)

0 -4

4

i.OY(rl) -+ 0.

In particular the complex (1:, v) is acyclic on X \Y. If y E Y, U E TXy, let 8uv(y) be the derivative of v in any holomorphic trivialization of v)

Jean-Michel Bismut

55

near y. Then by [B3, Theorem 1.2], 8uv(y) acts on H((C,v)b), the action only depends on the image z E Nyyx,y of U, and will be denoted by 6zv(y).

Let ir be the projection Ny1X -+ Y. Then by [B3, Theorem 1.2], there is a canonical isomorphism of holomorphic complexes on NyIX, (3.78)

(7r*H(C,v)1y),8zv) = (ir*(ANYlx 017),i)

,

where in the right-hand side of (3.78), appears a Koszul complex. Let g M

® g£' be a Hermitian metric on

rn

_ ®C;. Let gN}'/x, gn be Hermitian

metrics on Nylx,rl.

Put V = v + v*.

(3.79)

By finite dimensional Hodge theory, v) ly) ^- ker V y .

(3.80)

As a subbundle of gly, kerVly inherits a Hermitian metric. Let gH(f,v) ly be the corresponding metric on DEFINITION 3.33. We will say that g£0, ... , gc- verify assumption (A) with respect to gNY/x,g" if the identification (3.78) is an isometry. By [B3, Proposition 1.6], given metrics gNY/x,g''7, there exist metrics gf0, ... , gf" such that (A) is verified.

In the sequel we assume that (A) holds. Let Ve be the holomorphic Hermitian connection on . For t > 0, put (3.81)

At=V +V/t-V.

Let NH be the number operator of DEFINITION 3.34. For t > 0, put (3.82)

v).

at = `P Tr$ [exp(-A2)] it = So Tr6 [NH exp(-At )]

Of course, equations (3.10), (3.11) are still valid. Now we give a result of [B3, Theorems 3.2 and 4.3], which replaces Theorem 2.6 and (3.14) in this new situation. Let by be the current of integration on Y.

Theorem 3.35. As t -> +oo, at = Td-1(Ny1x,gNY/x) ch(rl,g")by + 0 (3.83)

\7/

7t = -((Td)-1)'(Nylx, gNY/x) ch(7,, g")by + 0 Ort)


56

REMARK 3.36. In(3.83), 0 (1) is taken in the adequate Sobolev space of currents. Also, the convergence is shown to be microlocal in the set of currents whose wave front set is conormal to Y.

By Theorem 2.4 and by (3.83), we see that (3.84)

ch(e) = Td-1(Ny/x) eh(i)3y in Heven(X, Q) .

This is exactly the content of Riemann-Roch-Grothendieck for immersions, which says that (3.85)

ch(i.r/) = Td-1(Nryx) ch(r/)Sy in Heven(X Q)

Using Theorem 3.35, we can now imitate Definition 3.5 and construct a current T(1',gf) on X by formulas (3.15), (3.16). The following result is proved in [BGS4, Theorem 2.5].

Theorem 3.37. The currentT(f,g4) is a sum of currents of type (p,p), and its wave front set lies in NY/x R. Moreover (3.86)

20T(f,9£) = Td-1(NY1x,9N1'/x) ch(ii,9n)5Y - ch(e,91)

Let PX be the set of currents on X, which are sums of currents of type (p, p), whose wave front set lies in NY/x R. We define Py '° as in Section 3 a), with the adequate condition on the wave front set of Q, ry. The following extension of Theorem 3.9 is established in [BGS5, Theorem 2.5].

Theorem 3.38. Let (gf,gNYfx,gq) and (g'£,g'NYix,gin) be triples of metrics verifying condition (A). Then (3.87)

-T(C,91) = (Td-1(NY/X,9Nv,x,9tNrix) ch(1?,9n)

+Td`1(Ny/x,g NY1x)ch(l1,9n,9'n))SY

in PROOF :

PrY/Px'o.

The proof of Theorem 3.38 is essentially the same as the proof

of Theorem 3.9.

f) Compatibility of the currents

to the composition of immersions. Let i : Y -> X, i' : Y' -* X be two complex submanifolds of X intersecting transversally. In particular dim Y + dim Y' > dim X. Let 77,77' be holomorphic vector bundles on Y, Y', let v), (a', v') be two holomorphic complexes of vector bundles on X which provide resolutions of i.r/, i;r/'. Then one verifies easily that if Y" = Y fl Y', if i" : Y" -4 X is

Jean-Dlichel Bismut

the corresponding embedding, then (t

57

y', v + v') provides a resolution of

Then we have the diagram

Y" --- Y

(3.88)

iiiX

Y'

Let (gNr1x,g'7,ge) and (gN1"/x,g"',0) be Hermitian metrics verifying (A). Then we equip EB Ny,IXiy with the metric 1Y,gnjY" One verifies easily that gN''lxls"' E9 gnir",g£") verifies (A).

Let PYUy Pr YbyYUY'.

be the obvious analogues of Pi y, PP '0 when replacing

The following result is established in [BGS5, Theorem 2.7].

Theorem 3.39. The following identity holds ch(t;',g{ )+

(3.89)

Td'1(NY/X, 9N'./x) ch(r1, 9n)T

g£ )Sy-

in PruY,/PYUY, .

g) Complex immersions and Quillen metrics. Assume now that X and Y are compact. Let gTX, 9TY be Kiihler metrics on TX,TY. Let gNYix,g' be Hermitian metrics on Ny IX,1, let m

gf = ® g£{ be a Hermitian metric on i=o

m

®ldi which verifies (A) with i=o

respect to gNvix,gn.

Put A(i7) = (detH(Y,rl))-1 , (3.90)

A(si) = (detH(X,e

))-1

,ri

1){

Aw =

.

i=0

We claim that there is a canonical isomorphism (3.91)

XW = A(r!) -

In fact let H(X,1;) be the hypercohomology of the sheaf Namely if S is the natural Oech coboundary, we consider the complex (OX (c), S+v). Needless to say, S and v are graded so that (3.92)

(6+V)2 = Sv + vS = 0.


58

Also if NOeth is the natural Cech number operator, we grade (Ox (C), 5 + v)

by NO., - NH so that b + v increase the total degree by 1. Then r (Ox (4), 5 + v) -* (Oy (wl), 5) is a quasi-isomorphism, so that

H(X, ) = H(Y, ri) .

(3.93)

Now there is a spectral sequence whose E1 is given by EiP,a)

(3.94)

= HQ(X-P)

By (3.94), we get (3.95)

(detH(X,g))-1

((ciet?') -1)

(-1)°

which is equivalent to (3.96)

(det H(X, c))-1 = 0

(A(ft))(-1)=

.

By (3.93), (3.96) we get (3.91).

and A(rl) are equipped with Quillen metrics II It is natural to compare these metrics. This question was first II IIA(n) varied by Gillet and Soule [GS3] in their program to prove a RiemannNow

Roch-Grothendieck formula in Arakelov geometry. In fact if A is the ring of integers of a number field k, if X -4 Spec(A) is an arithmetic variety, if is an algebraic vector bundle on X, then H(X,g)

then if is equipped with a is A-module. If A(C) = metric at places at infinity, \(e) has an Arakelov degree deg A(e). The idea in [GS3] is to precisely equip \(£) with Quillen metrics at the places at infinity.

In [GS3], Gillet and Soul gave a conjectural formula for deg A(s) in terms of arithmetic characteristic classes. Still, when calculating the ratio of two Quillen metrics on the same algebraic object, only the contributions at infinity remain, so that [GS3] suggests a comparison formula which should be valid for any complex Kahler manifold. Now we describe a result of Bismut-Lebeau [BL, Theorem 0.1] where the conjectured comparison formula was established. +00

First we introduce the Gillet-Soul6 series R [GS3]. Let c(s) _ > be the Riemann zeta function. DEFINITION 3.40. Let R(x) be the power series (3.97)

R(x) _

C2 C(-n) + n odd

E j) C(-n) ni .

y

Jean-Michel Bismut

59

q

We identify R(a) with the additive genus

R(xi).

E p}'/P '° be the Bott-Chern class,

Let such

(3.98)

2 Td-(TXIy,9Ty,9TX'Y,9NYIx)=Td(TXIy,9TX1Y)Td(TY, 9TY )

9NYix).

Then the result of Bismut-Lebeau [BL, Theorem 0.1] is as follows. Theorem 3.41. The following identity holds log (II

(3.99)

Td(TXIy,9Ty,9TXIY,9N'"fx) Td(Ny/x,9N'.,x)

+ y

pp

fx Td(TX)R(TX) ch(e) + [YTd(TY)R(TY) ch(rI) . Some details on the proof of Theorem 3.41 are given in Section 3 i)- 3 k). 0 PROOF :

h) Analytic torsion forms, analytic torsion currents and the composition of an immersion and a submersion. Let now i : W -+ V be an embedding of complex manifolds, let av V -+ S be a holomorphic submersion with compact fibre X, which restricts

to a submersion 7rw : W -+ S with compact fibre Y. Then we have the diagram (3.100)

Y i

X -V -=S 1

4

r

.A.Y r

Let n be a holomorphic vector bundle on W, let (l;, v) be a holomorphic complex of vector bundles of X which resolves i.p. Of course, fibrewise, the situation is the same as in Section 3 g). Equivalently, the case where S is a point is just the case considered in Section 3 g).

Assume that R7rw.77 is locally free. Then R7rw.t7 = H(Y,rlly). By (3.93), Rirv.g is also locally free.

Let Yv (resp. wn') be a (1, 1) closed form on V (resp. W). Let m

9f = ®9f+, 9r'Yix , g'" be Hermitian metrics on i=o

m

®Si, Ny/x, ,, which i=o

verify assumption (A) (keeping in mind that Ny1X = Nw/v).


60

Let T(ww,g'7) E PS be the analytic torsion forms constructed in Section 3 b). They verify the equation (3.101)

BeT(wv',9'i) 2

= ch

(H(y,nlY),9H(Y,OIr))

-xw, [Td(TY,gTr) Ch(17,9")]

Similarly, to the family of double complexes (12(X, Ix), 8X + v), we can associate analytic torsion forms T (wv, gf) E PS, which verify the equation (3.102)

7rv. [Td(TX,gTX) ch(t;,gf)]

Since H(X, flx)

= H(Y,711y), the Bott-Chern class ch(H(Y,771y),

9H(x,4Ix),9H(YMIv)) is well-defined.

The main result proved in [B5, Theorem 0.1], which extends Theorem 3.41 to the relative situation, is as follows. Theorem 3.42. The following identity holds (3.103)

f

-Jx

r

Y

Td (TXlw,9TY,9TXIW,gNyl.) Td(Ny/x,9N'.ix)

-Td(TX)R(TX) ch(.)+f[ Td(TY)R(TY) ch(i) = 0 Assume now that for j > 0,

pS/ pS,o

in

Ix

0, (0 < k < m), Rjirw. i = 0.

Then we have a holomorphic complex of vector bundles K on S, (3.104).

K: 0 -3 H°(X,fm) ->

H°(X, Ix) -+ 0

Let ch(K,9K) E ps/pS,o be the Bott-Chern class such that (3.105)

8a 7-r

ch(K,9K) = ch M

ch - E(-1)` i=O The following result is proved in [B5, Theorem 0.2].

Theorem 3.43. The following identity holds m

(3.106) T(wt',gf)-E(-1)iT(w1',gf')-ch(K,g')=0 i=o

in

PS/PS.O.

Jean-Michel Bismut

61

REMARK 3.44. Needless to say, Theorems 3.39, 3.41-3.43 are compatible to each other.

i) A sketch of the proof of Theorem 3.41.

Using the anomaly formulas of Theorem 3.23 as in (3.63), and also Theorem 3.38, to establish Theorem 3.41, we may and will assume that are the metrics induced by 9T' on TY, Nryx . Put (3.107)

E = CO0(X,

We define the total Z-grading on E by the operator NT.- - NH. Then aX + v

acts on E and (dk +v)22 = 0.

(3.108)

Therefore (E, ax+v) is a Z-graded complex, whose hypercohomology H(E, v) is finite dimensional. Dolbeault's theory shows that (3.109)

H(E,

aX

aX

+ v) = H(X, ) .

Set (3.110)

F = C°° (Y A(T ` (o,')Y) ®77)

aX + The restriction map r : gory ->'i extends to a map of complexes r : (E, v) -> (F, v'). By [BL, Theorem 1.71, it induces the canonical identification

H(X, e))-' . Then by (3.93), (3.96) (3.112)

( ) = )() = A(W -

Moreover by imitating Definition 3.25, we can equip a(g) with a Quillen IIa(O. A first step in the proof of Theorem 3.41 is the simple fact, established in [BL, Theorem 2.11, that metric II

(3.113)

II

1 A(£) = II

IIA(o 2 11

To establish Theorem 3.41, we must then compute Log (II

III({) II,

(>

+


62

Put DX = aX +aX*,

(3.114)

V=v+v*.

For T > 0, it is clear that (aX +Tv)z = 0.

(3.115)

Also for T > 0

+Tv) =H-(E,BX +v).

(3.116)

For T > 0, put

AT = DX +TV.

(3.117)

Then by Hodge theory,

kerAT =

(3.118)

Tv),

so that for T > 0, (3.119)

ker(AT) = H-(E,3X + Tv) = H-(F,9).

Set

DY=aY+BY*.

(3.120)

Then (3.121)

ker DY = H(Y, 17).

Now we describe in some detail a few difficulties which appear in the evaluation of Log

II

III({)

given in [BL]. We will here take a toy object

(n)

to describe these difficulties. Let x(ti) be the Euler characteristics of Ox (g), Oy(rl). Then, by the Mc Kean-Singer formula [McKS], for t > 0, T > 0, (3.122)

Tr. [exp(-(tAT)z] , X(rl) = Tr8 [exp(-(tDY)z]

Of course (3.123)

X(O) = X(q)

Let Pt (x, x') (x, x' E X), Qt (y, y') (y, y' E Y) be the smooth kernels for exp(-(tAT)z), exp(-(tDY)z) with respect to dvx(x'), dvy(y'). Here Tr$ [PtT(x,x)] dvx(x), Tre [Qt (y, y)] dvY (y) will be be considered as currents on X. In IBL. Sections 9 and 13], the following result is proved.

Jean-Michel Bismut

63

Theorem 3.45. For any t > 0, (3.124)

Tlimo Tis [Pa (x,x)] dvx(x) _ T's [Qt(y,y)] dvy(y)

Also, by the local index theorem given in Theorem 1.2, ch(e,g{)}max

lim's [PT(x,x)] dvx(x) _ {Td(TX,gTX)

(3.125)

,

t-,o lim Trs [Qt(y,y)]dvy(y) _

We then have the non commutative diagram (3.126)

TTs [PT (x,x)]dvx(x)

to

{Td(TX,gT-) ch(C,g£)}max

jT_+oo ms's [Qt(y, y)]dvy(y)6Y -e-:o {Td(TY,gTY) ch('7,9")}max5Y

Needless to say, (3.126) fits with (3.122), (3.123), because of (3.86). Observe that

tAi = tDX + TV.

(3.127)

We use the notation in Section 3 e). A simple application of local index techniques shows that (3.128)

m Trs [Pt lt(x,x)] dvx(x)

=

{Td(TX,gTx)aT2}ma"

.

In view of (3.83), (3.124), (3.128), we have the new diagram (3.129)

Trs [Pt'/t(x,x)]dvx(x)

two

{Td(TX,gTx)aT2}max T-++oo

T-++oo

Trs [Qt (y, y)]dvY(y)SY

{T (NY/xl

(,i,9n)}ma"SY

t-;o - {Td(TY, gTY) ch(i,gn)}maxa

The whole point is now to find how to close the gaps in the diagrams (3.126), (3.129) at the level of currents (the gap is of course 0 in cohomology). Clearly (3.130)

tAT/t2 = tDX +

7V .

Local index theory, eta invariants, and holontorphic torsion

64

Consider the exact sequence of holomorphic Hertuitian vector bundles on Y

0-+TY-TXI. -+Nyl->0

(3.131)

and more generally any short exact, sequence as in (3.131). In [B4], for T > 0, a form 6T E Py is constructed, such that Td(TXIy, 9TxiV )

T-40 6T = Td(Ny'/x,yN,/x)'

(3.132)

lira bT = Td(TY,9TY)

T-i+oo

Then, in [BL, Section 12], it is shown that for T > 0, (3.133)

lizn Trs

[PT/t' (x, x)] dvx (x) = {ST ch (,/, 9"))"' by.

Then we have the new diagram

(3.134)

Td(Tx,9"V) Td(N?( 7x)

p

[Pt (x,x)]dvx(x) T-+O

IT->0

t=

Pr [p /t= (x,x)]dvx(x)

Ch (r/,9") bY

,p

{bTch(r1,gq)}"naxby.

IT-++oo

I T-++oo

Trs[Qt(y,y)]dvy(y)6y t=,o

{Td(TY,9TY)ch(r/,9",)}"nax6y,

Let Pol be the orthogonal projection operator on ker(AT). Then P,To is given by a smooth kernel PT (x, x') on X. Similarly let Q.,, be the orthogonal projection operator on ker(Dy), and let Qw(y, y') be the corresponding kernel on Y. Then by [BL, Section 10], we have the diagram (3.135)

Trs [Pt (x, x)]dvx (x) t-,+00 Trs [P (x, x)]dvx (x) I

Trs [Qt(y, y)]dvy(y)by

t

T->+oo

a. [Q0o (y, y)]dvy(y)6Y +-3

Part of the proof of Theorem 3.41 [BL] consists in putting together the diagrams (3.126), (3.129), (3.134) and (3.135).

Jean-Michel Bisinut

65

j) The Hodge theory of the resolution of the point. In the proof in [BL] of Theorem 3.41, or in the construction of the form JT in [B4], the following toy model of an embedding i : Y -s X appears naturally. Let VR be a real even dimensional vector space, let J be a complex structure on VR. Let V be the corresponding complex vector space, so that

VR ®R C = V ® V. Put n = dim V. Recall that if z E V. z represents

Z=v+vEVR.

Let i be the embedding {0} -+ V. If z E V, let i. be the interior

multiplication by z. Let (3.136) (AV*,

iz):O_+AnV*A"-1V*-4...

Ao(V*)=Ci0

be the obvious Koszul complex. Let r be the restriction map a E Ao(V* )10 -i i2) provides a resa E C. Then by [GrH, p. 688], the complex (AV*, olution of the sheaf i*C. Let (C- (V AV *®AV * ), a") be the Dolbeault complex over V of smooth sections of AV*®AV*.

Let Nv, Ny be the number operator of AV*, AV*. The Z-grading of the complex (Cl (V, AV * (9 AV* ), a1 + vf--li,) is given by Nv - NH.

Let r : CO°(V,AV* 0 AV*) -+ C be such that if a E C-(V,AP(V )), Q E COO (V, A9 (V*)), then (3.137)

r(a®,Q)=0if p+q>0, =aof3oif

P=0,q=0.

Observe that AVR OR C = AV ® AV*, and that r is nothing else than the restriction of a smooth form on V to {0}. Let C be the trivial complex, equipped with the chain map 8{0} _ 0 By the arguments of [BL, Theorem 1.7], the chain map (3.138)

r: (CO0(V,AV*®AV*),2;ti +iz) -a (C,-j{o})

is a quasi-isomorphism. In particular if 3 is the hypercohomology of (C°° (V, AV*

AV*), 8v+ (3.139)

99=0ifp34 0, = C if p = 0,

and r = 9l -+ C identify canonically 9l with C = Ho ({0}, C). Let now gv be a Hermitian metric on V. Let (, )A(W)®A(v*) be the corresponding Hermitian product on A(V)®A(V*) and let dvv be the associated volume form on V. Then we equip C°°(V,AV* 0 AV*) with the Hermitian product

1)vJ

(3.140)

dvv .

66


Let 'J V. be the formal adjoint of a4. Then 5V* - /Ti*. is the formal adjoint of aV + Vf--iiz. Let 0 be the Kahler form of VR,, i.e.

0(X, Y) _ (X,JY)vv .

(3.141)

Let L = GA, and let A be the adjoint of L. Put

S = -(L +A).

(3.142)

By [B4, Proposition 1.4], (3.143)

(8V +iz + 8V* iz).2

0

+

12I2

+S.

Then the Laplacian (3.143) is an harmonic oscillator. The following elementary result is proved in [B4, Theorem 1.6]. Theorem 3.46. Let ,Q E CO°(V,A(V*)®A(V*)) be given by (3.144)

/3=exp 0-

2 IIZ2I

Then /3 has total degree 0, and moreover (3.145)

IIQIIL2 = 1.

Also

(aV +

iZ)Q=0,

13146)

(OV*-V-liz)N=0. Moreover/3 spans the 1-dimensional kernel of (aV+ Finally (3.147)

i

iZ+av -f i!)2.

r/3 = 1

i.e. /3 represents canonically 1 E H°({0},C) in C-(V,A(V*)®A(V*)).

REMARK 3.47. Several remarks are in order here. First note that 0, as a (1,1) form, has total degree 0, so that indeed /3 is of total degree 0. Also observe that IIQIILZ = 1 and r/3 = 1, so that 1 -+ /3 is an isometry.

Jean-Michel Bismut

67

Now we go back to the formalism of Section 3 h). Let it' : Ny1X -- Y be the obvious projection. Consider the embedding V: Y -a Ny1X (where Y is identified with the zero section of Ny1 ).

A fundamental fact, established in [BL, Section 10] is that for T -+oo, ker(AT) = H* (F, aY) can be asymptotically described as follows. *Put (3.148)

DY=sY +aY*.

Then (3.149)

ker(DY) = H(Y, q)

.

Take a E ker(DY) = H(Y, rl). Then by [BL, Section 10], the element 7T E ker(AT) = H(X, C) canonically identified with a can be described asymptotically in a tubular neighborhood of Y by (3.150)

7T = ir'*,6exp 10N,x -

T I2 IZ

Of course 1T can be viewed locally as a smooth section of A(T*(o'l)X)®g because of (3.80).

k) The forms JT : A toy model for the analytic torsion forms. Let Y be a complex manifold. Let (3.151)

0-*L-.MAN-30 3

4

be a short exact sequence of holomorphic vector bundle on Y. Let 9M be a Hermitian metric on M, let gL be the induced metric on L. By identifying N to L1, let gN be the metric induced by gm on N. Let L, M be the total spaces of L, M. Then we have the diagram (3.152)

M

SM/S

S

Also on M, the Koszul complex (ir,*uAN*, /Tij( )) is a resolution of the constant sheaf i*C. Let V L, OM, VN be the holomorphic Hermitian connections on L, M, N, and let RM, RL, RN be their curvatures. Then the connections VL, VM define horizontal subbundles THG, THM. Put iaa Iz1

(3.153)

2

wAtA =

iaaIx1M 2

Local index theory, eta invariants, and holoinorphic torsion

68

Clearly (3.154)

Also wf-,WM induce the tautological Kahler forms along the fibres L, M.

Moreover one verifies easily that THI,THM are exactly the orthogonal bundles to L, M with respect to wc, Then we are in a situation formally similar to the one we iuet in SecWM.

tion 3 g). We will then construct the associated Levi-Civita supercounection. In our context, if U, V E TRS,

TC(UH,VH) = RL(U,V)Z, TM(UH,VH) = RM(U,V)Z.

(3.155)

Let E, F be the bundles on Y of smooth sections of A(RI *)®A(N*), A(N* )

along the fibres M, L. Let DE, V1 be the connections on E, F constructed as in Definition 3.11. DEFINITION 3.48. For T > 0, let BTM be the Levi-Civita superconnection on E, (3.156)

_

BM=(aM+C7*\+vl (Y---1ij(x)+(V+VE-c(

hr)

Similarly by making T = 0 in (3.156), we can construct the superconnection BN. Recall that A(*), A(N*) are c(MR), c(NR) Clifford modules. Let c, F. denote the corresponding Clifford actions. We have the CO° identification M = L ® N. Put A=VM-VL®VN.

(3.157)

Then A exchanges LR and NR. Let PL : M -+ L, PN : M -+ N be the orthogonal projection operators. If (fe) is a basis of TRS, put

= -E f ° (A(fa)PLZ) . Let e1,... , e2,, be an orthonornal basis of NR. Put

(3.158)

-C(APLZ)

S= Vzi (3.159)

2

2n

> c(e,;)c(e;).

Then by [B4, Theorem 3.10] (or by the more general curvature identity of [B2, Theorem 3.5], if en, ... , elm is an orthonormal basis of MR, (3.160)

B'2 = -2

2m

(Ve, +

2

(RMZ,es))2

1

12NZI2

+T

+ fTS +

72

c(APLZ)

+ ; Tr [RM] + R^(N*) .

Jeau-Michel Bisinut

69

In particular, rn

(3.161)

Bo .2_-2E(Ve; +

With some surprise, we see that Bo 112 is nothing else than the Getzler

operator [Gel,2] [BeGeV, Proposition 4.19] in local index theory. The fact that the Getzler operator (obtained by resealing the square of the Dirac operator) is itself a square is a surprise. Another surprising feature of BT '2 is that its matrix part only acts on A(N*)®A(N*) and not on A(M)®A(N*).

For T > 0, BT is essentially a perturbation of the harmonic oscillator in (3.143). For T > 0, let ST(Z, Z') (Z, Z' E A1,1) be the smooth

kernel of the operator exp (-BT'2) with respect to

m dr

.

Then

ST(Z, Z') E A(TRY)® End (A(N)®A(N*)). Then a simple fact proved in [B4, Theorem 4.2] is that Z -i Trs [ST(Z, Z)] only depends on j (Z) E Na.. Also one verifies easily that if Z E Na.,

I ST(Z, Z)j < c(T) exp(-C(T) IZ12).

(3.162)

The operator exp(-BT) is in general not trace class. Still we can define a generalized supertrace as follows. DEFINITION 3.49. Set (3.163)

Tr. [exp ,l Na

dvN(Z) T [ST(Z, Z)] (21r)dimN

Put (3.164)

iiT = 41 Trs [exp (-BT,2)1

Then we have the result of [B4, Theorems 4.8 and 7.7].

Theorem 3.50. The forms 6T lie in pB, they are closed, and their cohomology class does not depend on T. Moreover as T -+ 0, (3.165)

6T =

Td(Mgm) +O(T), Td(N,g")

and as T -+ +oo (3.166)

aT =

Td(L,gL) +0 (;)

Let NH be the number operator of A(N*). Set (3.167)

ET = § Tre [NH exp(-,i3Z.)]

Local index theory, eta invariants, and hololnorphic torsion

70

(where again, the right-hand side of (3.167) is a generalized supertrace). By [B4, Theorem 4.6]

a

(3.168)

aT

ET

as ET

= 2iir T

Also by [B4, Theorem 7.7), as 'T -4 +oo, (3.169)

CT =

DEFINITION 3.51. For s E C, 0 < Re(s) < 1/2, put B(s) = r(s1 )

(3.170)

f"T"-1(e,-e,,.)dT.

Then B(s) extends holomorphically near s = 0. DEFINITION 3.52. Put

B(L,M,9M) =

(3.171)

asB(s)1,=o

Clearly B(L, M, gm) is a generalized analytic torsion form on Y. The following result is established in [B4, Theorem 8.3].

Theorem 3.53. The form B(L, M, gm) lies in PY. Moreover (3.172)

aoB(L,M,gM) NIr

=Td(L,gL) _

Td(M,g_)

Td(Ng )

The construction of B(L,M,gM) is functorial. In view of (3.172), a natural question is to evaluate (3.173)

B(L,M,gm)+Td-1(N,gN)Td-(L,M,gM).

in PY/PYO. Using equation (3.172), it is enough to calculate (3.173) in a split situation. DEFINITION 3.54. Let D(x) be the formal power series (3.174)

r, (1) + S(( n)) +

D(x) _ nodd

S(-n) n

Jean-lM-Iichel Bismut

71

We identifyD(x) to the corresponding additive genus. Let Td (L), D(N) be the classes of Td(L,gL), Td(N.g^) in PY/PI'00. Clearly they do not depend on gL, gN

Then we have the result of [B4, Theorem 8.5]. Theorem 3.55. The following identity holds (3.175)

B(L,M,gM) _ -Td-1(N,g`v)Td(L,11,gA1)+Td(L)D(N) in PY/PY,0 PROOF : As explained in (3.173), it is enough to evaluate B(L, M, g") in the case where the exact sequence splits holomorphically and metrically. Let cp(T,x) be the function

(3.176)

p(T, x) =

T

sink I x + 4z + 4T ) sink I -x +

4

Z + 4T

0 We identify (T, x), as a function of x with the corresponding additive genus. Then in [B4, eq. (8.26)-(8.28)], it is shown by explicit computation that in the split case, for T > 0 (3.177)

ET - e _ - Td (L, gL)

(T N, 9N )

Set

C(s, x) = (3.178)

-

f

+oo

Ts-1 _ (T, x)dT,

D(x) = "s (O, x) .

We identify D(x) with the corresponding additive genus. Then by (3.170), (3.173), (3.178), when (3.151) splits, (3.179)

B(L, M, 9M) = Td(L,9L)D(N,9N)

Now we have the easy expressions for W(T, x) (3.180)

tp(T, x) _ +00 11 Cl +

ix

T 2k7r + 4k Ir2/

T

2kir + 4k 2?r2)

which makes the computation of a/ax (T, x) quite pleasing. In [B4, Appendix], Bisrnut and Soule obtain the expression of D(x) given in (3.174) by using (3.180) and the functional equation for ((s). By (3.97), (3.174),

D(x) = R(x) + I''(1)

Al

(X).

In [BL], the term related to r,(1)-&(z) disappears in the final result, because A in T(g, g{). it is killed by a corresponding term

72

Local index theory, eta invariants, and holorrrorphic torsion

REMARK 3.56. In [GS3], Gillet and Soule have obtained the genus R by evaluating the analytic torsion of equipped with the Fubini study metric, and by calculating the degree over P,,(Z) of their Todd genus 1d. They obtained the R-genus as a defect in their conjectured Rietuann-Roclh formula in Arakelov theory. It has in fact been made clear in the work of Bost [Bos] and Roessler [Ro] that the evaluation of the analytic torsion of P (C) can be obtained as a consequence of [BL]. Using the results of [BGS5], the formula of [GS3] for can then be also obtained as a consequence of [BL]. The amazing almost coincidence of the genera R of [GS3] and D of [B4] is then explained.

Jean-Michel Bismut

73

References [A] Atiyah, M.F. : K-Theory. New-York-Amsterdam: Benjamin 1967. [ADS] Atiyah, M.F., Donnelly, H., Singer, I.M. : Eta invariants, signature defects of cusps and values of L functions. Ann. of Math. 18, 131-177

(1983). [Al)

Alvarez-Gaume, L. : Supersymmetry and the Atiyah-Singer index theorem. Comm. Math. Phys. 90, 161-173 (1983).

[ABoP] Atiyah, M.F., Bott, R., Patodi, V.K.: On the heat equation and the index theorem. Invent. Math. 19, 279-330 (1973). [APS1) Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry, I. Proc. Cambridge Philos. Soc. 77, 43-69 (1975). [APS2] Atiyah, M.F., Patodi, V.K., Singer, I.M. : Spectral asymmetry and Riemannian geometry, III. Proc. Cambridge Philos. Soc. 79, 71-99 (1976).

[AS1] [AS2] [AS3]

[BeV]

[BeGeV] [BerB]

Atiyah, M.F., Singer, I.M.: The index of elliptic operators. III, Ann. of Math. 87, 546-604 (1968). Atiyah, M.F., Singer, I.M. : The index of elliptic operators. IV, Ann. of Math. 93, 119-138 (1971). Atiyah, M.F., Singer, I.M. : Index Theory for skew-adjoint Fredholm operators. Publ. Math. IHES, 37, 5-26 (1969). Berlin, M., Vergne, M.: A proof of Bismut local index theorem for a family of Dirac operators. Topology 26, 435-463 (1987). Berline, N., Getzler E., Vergne, M. : Heat kernels and the Dirac operator. Grundl. der Math. Wiss. Band 298. Berlin-Heidelberg-New-York: Springer 1992.

Berthomieu A., Bismut, J.-M.: Quillen metrics and higher analytic torsion forms. J. reine angew. Math. 457, 85-184 (1994).

[Bi] Bismut, J.-M. : The Atiyah-Singer index theorem : A probabilistic [B2] [B3) [B4]

[B5] [B6]

[B7]

approach. I. The index theorem. J. Funct. Anal. 57, 56-99 (1984). Bismut, J.-M. : The index Theorem for families of Dirac operators : two heat equation proofs. Invent. Math. 83, 91-151 (1986). Bismut, J.-M. : Superconnection currents and complex immersions. Invent. Math. 99, 59-113 (1990). Bismut, J.-M. : Koszul complexes, harmonic oscillators and the Todd class. J.A.M.S. 3, 159-256 (1990). Bismut, J.-M. : Holomorphic families of immersions and higher analytic torsion forms. Asterisque 244. Paris: SMF 1997. Bismut, J.-M. : Families d'immersions et formes de torsion analytique en degre superieur. C.R. Acad. Sci. Paris, Serie 1, 320, 969-974 (1995). Bismut, J.-M. : From Quillen metrics to Ray-Singer metrics. Some aspects of the Ray-Singer analytic torsion. In Topological methods in

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modern mathematics. L. Goldberg and A. Phillips eds. p. 273-324. Houston : Publish or Perish 1993. Bismut, J: M., Cheeger, J. : 77-invariants and their adiabatic limits. J.A.M.S. 2, 33-70 (1989). Bismut, J: M., Cheeger, J. : Families index for manifolds with boundary, superconnections and cones, I. J. F. Anal. 89, 313-363 (1990). Bismut, J: M., Cheeger, J. : Families index for manifolds with boundary, superconnections and cones, II. J. F. Anal. 90, 306-354 (1990). Bismut, J: M., Cheeger, J. : Remarks on the. index theorem for faru-

ilies of Dirac operators on manifolds with boundary, In Differential Geometry. F. Cheeger et al. eds, 59-83, Harlow Longman (1991). [BC5] Bismut, J: M., Cheeger, J. : Transgressed Enter classes of SL(2n, Z) vector bundles, adiabatic limits of eta invariants and special values of L functions. Ann. Scient. Ec. Norm. Sup. 25, 335-391 (1992). [BF1] Bismut, J: M., Freed D.S. : The analysis of elliptic families I, Metrics and connections on determinants bundles. Comm. Math. Phys. 106, 159-176 (1986).

[BF2] Bismut, J: M., Freed D.S. : The analysis of elliptic families II, Dirac

operators, eta invariants and the holonomy theorem. Comm. Math. Phys. 107, 103-163 (1986). [BGS1] Bismut, J: M., Gillet, H., Soule, C. : Analytic torsion and holomorphic determinant bundles, I. Comm. Math. Phys. 115, 49-78 (1988). [BGS2] Bismut, J.-M., Gillet, H., Soule, C. : Analytic torsion and holomorphic determinant bundles, II. Comm. Math. 115, 79-126 (1988). [BGS3] Bismut, J: M., Gillet, H., Soule, C. : Analytic torsion and holomorphic determinant bundles, III. Comm. Math. Phys. 115, 301-351 (1988). [BGS4] Bismut, J: M., Gillet, H., Soule, C. : Bott-Chern currents and complex immersions. Duke Math. Journal 60, 255-284 (1990). [BGS5] Bismut, J: M., Gillet, H., Soul6, C. : Complex immersions and Arakelov geometry. The Grothendieck Festschrift, P. Cartier and al. ed. Vol. I, pp. 249-331. Progress in Math. n° 86. Boston : Birkhaiiser 1990. [BK] Bismut, J: M., Kohler, K. : Higher analytic torsion forms for direct images and anomaly formulas, J. Alg. Geom. 1, 647-684 (1992). [BL] Bismut, J: M., Lebeau, G. : Complex immersions and Quillen metrics, Publ. Math. IHES, 74, 1-297 (1991). [Bos] Bost, J.-M. : Analytic torsion of projective spaces and compatibility with immersions of Quillen metrics. I.M.R.N. 8, 427-435 (1998). [BoCh] Bott, R., Chern, S.S. : Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections. Acta Math. 114, 71-112 (1965).

[Cl] Cheeger, J. : On the spectral geometry of spaces with cone-like singu[C2]

larities. Proc. Math. Acad. Sci. USA 76, 2103-2166 (1979). Cheeger, J. : Spectral geometry of singular Riemannian spaces. J. Diff. Geom. 18, 575-657 (1983).

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Cheeger, J. : On the Hodge theory of Riemannian pseudomanifolds. In Proceedings of Symposia in Pure Math., vol. 36, 91-146, Am. Math. Soc. Providence R.I. 1980. [C4] Cheeger, J. : 77-invariants, the adiabatic approximation and conical singularities. J. Diff. Geom. 26, 175-221 (1987). [CSi] Cheeger, J., Simons, J. : Differential characters and geometric in[C3]

variants. In Geometry and Topology, J. Alexander and J. Harer ed., Lect. Notes in Math. n° 1167, p. 50-80. Berlin-Heidelberg-New-York: Springer 1985.

Chern, S.S., Simons, J.: Characteristic forms and geometric invariants. Ann. of Math. 99, 48-69 (1974). (Cho] Chou, A.W. : The Dirac operator on spaces with conical singularities and positive scalar curvature. Trans. Am. Math. Soc. 289, 1-40 (1985). [D] Dai, X. : Adiabatic limits, non multiplicativity of signature and Leray spectral sequence, J.A.M.S. 4, 265-321 (1991).

[ChS]

[Gel] Getzler, E.: Pseudodifferential operator on supermanifolds and the Atiyah-Singer index theorem. Comm. Math. Phys. 92, 163-178 (1983). Getzler, E.: A short proof of the Atiyah-Singer Index Theorem, Topology 25, 111-117 (1986). [Gil] Gilkey, P. : Curvature and the eigenvalues of the Laplacian for elliptic complexes. Adv. Math. 10, 344-382 (1973). [Gi2] Gilkey, P. : Invariance theory, the heat equation and the Atiyah-Singer index theorem. Washington : Publish or Perish 1984.

[Ge2]

[GS1]

Gillet, H., Soul6, C. : Arithmetic Intersection Theory. Publ. Math.

IHES, 72, 93-174 (1990). [GS2] Gillet, H., Soule, C. : Characteristic classes for algebraic vector bundles with Hermitian metrics. Ann. Math. I. 131, 163-203 (1990). II. 131, 205-238 (1990). [GS3] Gillet, H., Soule, C. : Analytic torsion and the arithmetic Todd genus. Topology, 30, 21-54 (1991). [GS4]

[GrH] [Hir]

Gillet, H., Soul6., C.: An arithmetic Riemann-Roch theorem. Inv. Math., 110, 473-543 (1992). Grif$ths, P., Harris, J. : Principles of algebraic geometry, Wiley, New York 1978. Hirzebruch, F. : Hilbert modular surfaces. Enseign. Math. 183-281 (1973).

Hitchin, N.J. : Harmonic spinors. Adv. in Math. 14, 1-55 (1974). Kleiman, S. L. : The development of intersection homology theory. A century of Mathematics in America. Part II. History of Mathematics 2, 543-585. Providence : A.M.S. 1989. [KMu] Knudsen, F.F., Mumford, D. : The projectivity of the moduli space of stable curves, I : Preliminaries on "det" and "div". Math. Scand. 39, [Hi] [Kl]

19-55 (1976).

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Ma, X. : Formes de torsion analytique et families de submersion. C.R.A.S. 324 (S6rie I), 205-210 (1997). [MazMe] Mazzeo, R., Melrose, R. : The adiabatic limit, Hodge cohomology and Leray's spectral sequence of a fibration. J. Diff. Geom. 31, 185-213 [Ma]

(1990).

[McKS] McKean, H., Singer, I.M. : Curvature and the eigcufortns of the Laplacian. J. Diff. Geom. 1, 43-69 (1967). [Me] Melrose, R. : The Atiyah-Patodi-singer index theorem. Wellesley: Peters 1993. [MeP1] Melrose, R., Piazza, P. : Families of Dirac operators, boundaries and the b-calculus. To appear in J. of Diff. Geom. [MeP2] Melrose, R., Piazza, P. : An index theorem for families of Dirac operators on odd dimensional manifolds with boundary. To appear in J. of Diff. Geom.

Mathai, V., Quillen, D. : Superconnections, Them classes, and equivariant differential forms. Topology 25, 85-110 (1986). [Mill] Miiller, W. : Signature defects of cusps of Hilbert modular varieties and values of L series at s = 1. J. Diff. Geom. 20, 55-119 (1984). [Mu2] Miller, W. : Manifolds with cusps of rank 1, spectral theory and L2 index theorem. Lecture Notes in Math. a° 1244. Berlin-Heidelberg[MQ]

New-York: Springer 1987.

[Pl] Patodi, V.K. : Curvature and the eigenforms of the Laplace operator. [P2]

[Ql] [Q2]

[RS] [Ro]

[St] [W]

J. Diff. Geom. 5, 233-249 (1971). Patodi, V.K. : An analytic proof of Riemann Roch Hirzebruch theorem for Kiihler manifolds. J. Diff. Geom. 5, 251-283 (1971). Quillen, D. : Superconnections and the Chern character. Topology 24, 89-95 (1985). Quillen, D. : Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl. 14, 31-34 (1985). Ray, D.B., Singer, I.M.: Analytic torsion for complex manifolds. Ann. of Math. 98, 154-177 (1973). Roessler, D. : An Adams-Riemann-Roch theorem in Arakelov geometry. To appear in Duke Math. J. (1999). Stern, M.: L2 index theorems on locally symmetric spaces. Inventiones Math. 96, 231-282 (1989). Witten, E. : Global anomalies in string theory. In Proceedings of the Symposium on anomalies, Geometry and Topology of Chicago (1985). W.A. Bardeen, A.R. White Eds, pp. 61-99. Singapore : World Scientific 1985.

Thurston's Hyperbolization of Haken Manifolds

by Jean-Pierre Otal

Juin 1997 CNRS-UMR 128, UMPA, ENS LYON

46, A11ee. d'Italie 69364 Lyon, cedex 07, France [email protected]

A Paule Tilouine in memoriam

77

INTRODUCTION

In the early 1970's, useful connections between 3-manifolds and Kleinian groups began to emerge and set the scene for Thurston's hyperbolization theorem.

- On the one hand, techniques from 3-dimensional topology improved the ua derstanding of Kleinian groups, i.e. discrete torsion-free subgroups of PSL2(C) , the group of orientation preserving isometries of the hyperbolic space HI3 . A. Marden recognized in [Mardi some important consequences of a theory of Waidhausen for the study of geometrically finite groups (cf. §1). A fundamental result of Waldhausen gives a necessary and sufficient condition under which a homotopy equivalence between Haken manifolds (see below) can be deformed to a diffeomorphism [Wa2]. Using this theorem, Marden obtained a necessary and sufficient condition under which an abstract isomorphism between two geometrically finite groups is induced by a quasiconformal homeomorphism of S2, the formal boundary of HB3 . This condition was a step to fit the geometrically finite groups into Ahlfors-Bers theory of quasi-conformal

deformations. It is also probably Marden who first posed in print the problem of giving conditions on a compact 3-manifold to be hyperbolic [Mard, p. 461]. We say that a compact orientable 3-manifold M is hyperbolic if its interior is diffeomorphic to the quotient of R3 by a geometrically finite group. Another equivalent definition is the following. Let M be a compact orientable 3-manifold and let P be the union of the tori contained in 9M. We say that M is hyperbolic if M - P carries a hyperbolic metric i.e. a complete metric of constant curvature -I such that M is locally outwardly convex along aM-P (cf. §1). Marden observed that the irreducibility of M (see below) and the triviality of the center of 9r1(M) are necessary conditions. B. Maskit developed a construction for Kleinian groups from simpler ones. In particular, his Combination theorems provide sufficient conditions under which two

Kleinian groups can be amalgamated in order to produce a new Kleinian group [Masi]. The topological description of this amalgamation at the level of the quotient 3-manifolds is the gluing of the two corresponding 3-manifolds along a subsurface contained in their boundaries. This is parallel to the key construction used in

78

JEAN-PIERRE OTAL

the study of Haken manifolds, namely as the gluing of two simpler ones along incompressible parts of their boundaries. R. Riley, exploiting different ideas, wrote a computer program to find discrete and

faithful representations into PSL2(C) of certain knot groups. Using this program, he gave explicitly the representation of the fundamental group of the figure 8 knot [R11. A fibering theorem of Stallings [Stall then implied that the quotient of H3 by this Kleinian group was diffeomorphic to the complement of the figure 8 knot in S3. Since the quotient of H3 by a Kleinian group is a complete hyperbolic 3-manifold which conversely determines the Kleinian group up to conjugacy, 3-manifolds were inevitable side products of Kleinian group theory; however, topologically interesting examples were slow to be discovered. F. Lobell provided in 1931 what was perhaps the first example of a closed hyperbolic manifold [L8J. In 1970, E. Andreev succeeded in giving a complete combinatorial classification of 3-dimensional hyperbolic Coxeter groups with compact fundamental domains [An]: this result provided a huge family of closed hyperbolic 3-manifolds for it was known by a theorem of Selberg [Sell that each such Coxeter group contains a finite index subgroup which is torsion-free (cf. [Bo] for examples of closed quotients of an arbitrary simply connected symmetric space). But the foremost example of a hyperbolic manifold to have been appraised as a topological manifold is probably the hyperbolic dodecahedral space, which appeared in 1933 [WSJ. In this paper, H. Seifert and C. Weber describe this manifold as a 5-fold cover of S3 ramified over a link with two components and they compute its first homology group showing that it is a torsion group [WS, p. 252]. Furthermore, they observe that this manifold is not a Seifert fibered space [Sell, as a direct consequence of the triviality of the center of a cocompact Kleinian group [WS, p. 2491.

- On the other hand, the progress made in understanding 3-manifolds, especially Haken manifolds, very gradually led to hyperbolic geometry. An irreducible manifold is a 3-manifold in which any embedded 2-sphere bounds a 3-ball. By a theorem of

H. Kneser, any compact 3-manifold M without 2-spheres in the boundary, can be written as the connected sum of irreducible manifolds and copies of S2 x SI [Kn]. J. Milnor [Mill proved that this decomposition is unique up to diffeomorphisms when M is closed and orientable; see (Hell for the generalization to the case with non-empty boundary. This justifies restricting the study of compact orientable 3-manifolds to irreducible orientable manifolds.

The Sphere theorem was proven by C. Papakyriakopoulos in the mid 1970's [Pal: it says that if M is a 3-manifold with 7r2(M) # 0, then M contains an embedded 2sphere which represents a non-zero element of ire (M). Therefore, if M is irreducible, then 7r2(M) = 0. It follows that the universal cover M of an irreducible manifold M with infinite fundamental group is contractible. It was then natural to ask what the topological type of M was. For instance, when M is a closed irreducible manifold with infinite (or torsion free) fundamental group, is M homeomorphic to Ra ? And

if the answer is yes, how does rrl(M) act? When M is a Seifert fibered space, the answer to the first question is positive. Any Seifert fibered space with infinite fundamental group has a finite cover which is diffeomorphic to the product of a dosed surface by the circle [Sei]. Therefore, the universal cover of an irreducible

EYPERBOLIZATION OF 3-MANIFOLDS

79

Seifert fibered space M with infinite fundamental group is homeomorphic to HR3. Before Thurston intervened, it seems not to have been realized that the corresponding action of r1 (M) is always geometric. Excepting the Seifert fibered spaces which were thoroughly studied in the 1930's [Sell, nothing was known about the topological type of a contractible cover of an irreducible manifold. For example, it was plausible that the Whitehead manifold (a contractible open 3-manifold which is not homeomorphic to H3 [Wh]) should cover a compact 3-manifold (as a matter of fact, this was shown only in the late 1980's: the Whitehead manifold covers no manifolds but itself [My]). Great progress on this question was made in the late 1960's by F. Waldhausen [Wa2j. His methods provided a deep understanding of the vast class of Haken manifolds. A Haken manifold is an irreducible manifold M which contains an incompressible surface, i.e. a properly embedded connected surface S such that (i) the fundamental group rl(S) injects into rl(M) and the relative fundamental group r1(S,8S) injects into r1 (M, M), and

(ii) S cannot be isotoped into a component of OM (cf. §7).

Among other results, Waldhausen proved that the universal cover of a Haken manifold M is homeomorphic to 1$3 - F, where F is a closed subset of S2 = 8H3 [Wa2, p. 86]. In reality, he offered a picture of k which is reminiscent of the complement of the limit set of a Kleinian group in the compactified hyperbolic space. A new light on 3-manifolds came through the Torus decomposition. Any compact irreducible orientable 3-manifold M contains a (possibly empty) finite collection 7 of disjoint incompressible tori such that any component V of the manifold obtained

by splitting M along 7 either is a Seifert fibered space or does not contain any incompressible torus. This collection T is well defined up to isotopy, once we require that it satisfy a minimality condition and the pieces obtained by splitting M along 7 form the Torus decomposition of M. The existence of the Torus decomposition appears in a cryptical announcement of Waldhausen [Wa3] which to some extent

guided the development. The complete proof along with important applications was established by W. Jaco and P. Shalen ([J81], [JS21), and independently by K. Johannson ([Johl], [Joh2J). Although the Seifert fibered spaces were well understood topologically [Sell, there were no general methods to describe the pieces of other type in the Torus decomposition. Towards the mid 1970's, a number of algebraic properties of Haken manifold fundamental groups were established which may well have helped to convince Thurston that the non-Seifert pieces in the Torus decomposition of a Haken manifold are in fact hyperbolic. The most important is probably the Torus theorem ([Wa31, (Fe]). This theorem asserts that, if M is Haken, any non-Seifert piece V in the Torus decompo-

sition of M is atoroidal, i.e. any Z+Z-subgroup of arl(V) can be conjugated into the fundamental group of a component of OV. Any Z + Z-subgroup of a Kleinian group is parabolic (cf. §1); it follows that any hyperbolic manifold is atoroidal. The fundamental group of a Haken manifold M shares other common properties with Kleinian groups.

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(i) Each non-zero element of ar (M) is uniquely divisible, i.e. it has a unique positive root of maximal order [Sh], and (ii) if M is atoroidal and acylindrical (see below), then the group of outer automorphisms of irr, (M) is finite [Joh2].

Property (i) is easily seen to be satisfied when M is a hyperbolic manifold. When M is a closed hyperbolic manifold, (ii) follows from the Mostow rigidity theorem [MOs].

In the spring of 1977, in his lectures on hyperbolic 3-manifolds, Thurston announced his Hyperbolization theorem for Haken manifolds. The printed announcement came several years later, along with generalizations that won't be treated in the present article [Thu3]. This theorem gives a necessary and sufficient condition for a Halcen manifold to be hyperbolic.

Thurston's hyperbolization theorem [Thu3]. - Let M be an irreducible and atoroidal manifold. If M is Haken, then M is hyperbolic. Furthermore, at about the same time, Thurston formulated his Geometrization conjecture. This conjecture says that each piece of the Torus decomposition of an irreducible manifold is modeled locally on one of the following eight geometries. the

three constant curvature ones: R3, R3 , S3, and the five fibered ones: H2 x R, S2 X R, PSL2(R), Nil and Sol (cf. [Scol for a detailed survey on these geometries). This conjecture is satisfied by Haken manifolds: in view of the Hyperbolization theorem above, the proof amounts to merely observing geometric structures on Seifert fibered spaces. Thus, Thurston's hyperbolization theorem, as a particular case of the Geometrization conjecture was in harmony with the recent Torus decomposition theorem. Thus, for Haken manifolds, Thurston completely settled the Geometrization conjecture. Recall that every compact irreducible manifold whose boundary is non-empty is necessarily Haken; again, every compact irreducible manifold with infinite first homology group is Haken (cf. [Hell, [Jai). However, some of the closed hyperbolic manifolds that had been considered before Thurston were already known to be nonHaken: for instance, Haken himself [He2], and also Waldhausen knew this for the hyperbolic dodecahedral space. Thurston went on to prove that in some sense, most 3-manifolds with finite first homology group are non-Haken. His Hyperbolic Dehn's surgery theorem implies that, with only a finite number of exceptions, the manifolds obtained by Dehn surgery on a knot in S3 whose complement is hyperbolic and does not contain any closed incompressible surface are hyperbolic. At the same time, only finitely many of them are Haken ([Thai], [Hat]). (For the test case of 2-bridge knots which are not torus knots, see [lIT].)

According to the Geometrization conjecture, one should be able to replace, in the above Hyperbolization theorem, the phrase "is Haken" by "has torsion-free fundamental group". The later is clearly a necessary hypothesis. The proof of Thurston's hyperbolization theorem distinguishes two cases according as M is fibered over the circle or not. The difference arises from the fact that, in

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the fibered case, the quasi-Fuchsian groups appearing in the construction degenerate, while in the non-fibered case, they remain quasi-Fuchsian. The fibered case can be formulated as the beautiful theorem below. Recall that a 3-manifold which fibers over the circle with fiber a surface S, is determined up to diffeomorphism by the isotopy class of its monodromy, which is an element of the mapping class group Mod(S) (cf. §1). To any diffeomorphism 0 E Mod(S), one can associate a 3-manifold M# , called its mapping torus which is defined as the quotient space of S x [0,1] by the relation which identifies (x, 1) with (ci(x), 0). We say that 0 E Mod(S) is pseudo-Anosov when its action on the set of conjugacy classes of TI(S) does not act periodically on any non-trivial element [Thu2J.

Hyperbolization theorem for manifolds which fiber over the circle ([Tlu5J, [8u]). - Let S be a closed surface of genus greater than 2 and let ¢ E Mod(S) . Then MM is hyperbolic if and only if 0 is pseudo-Anosov.

The proof of this particular case of Thurston's hyperbolization theorem is completely different from the proof in the non-fibered case and it has been already quite well explored (cf. [McM3], [01). For this reason, we will restrict our attention in this article to the manifolds which are not fibered. The proofs of the two halves of the Hyperbolization theorem may nevertheless overlap, as Thurston himself observed. For example, there is the following still unsettled question: does every compact 3-manifold fibered over the circle have a finite cover that contains an incompressible surface which is not a fiber of a fibration over the circle? If this were true, the results of the present article alone would suffice to completely prove the Hyperbolization theorem. In the proceedings of the Smith Conjecture Symposium [BMJ, J. Morgan gave a survey of a part of Thurston's original proof and the book of M. Kapovitch [KaJ is also devoted to the original approach. However, in this article, we will detail the proof from another viewpoint. The crucial part relies more on Teichmiiller theory than Thurston's proof. It is due to C. McMullen [McM2J. We will here prove the Hyperbolization theorem only under an assumption which

is stronger than the assumption "atoroidal". Namely, we will assume that al(M) does not contain any Z + Z -subgroups. The main consequence is that we can completely avoid the study of Kleinian groups containing parabolic elements. (The general case is more complex, partly due to heavier notations, but it is not much more difficult.) It is in a similar spirit that this article excludes all mentions of non-orientable manifolds (see [To]).

A crucial property of a Haken manifold is the existence of hierarchies, discovered by Haken. By definition, a Haken manifold M contains an incompressible surface S, and by splitting M along S, we obtain a new manifold Ms. The incompressibility

of S implies that MS is irreducible. Moreover, if MS is not a disjoint union of 3-balls, it is Halcen (cf. §7). Therefore, the splitting process can be iterated. It is a fundamental observation of W. Haken that this process ends up, after a finite number of steps, with a disjoint union of 3-balls (cf. [Hal). This sequence of manifolds is called a hierarchy for M and the number of terms in this sequence is called the length

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of the hierarchy. For technical reasons, we prefer to use hierarchies of another type, called special hierarchies (cf. [Jal): this will allow us to prove the Hyperbolization theorem by induction on an integer, which is defined as the greatest length of a special hierarchy, and denoted by t(M) (cf. §7). Like other theorems on Haken manifolds, such as Waldhausen's well-known theorem that homotopy equivalences between Haken manifolds respecting boundary can be deformed respecting boundary to diffeomorphisms, the proof of Thurston's hyperbolization theorem in the non-fibered case involves a finite induction using a hierarchy for M. There is a further parallel between the hyperbolization procedure of Thurston and the proof of this theorem of Waldhausen. Both proofs consist of two distincts parts, one combinatorial and 3-dimensional, and the other 2-dimensional and more

geometrical. In each case, the 3-dimensional part is a hierarchical induction. In Waldhausen's theorem, the 2-dimensional part is a known theorem of Nielsen [Nie]. The 2-dimensional part of Thurston's theorem however, was an entirely new Fixed point theorem, involving the Teichmuller spaces of the gluing surfaces.

To carry out the proof of Thurston's theorem by induction on the length of M, we will enunciate before long a more general theorem, which applies to manifoldswith-corners. First, we explain the basic gluing procedure to which the inductive step will reduce.

Final gluing theorem.- Let N be a hyperbolic manifold with incompressible boundary. Let T be an orientation reversing involution of 8N which exchanges the boundary components by pairs. Suppose that N is not an interval bundle. Then if N/T is atoroidal, it is hyperbolic.

This theorem still holds if N is an interval bundle but the proof is entirely different. It corresponds to the case when NIT is fibered and the hyperbolic structure is obtained as a degeneration of a certain sequence of quasi-Fuchsian structures on

N. Note that the Final gluing theorem never directly provides a hyperbolic structure on a compact manifold with non-empty boundary. However it does so indirectly by a trick of Thurston. This trick makes the boundary of a 3-manifold invisible by covering it with mirrors: in some sense, it converts boundary points to interior points. We present an explication of this trick which is due to F. Bonahon who was first to observe that right-angled corners are sufficient. The following notion of manifold-with-corners is essentially equivalent to the notion of "manifold with (useful) boundary pattern", introduced by Johannson in his work on homotopy equivalences

([Joil, [Jo2l). A manifold-with-corners is a triple (M, 9,B°M), where M is a 3manifold, and 9 C 8M is a smooth trivalent graph such that (i)

each component of OM - 9 equals the interior of its closure, and

(ii) each component of 8°M is the closure of a component of 8M - 9.

The closure of a component of OM - 9 which is not in 8°M is called a mirror of (M, 9, 8°M) , and 8°M is called the boundary of (M, 9, 80M). One should think of (M, 9, 8°M) as a differentiable structure with corners on M, i.e. an atlas

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of class C' on M with charts modeled on open subsets of (R+)3. Then, the graph 9 corresponds to points which have a neighborhood diffeomorphic to the neighborhood of a point of (1R+)3 with 2 or 3 coordinates equal to 0, and the boundary of (M, 9, 8°M) corresponds to distinguishing a set of disjoint mirrors. Such a differentiable structure depends only on the pair (M, 9) : this follows from [Ce[ and [Do[.

The notions of irreducibility and atoroidality can be extended to manifolds-withcorners; one can also define the notion of a manifold-with-comers with incompressible boundary (cf §7). Rather than directly prove the Hyperbolization theorem for manifolds possibly with boundary, we will proceed by proving a Hyperbolization theorem

for manifolds-with-corners that have empty boundary -which will turn out to be just as strong. A manifold-with-corners (M, 9,01M) is hyperbolic when there is a hyperbolic metric on M such that (i)

the mirrors are totally geodesic,

(ii) M is locally outwardly convex along 8°M, and (iii) the components of 8M - 9 meet at right-angles along the edges of 9

.

Let (M, 9) be a hyperbolic manifold-with-corners having empty boundary and let S' C OM be a surface which is a disjoint union of mirrors. Let 9' be the graph obtained from 9 by erasing the edges whose interior is contained in the interior of S'. Then, after rounding the corners along the erased edges, (M, 9', S') becomes a manifold-with-corners and it follows almost directly by taking 6 -neighborhoods in the

ambient complete hyperbolic manifold that (M, 9', S') is hyperbolic. In particular, if (M, 9) is a hyperbolic manifold-with-corners having empty boundary, then M is hyperbolic.

Hyperbolization theorem for manifolds-with-corners. - Let (M, 9)

be

a compact irreducible oriented and atoroidal manifold-with-corners having empty boundary. If M is Haken, then (M, 9) is hyperbolic. Any irreducible and atoroidal manifold M having non-empty boundary can be easily promoted to a manifold-with-corners (M, 9) that has empty boundary and is irreducible and atoroidal: this is the mirror trick (absolute version) see §7. Therefore, Thurston's hyperbolization theorem is a consequence of the theorem above. This theorem is proven by induction on the special length of M, viewed as a manifold without corners. The most important advantage of the introduction of manifoldswith-corners is above all that the proof of the inductive step will be in strict parallel with that of the Final gluing theorem and is in the final analysis a consequence of it. Suppose that (M, 9) is an irreducible and atoroidal manifold-with-corners with empty boundary. When M is Haken, we show in §7 the existence of an incompressible surface S which is a good splitting surface. This means that the manifold-

with-corners (Ms, 9S, S') obtained by splitting (M, 9) along S has incompressible boundary. We show also that SS can be extended to a graph 9' by adding

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edges contained in S' so that (MS, 9) is an irreducible and atoroidal manifoldwith-corners with empty boundary. Suppose that (MS, Ss) is hyperbolic. Then, as observed above, (MS, Ss, S') is also hyperbolic, and further the hypotheses of the next theorem are satisfied.

Gluing theorem for manifolds-with-corners. - Let (M, 9) be an irreducible and atoroidal manifold-with-corners having empty boundary. Let S be a good splitting surface for (M, 9) and let (MS, Ss, S') be the manifold-with-corners obtained from

(M, 9) by splitting along S. Suppose that (MS, Ss, S') is hyperbolic. Then (M, 9) is hyperbolic.

This gluing theorem can be deduced from the statement of the Final gluing theorem (cf. §8). We now apply the Gluing theorem for manifolds-with-corners to prove the Hyperbolization theorem for manifolds-with-corners. The proof is by induction on the special length £(M) of M.

The induction starts at 8(M) = 0. Then M is an handlebody and MS is diffeomorphic to B3 (cf. §7). Thus (MS, Ss) can be interpreted as a polyhedron. Saying that (MS, Ss) is hyperbolic means that this polyhedron can be realized in H3 with all dihedral angles equal to v/2. A characterization of the compact polyhedra which can be embedded in H3 with prescribed acute dihedral angles is provided by the theorem of Andreev already mentioned [An]. In the case of a rightangled polyhedron, the hypothesis of this theorem turn out to be equivalent to the irreducibility and atoroidality of (MS, Ss) . Therefore the Andreev theorem asserts that (Ms, Ss) is hyperbolic. (Incidently, note that the problem of realizability of a polyhedron in H3 with various sorts of prescribed data is currently a field of intensive

study [HR].) Thus, by the Gluing theorem for manifolds-with-corners, (M, 9) is hyperbolic when t(M) = 0. The inductive step reduces similarly to the Gluing theorem. This proves the Hyperbolization theorem for manifolds-with-corners.

Now we sketch the logic of the proof of the Final gluing theorem. Let N be a hyperbolic manifold. Let G be a geometrically finite group such that N is diffeomorphic to the Kleinian manifold M(G) associated to G (cf. §1). The space 93(G) of the geometrically finite groups isomorphic to G can be parametrized by the Teichmiiller space T(ON) : this is one important application of Alilfors-Bers theorem on the existence and uniqueness of solutions to the Beltrami equation. Using this parametrization and the Maskit combination theorem (cf. §2), Thurston showed how to reduce the Final gluing theorem to the problem of finding a fixed point for a certain map on Teichmnller space. This map is the composition T* o Or of two

maps: r' : T(W) - T(ON) is the action induced by the (orientation reversing) diffeomorphism r and a : T(ON) -# T(FN) is the skinning map (see below). This translates the final gluing theorem into the following:

Thurston's fixed point theorem. - Let N be a hyperbolic manifold with incompressible boundary which is not an interval bundle. Let r be an orientation reversing involution of ON which permutes the components by pairs. Then, if N/r is atoroidal, r' o a has a fixed point.


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To prove this theorem, we will adopt the approach which was given by McMullen [McM2]. This approach originated in an observation that J. Hubbard made shortly after Thurston enunciated the Fixed point theorem. Hubbard noticed that the formula for the coderivative of the skinning map involved a well-known operator in Teichmuller theory, the Poincare series operator. As the proof of Thurston's fixed point theorem presented here relies mostly on complex analysis, we must recall briefly the definition of the Poincare series.

Let Y -+ X be a cover of Riemann surfaces. A holomorphic L' -integrable quadratic differential 0 on Y can be summed over the sheets of the cover to define

a holomorphic integrable quadratic differential 6y/xo on X. If we denote by Q(X), Q(Y) the space of integrable holomorphic quadratic differentials on X and Y respectively, this defines a map 6y/x : Q(Y) Q(X), called the Poincare series operator or Theta operator. When Q(X) and Q(Y) are endowed with their respective L' -norms, the norm of ey/X is less than or equal to 1.

In view of the formula for the coderivative of a, Hubbard suggested that the existence of a fixed point for r' o a would be easier to establish if one could prove a conjecture of Kra [Kr]. This conjecture asserts that the norm of the Theta operator associated to the universal cover of a finite volume hyperbolic Riemann surface X is strictly less than 1. It was McMullen who, in 1989, succeeded in carrying out the program of Hubbard. In [McM1] he proves a generalized version of Kra's conjecture, giving a necessary and sufficient condition on a cover Y X for the norm of ey/X to be strictly less than 1. In (McM2] he shows how this result applies to give a new proof of Thurston's fixed point theorem. §2 begins with a proof of the particular case of the Maskit combination theorem we need in order to show the equivalence between the Final gluing theorem and Thurston's fixed point theorem. Next, we study the skinning map which is defined as follows. By the Ahlfors-Bern theorem, there corresponds to any point s = (8 , , ak) E T (ON) a Kleinian manifold N' diffeomorphic to N, such that ON' = s (cf. §1). By taking the cover of N' associated to the component Si of ON, we obtain a quasi-Puchsian structure on Si x [0,1] : the Ahlfors-Bers parameters of this structure are (si, si) , where si is a complex structure on Ti , the surface Si

with the reversed orientation. Then, the skinning map assigns to ON' the point whose i-th coordinate is si . We see spots on the Riemann surface a(&N') : in the cover of N' associated to Si they are the components of the preimage of 8N' others than the canonical lift of Si S. In particular, to each spot U is associated a cover U -+ XU of a component Xu of ON'. The topological o (ON8) E

configuration of the spots on ON' and the topological type of the covers associated to them are independent of s . The cover U -+ XU associated to a spot is geometric: it arises from a compact incompressible surface contained in XU . Also the shape of the spots reflects some important topological properties of N. The basic one is that each curve on U which is not homotopically trivial projects under the covering U -+ XU to a curve on ON which is one boundary component of an essential annulus in N. In particular, the spots are all simply connected if (and only if) N is acylindrical, i.e. if N does not contain essential annuli. In §2, we compute also the

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coderivative of o : it is a convex linear combination of the Theta operators associated

to the spots.

§4 and 5 are devoted to prove the McMullen theorem that the Theta operator 6Y/X associated to a geometric cover Y -* X has norm bounded away from 1 by a constant depending only on X(X) and on the systole of X, i.e. the length of the shortest closed geodesic of X . We will follow the approach given by Barrett and Diller (BD].

§3 contains auxiliary results on Riemann surfaces which are used during the proof. The most important is Theorem 3.1 which is due to McMullen [McMIJ. It establishes a property of convergence for a sequence of triples (Xi, xi, 0i) where 0, is an integrable holomorphic quadratic differentials on the Riemann surface Xi and xi E Xi. Recall first what it means for a sequence of pointed hyperbolic Riemann surfaces (Xi, xi) having a fixed topological type to converge to a Riemann surface (X, x) . There are two cases to consider, according the behaviour of the injectivity radius inj(xi) at xi. If inj(xi) remains bounded away from 0, it means that (Xi, xi) converges to (X, x) for the Hausdorff-Gromov topology on pointed metric spaces: in this case, X is a hyperbolic Riemann surface with finite volume. If inj(xi) tends to 0, it means that (Xi, xi) with the hyperbolic metric resealed by 1/ inj(xi) converges to (X, x) : in that case, X = C with a complete flat metric. Then, if Oi E Q(Xi) and if ¢ is a holomorphic quadratic differential on X, we say that ((bi) converges uniformly to 0, when the local expression of ¢i (in a chart for Xi which converges to a chart for X) converges uniformly to the local expression of 0. Theorem 3.1 asserts that, if (Xi, xi) converges to (X, x) and if (q5) is a sequence in Q(Xi) with I 10iI I =1, (ttii) converges uniformly to a non-zero holomorphic quadratic differential

0 on X, up to multiplying 4i by a constant and up to extracting a subsequence. This theorem is proved in two steps. First, we produce a sequence of non-zero 9i E Q(XX) which converges uniformly to a holomorphic quadratic differential 0 in the above sense. This reduces the uniform convergence of ¢i = (0i/9i)9i to the uniform convergence of (the functions) (0i/9i). This follows from the classical theorems of Montel and Picard. The proof we give here of the first step is slightly different from the original one and relies on less advanced machinery than [McM1J.

§4 concerns the solution of a certain 0-problem on an open Riemann surface Y; an open Riemann surface is a Riemann surface with finite topological type but with infinite volume. On any open Riemann surface Y, the hyperbolic volume form dv is exact. Moreover, since any open Riemann surface is a Stein manifold, dv is 8-exact. Theorem 4.1, which is due to Diller [DiL, provides a well-behaved solution to the equation 8n = dv . This solution n is well-behaved in the sense that it is a 1-form of type (1, 0) whose hyperbolic norm is finite, bounded by a function of X(Y) and of the systole of Y . We consider only the case when Y has no cusps. Then Y is the union along the boundary of a compact surface Yo with geodesic boundary and a finite collection of half-infinite annuli. Using the formula for the hyperbolic metric on an annulus, one can define an explicit 1-form no of type (1,0) which is supported on Y - Yo and which solves 8% = dv on a neighborhood of the ends of Y. By construction, the hyperbolic norm of no is independent of Y, and therefore,

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87

we are led to find a well-behaved solution 77' of Or' = dv - aqo , for then 17 = 270 + will be the required solution. Since the 2-form dv - 6770 has compact support, this equation can be solved using the Green's function on Y. Estimates on the circular averages of this Green's function provide then the required bound on the hyperbolic norm of 77' .

Also in §4, we prove a refinement of Theorem 4.1, in which one we no longer have

control on the systole of Y . Let e a constant smaller than the Margulis constant. Rather than finding a well-behaved solution to the equation bh7 = dv which is defined on the entire surface Y, we find such a solution on the unbounded (i.e. non-compact)

components of the a-thick part Yl`'°°i of Y, and whose norm is bounded by a function of e and X(Y). In [BDI, D. Barrett and J. Diller show how to deduce from Theorem 4.1 the McMullen theorem about the norm of the Theta operator Gy/X associated to a geometric cover Y -+ X. This short proof is explained in §5. For the applications, one further needs to control how the norm of Gy/X can approach 1, when the topological type of the cover Y -. X is fixed but when there is no information on the systole of X X. This control can be formulated in terms of the a -amenable part of the cover Y -+ X. Recall that the cover Y -+ X arises from a proper incompressible surface S C X. Thus, any component of X10'`) or of XI`'°OI which can be isotoped into S can be lifted homeomorphically to a surface contained in Y. The c -amenable part of the cover Y -+ X is then defined as the union of the preimage of X 10,s) and the lifts of the components of Xi®'°°l which can be isotoped into S. Theorem 5.1 is made more precise by the next statement (Theorem 5.3): if Ilex/X011 is more than

1 - b for a unit norm 0 E Q(Y) , then the ¢-mass of the a -amenable part of the cover Y -+ X is more than 1- c(b), where c(b) depends on e and tends to 0 with 6. This result is also due to McMullen, who deals not only with geometric covers, but also with non-amenable covers [McM1J.

In §6, we prove, following McMullen (McM2j, Thurston's fixed point theorem.

The existence of a fixed point for r` o a is related to a contraction property for z-` o or with respect to the Teichmiiller distance. Since r' is an isometry, the contraction properties of 'r o a follows from the contraction properties of a. The results of §5 have direct consequences for the norm of d'a at s E 7(ON) . From Theorem 5.1, it follows that Ildeall is bounded away from 1 by a constant which depends only on X(ON) and on the systole of s (Proposition 6.1). Proposition 6.2 is also a consequence of Theorem 5.3: it says that if I1d,"4II > 1 - d for a unit norm 0 E Q(a(8N3)), then the ¢-mass of the e-amenable part of a(8N°) is more than 1 - e(b) where c'(b) tends to 0 with d (the e-amenable part of or(ON) is XU associated to the spots the union of the e-amenable parts of the covers U U C a(8N3) ). However, these two results of 2-dimensional nature, don't suffice to solve the Fixed point problem and another argument is needed. For this, we observe that the e-amenable part of a(8N8) can be decomposed into the union of the simply connected components of the preimage of XUO'`l and the lifts of the components of X10") or XU'OO' : the (possibly empty) union of these lifts forms a compact surface, called the e -liftable part of a(0N°). With this terminology, Proposition 6.3 asserts

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that, for sufficiently small e, if IIdBaoII >- 1- S for a unit norm 0 E Q(a(ON8)),

then the 0-mass of the c-liftable part of a(8N8) is more than 1 - c"(S) where c"(S) only depends on a and tends to 0 with S. One deduces Proposition 6.3 from Proposition 6.2 and from Proposition 6.4, whose proof rests on a more global argument: after a normalization of the limit set of G8 in C (which uses the geometry of the 3-dimensional Margulis tubes) it is a consequence of the compacity theorem for holomorphic quadratic differentials (Theorem 3.1).

To prove the Fixed point theorem, we exploit the geometric consequences of Proposition 6.3. There are two cases to consider according as N is acylindrical or not. When N is acylindrical, all the spots are simply connected, and in particular, the e-liftable part is empty. This implies that or contracts uniformly the Teichmaller distance and therefore r* o a also (for any gluing data r). Since Teichmbller space is complete, r* o a has a fixed point.

When N is cylindrical, some gluing data r may produce non-atoroidal manifolds, thereby forbidding the existence of a fixed point for r* o a. This occurs for instance when r maps one boundary component of an essential annulus to the other. Therefore, we must take into account r in proving the Fixed point theorem. In the cylindrical case, we won't prove that r* o a is uniformly contracting, but only that some iterate (r* o a)K is uniformly contracting on a certain r* o or-invariant closed subset of 7(ON) (since (r* oa)K and r* oa commute, this suffices to prove the Fixed point theorem). The principal geometric consequence of Proposition 6.3 and indeed, the only one necessary to prove the Fixed point theorem is: if IId,*aVI is sufficiently near to 1, there is an essential annulus in N which joins two curves a and ry such that a is shorter than e for the metric s and such that 'y is shorter than a for the metric or(s). If we suppose that 11d,' (r* o a)KII is near 1, then since a is contracting and since r* is an isometry, the norm of d*a at r*(r* o a)' (s) is also near one, for all 0:5 k:5 K -1. Therefore, by successive applications of Fact 6.14, we produce a sequence of K essential annuli A2 in N with boundary the union of two simple closed

curves a,, and yi, such that ai+, = r(-y) and such that the curves ai are shorter for the hyperbolic metric s than the Margulis constant. Then, by the well known Margulis lemma, each of the curves ai is homotopic to one of finitely many disjoint simple closed curves on ON of length less than the Margulis constant. Therefore, if we choose K bigger than the maximal number of disjoint pairwise non-homotopic simple closed curves on ON, two of the curves ai are homotopic on 19N. It is then easy to produce an essential singular torus in N/r, contradicting the hypothesis of

atoroidality on N/r.. §3, 4, 5, 6 are completely self-contained. They form the main part of this paper and give a complete and detailed proof of Thurston's fixed point theorem. §7 develops the theory of manifolds-with-corners which was sketched above. It is the "3-dimensional core" of the proof. Proofs are given with details, but to shorten the exposition, we use the equivariant versions of the Dehn lemma, the Sphere theorem and the Torus theorem.


89

In §8, we explain the equivariant machinery which allows one to deduce the Final gluing theorem for manifolds-with-corners from the Final gluing theorem. Then, we deduce the Hyperbolization theorem.

Thus, granting the by now standard material on Kleinian groups laid out with appropriate references in §1, the above three equivariant theorems, and the case of Andreev's theorem for right-angled compact polyhedra in H3, this survey comprises a complete proof of Thurston's hyperbolization theorem for non-fibered Haken manifolds whose fundamental group does not contain Z+Z. The extension to the general case, i.e. to atoroidal manifolds, or to "pared manifolds" (cf. [Mor]) does not encounter any difficulties that are unfamiliar or deep. The reader is invited to find the modifications necessary to establish this general case. For this, he or she needs to extend the results concerning the norm of Theta operators to geometric covers of finite volume Riemann surfaces. The topological results of §7 also need to be extended to deal with the case of "pared manifolds-with-comers". The Hyperbolization theorem for pared manifolds is covered by the (still informal) notes [OP]. I wish to thank Peter Shalen and John Stallings who have told me their memories of the Hyperbolization theorem, some of which I have tried to evoke in the first part of this introduction. Larry Siebenmann clarified their points of view and induced me to investigate the earlier literature. I thank him for being so exacting. I felt obliged to reorganize the combinatorial part of this article after he convinced me that the most direct way from Thurston's fixed point theorem to his Hyperbolization theorem goes via the natural generalization to compact manifolds-with-comers of the case for right-angled polyhedra of Andreev's hyperbolization theorem. This approach nicely complements Francis Bonahon's observation [OP] that right-angles are sufficient. The pictures which illustrate this article were drawn by Greg McShane who also patiently commented on early drafts. With a lot of criticism, Saar Hersonsky helped to bring

the all analytical part of this article to its present form. During the writing, I benefitted also from several fruitful discussions with Federic Paulin. In particular, §7 and §8 emerged from chapters he contributed to the notes [OP].

90

CHAPTER I

Kleinian groups and Teichmuller theory

We refer the reader to the books [Bea[, [BP[ and (Ra[ for more details on the first section.

1.1 Kleinian groups The hyperbolic space of dimension n is the complete and simply connected ndimensional Riemannian manifold of constant curvature -1 (we think of n as being equal to 2 or 3). This manifold has two well-known isometric models, the Riemann model

E O}, endowed with the Riemannian metric

z_ i+...+dxn x2n

Any isometry of Hn extends continuously to the boundary RR"-1-1 where it induces a conformal or anticonformal map according to whether it preserves the orientation

or not. Let Isom(Hn) be the group of orientation preserving isometries of Hn. Therefore, Isom(H2) and Isom(H3) can be identified with PSL2(R) and PSL2(C) respectively.

Definition. - A Kleinian group is a discrete, finitely generated subgroup of Isom(H").

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91

Let y be an isometry of HH" which is different from the identity. It is well known that y is either hyperbolic (it has exactly two fixed points in 8HH") or parabolic (it has a unique fixed point in OM' ), or elliptic (it has a fixed point in HH" ). When y is hyperbolic, it leaves invariant the geodesic of HH" joining its two fixed points. This geodesic A(y) is called the axis of y . The isometry y acts on A(y) as a translation of a certain distance t(y) called the translation distance of -y. From now on, all the Kleinian groups will be supposed to be torsion-free.

Let G be a Kleinian group. Then G does not contain elliptic elements and its action on HH" is properly discontinuous. The quotient space M(G) = HH"/G is a complete Riemannian manifold of constant curvature -1.

The limit set and the domain of discontinuity of a Kieinian group. Definition. - A group is elementary if it contains an Abelian subgroup of finite index.

One can show that elementary Kleinian groups are characterized among all Kleinian groups as those which act on Mn by fixing one or two points. Let G be a non-elementary Kleinian group.

Definition. - The limit set of G is the smallest non-empty closed subset of 8W' which is invariant under G. It is denoted by L(G). One can also define L(G) as the closure in 8W' of the set of fixed points of the non-zero elements of 0. It is a perfect subset of OR.

Let C(G) be the smallest closed convex subset of if' whose closure in Un contains L(G). It is a convex subset invariant by G. The quotient space N(G) = C(G)/G is contained in M(G) and is called the Nielsen core of M(G). It is the smallest closed convex subset of M(G) such that the inclusion into M(G) is a homotopy equivalence [Thi[.

For n = 2, 8N(G) is totally geodesic, but for n = 3, N(G) is not a differentiable submanifold of M(G) in general: its boundary is "bent" along certain geodesics. One

way to avoid this problem is to replace N(G) by its neighborhood of radius b in M(G). Denote this neighborhood by N6(G) . It is not difficult to see that, for any b > 0, N6(G) is a submanifold of M(G) of class CI which is strictly convex (i.e. any geodesic arc of M(G) joining two points of N6(G) is contained in the interior of N6(G) , except possibly its endpoints). Furthermore, N6(G) does not depend on S > 0 up to diffeomorphism. It is called the thickened Nielsen core.

Although 8N(G) is not a differentiable submanifold of M(G), the convexity of N(G) allows to consider the path metric that the metric of M(G) induces. When n = 3, a basic property of this distance is that it is "hyperbolic": with this induced metric, 8N(G) is locally isometric to HH2 (cf. [Thul, [Ro[). This important property will be used at the end of §6.

Definition. - The domain of discontinuity of G is the complement of L(G) in OH'. It is denoted by I1(G).

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When fl(G) 36 0, the action of G on H" U 1(G) is properly discontinuous. This can be seen using the nearest point retraction. The map which assigns to

x E H" the point of (the closed subset) C(G) which is nearest to x extends continuously to a map r : H" U S1(G) -+ C(G) called the nearest point retraction. The map r commutes with the isometries of H" which leave C(G) invariant. In particular r commutes with the elements of G. Therefore, since G acts properly and discontinuously on C(G) C H', G acts properly and discontinuously on H" U 11(G)

also and one can form the quotient space of H" U 12(G) by C. It is a smooth (analytical) manifold with boundary denoted by M(G) whose interior equals M(G). For n = 3, its boundary is a Riemann surface.

Since r commutes with the elements of G, it induces a retraction r N(G). We can define in a similar way a retraction r6: M(G) --+ N6(G) . For 6 > 0, it follows from the strict convexity of N6(G) that r61(8N6(G)) is diffeomorphic to 8N6(G) x [0, 11. Thus, M(G) is diffeomorphic to N6(G) .

Thus one can associate to G three manifolds: (i)

the manifold with boundary M(G), whose interior is

(ii) the complete Riemannian manifold M(G) with constant sectional curvature

-1, and (iii) the Nielsen core N(G) or its 6-neighborhood N6(G).

The Margulis decomposition. Let G be a non-elementary Kleinian group.

Definition. - Let e > 0. The a -thin part of M(G) is the set of points x E M(G) through which goes a geodesic loop of length less than or equal to e. We denote the a-thin part by M(G)10,']. Equivalently M(G)lo.el is the set of points where the injectivity radius is less than a/2. The closure of the complement of M(G)lo-el in

M(G), denoted by M(G)le*, is called the a-thick part of M(G).

Margulls lemma [Marg[.- There exists a constant e(n) > 0 such that, if G C Isom(H") is a Kleinian group and if x E H", then the subgroup of G generated by the elements which move x a distance smaller than e(n) is elementary. The constant e(n) depends on n but not on G. It is called the Margulis constant As a consequence of Margulis lemma, we describe now the geometry of M(G)le'el

for n = 3 when G has no parabolic elements and for n = 2 (cf. [Th1[). Let G C PSL2(C) be a Kleinian group without parabolic elements. Any elementary subgroup of G is then a cyclic group generated by a hyperbolic isometry. Let x E M(G)lo'el . For any x' in the preimage of x in H3, there is an isometry in G other than the identity, which moves a a distance less than or equal to e. Let g E G be a hyperbolic isometry. The set of points in H3 which are moved by g a distance less than or equal to a is non-empty only when the translation distance of g is less than or equal to c. Then, by reasons of symmetry, it is a neighborhood of constant radius of the invariant axis A(g) of g. We denote this neighborhood by ne(g). Let ((g)) be the maximal Abelian subgroup of G containing g. Since

93

J: P. OTAL - HYPERBOLIZATION OF 3-MANIFOLDS

g is an hyperbolic isometry ((g)) is a cyclic group generated by a root of g. Let N`(g) be the union of the neighborhoods n(h) over all non-zero elements h in ((g)) . Suppose e:5 e(3). By the Margulis lemma, the restriction of the covering map

p : H3 -a M(G) to W(g) identifies points only when they are in the same orbit of ((g)). Hence p(f(g)) C M(G) is a solid tube diffeomorphic to NE(G)/((g)). This image is a neighborhood of constant radius of the (embedded) closed geodesic p(A(g)) = A(g)/((g)). It is called the Margulis tube around p(A(g)). By Margulis lemma again, M(G)J0,E1 is a disjoint union of Margulis tubes. It is a nice exercice to show that the radius of the Margulis tube around a (very short) geodesic of length £(g) is equivalent to log(e/t(g)), independently of G ([Th5J, [OPI). Qualitatively, the shorter the geodesic, the larger the Margulis tube around it.

When G is a Kleinian group contained in PSL2(R), M(G)10'`1 can be similarly described. The set of points which are moved a distance less than or equal to e by a hyperbolic isometry g E G is a regular neighborhood of the axis of g. For c:5 e(2)

the quotient of this neighborhood by ((g)) embeds in M(G). Its image, called a Margulis tube, is diffeomorphic to an annulus. The set of points which are moved a distance less than or equal to e by a parabolic isometry 9 E PSL2(R) is an horoball. Fore 0, 11(G) has only a finite number of components with a diameter bigger than rl .

Hyperbolic annuli. We give some formulas for the hyperbolic metric on annuli, which will be used in §3 and §4.

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1 KLEINIAN GROUPS AND TEICHMOLLER THEORY

For r < r, we denote by Aer e,., the annulus: Ae,,e., = {z E C,

er < IzI < e°' }.

Any hyperbolic annulus H2/(ry) , where y is an hyperbolic isometry, is conformally equivalent to an annulus A,-R eR . The hyperbolic metric on Ae-R eR is given by: ds =

irldzj 2RIzI cos(1rV2'

The circle of radius 1 is the only embedded closed geodesic of this annulus. Its hyperbolic length is f(ry) = ire/R. For r < R, the two circles of radius e'' and a-' are equidistant curves to the circle of radius 1 at distance D such that tanhD = sin(,r 2R ). The hyperbolic length of these two circles is 72

Rcos(7r2R).

The injectivity radius of the hyperbolic metric of Ae-R,en is constant on the circle of radius e': if it is bounded above, say by e(2), the injectivity radius on the circle of radius e' is equivalent to the hyperbolic length of this circle, independently of R.

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CHAPTER 2

The fixed point problem

Let N be a connected and orientable closed 3-manifold. Let S be a (not necessarily connected) closed, orientable, incompressible surface embedded in N. Suppose that

the Euler characteristic of each component of S is strictly negative. Denote by M the complement in N of an open regular neighborhood of S. We say that M is obtained by splitting N along S (cf. §7). Since S is incompressible, M has incompressible boundary.

The boundary of M is made of the union of two copies Sl and S2 of S. There is an orientation reversing diffeomorphism f : Sl -> S2 such that N is diffeomorphic

to the quotient of M by the relation: x =_ y if and only if x E Sl, y E S2 and y = f (x). Rather than to consider the diffeomorphism f : Si S2, it is more convenient to introduce the map r : SM -+ SM defined by T(x) = f (X) for x E S, and r(y) = f -1(y) for y E S2. Then r is an orientation reversing involution of SM which permutes the components by pairs. And N is diffeomorphic to MIT, i.e. to the quotient space of M by the equivalence relation

x_y if and only if xESM and y=r(x). The core of the proof of Thurston's hyperbolization theorem is the following result.

Final gluing theorem. - Let M be a hyperbolic manifold with incompressible boundary which is not an interval bundle. Let r be an orientation reversing involution of SM which permutes the components by pairs. If M/T is atoroidal, then it is hyperbolic.

In this chapter, we prove that if a certain map from T(SM) to itself has a fixed point, then the conclusion of the Final gluing theorem holds (M is assumed to satisfy the hypothesis in the Final gluing theorem).

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2 THE FIXED POINT PROBLEM

2.1 Maskit's combination theorem We keep the same notations as in the introduction of this chapter. But we suppose that S is connected and we don't suppose anymore that N is closed. Then M has two or one components according to whether S separates N or not. We suppose also that N is not an interval bundle (cf. §?).

Suppose that M is hyperbolic. Then there exists a geometrically finite group Gl (resp. two geometrically finite groups GI and G2) such that we can identify M with W(G1) (resp. with the disjoint union of M(G1) and M(G2)) in the case that S does not separate N (resp. separates N). Maskit's combination theorem provides a sufficient condition on G1 (resp. on Gl and G2) which implies that N is hyperbolic. Since S' = Sl U S2 is incompressible in M, Maskit's theorem (cf. §1) says that the images of ir1(S1) and 7r1(S2) in Gl (resp. in Gl and in G2) are quasiFhchsian groups. We still denote these images by iri(SI), ir1(S2). For i =1,2, we set Ni = M(irl(S2)) and Ni 7= M(7r1(Si)). Since irl(Si) is quasi-Fuchsian, Ni is diffeomorphic to Si x [0, 1]. Under this diffeomorphism, one component of 8Ni gets identified with Si and will be still denoted by Si. The diffeomorphism f induces a homotopy equivalence f : N1 -+ N2. With these notations we have:

Maskit's combination theorem [Masi]. - Assume that f is homotopic to an isometry 3: N1 -+ N2 whose extension to Ni satisfies J(S1) =0-2 - S2. Then N is hyperbolic.

Remark.- This theorem would not hold if M(G1) and M(G2) were twisted interval bundles.

Proof. - We consider the case when S separates N, the proof in the other case being similar. When one of the manifolds M(G1) and M(G2) is diffeomorphic to a trivial product, there is nothing to prove. We will suppose during the proof that say, M(G1) is not a twisted interval bundle over a closed surface. Hence, only M(G2) might be an interval bundle over a closed surface. Recall that M(G2) is an interval bundle if and only if irl(Si) has index one or two in Gi, if and only if fl(G1) has two components, see §7.

For i = 1,2, M(Gi) is diffeomorphic to N8(Gi). Consider the covering pi Ni - M(Gi) . The map pi extends continuously to a map Ni U Si -+ M(Gi) U Si whose restriction to Si is an embedding. Let fi be the (unique) harmonic function on Ni such that fi(x) tends to 1 when x tends to Si and to 0 when x tends to 8Ni - Si (the existence of such a function is obtained by solving the corresponding Dirichlet problem in H3 ).

Suppose that M(G1)) is not an interval bundle. Then fi has the following two properties: (i) the level surface fi 1(1/2) maps injectively into M(Gi) under the covering map pi . This follows from the maximum principle for harmonic functions and from the hypothesis on the index of 7rl (Si) in Gi.

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109

(ii) fi- 1(1/2) is compact. The reason is that fi(x) tends to 0 or 1 as x tends to oo in M(Gi).

(iii) for b > 0, fi-1(1/2) is contained in the interior of N6(Gi). This follows from the fact that fs 1(1/2) is contained in the Nielsen core N(Gi), by the maximum principle again.

Therefore, since pi is a local homeomorphism, for each regular value c of fi which is sufficiently close to 1/2, Ei = fi 1(c) is a compact surface embedded in the interior of N6(Gi) such that pilEi is an embedding.

Also, if c is chosen sufficiently close to 1/2, E2 = f2 1(1- c) is an embedded compact surface in the interior of N6(G2) which satisfies p21E2 is an embedding.

When M(G2) is an interval bundle, then (ii) and (iii) still hold, but (i) does not

anymore. Let t be the deck transformation of the cover N2 -+ M(G2). Then t leaves ff'(1/2) invariant (inducing a degree two cover) and exchanges f21([0,1/2[) with f21Q1/2,11) . Therefore, if c < 1/2 is a regular value of f2 sufficiently close to 1/2, E2 = f2 1(1- c) is a compact surface embedded in the interior of N6(G2) such that p2 restricts to f21([1- c, 1[) as an embedding. Since by hypothesis, the index of 7r1(S1) in GI is greater than 2, we may suppose for this choice of c, that E1= f2 1(c) is contained in the interior of N6(G1) and that pilE1 is an embedding. By the maximum principle, no sum of components of Ei can be homologous to 0 in Ni . Therefore since Ei separates the two components of 8131i it is connected.

Denote by Hi the submanifold of Ni bounded by Ei and whose closure in Ni contains Si. The covering map pi is an embedding when restricted to Ei and to the end of Hi approaching S. When 7rj(Si) has index greater than 2 in Gi, it follows that pilHi is an embedding. When M(G2) is an interval bundle, the same conclusion holds by the choice of E2 .

The hypothesis of the theorem and the uniqueness of the functions fI and f2 imply: f2 oS =1- fl. Therefore E2 = 5(EI) and S induces an orientation reversing diffeomorphism from El to E2 (Ei is oriented as boundary of Hi). MI'

Let us consider now the manifold N' obtained as the result of the gluing of = N6(G1) -p1(H1) and M2 =N6(G2 --p2(H2) identifying pi(E1) and p2(E2)

by the diffeomorphism P2 0 7 o pi 1. Then N' admits an atlas with charts modelled on convex sets of H3 (since a neighborhood of 8N' is isometric to a neighborhood of certain components of the boundary of the disjoint union of N6(GI) and N6(G2)) and with coordinate changes in PSL2(C). Therefore N' is a hyperbolic manifold. In order to prove the Maskit combination theorem, it remains to prove that N' is diffeomorphic to N. For this, we could invoke Waldhausen's theorem on homotopy equivalences between Haken manifolds (cf. (Mori). But since only a small part of this theorem is necessary, we prefer to explain this point.

Suppose first that p1(EI) is incompressible in Y. Then Ei is incompressible in Ni. Therefore, by a theorem of Stallings [Stall, Hi U Si is diffeomorphic to Si x [O,1] . Since pilHi is an embedding, this implies that Ms is diffeomorpbic to V(G2) . Under this diffeomorphism, p2 0 5 o pi 1 : El -+ E2 is homotopic to

110


Figure 2.1

f : S1 -+ S2. Thus, by a theorem of Nielsen [Niel, p2 0 7 o pi 1 is isotopic to f . So

N' is diffeomorphic to N. When pi(EI) is compressible in N', there exists, by Delm's lemma, a compression disc for p, (El) in N', i.e. there is a disc D' embedded in N' which intersects pI(EI) transversally and exactly along its boundary and such that the curve 8D' is not homotopic to 0 on pi(EI) (cf. §7). Suppose that D' is contained, say in Ml C M(GI). Then D' can be lifted isomorphically to a compression disc D for E1 in N1. Hence piI(E1 U D) is injective. Therefore, the restriction of p1 to the union of HI and a regular neighborhood N(D) of D is injective. The boundary of HI UN(D) contains two or one components according to wether OD does or does not

disconnect E1. Let Ei be the component of 0(HI U N(D)) which is homologuous to E1. If it exists, the other component, denoted by B1i is homologuous to 0. It bounds therefore a compact (connected) manifold Z1 in N1. The surface Ei cuts N1 into two components, one of which, denoted by Hl', contains H1. We prove that p1JH1 is an embedding. Clearly, Hi equals the union along B1 of H1 U N(D) and Z1. The surface P1(B1) cuts M(GI) into two components: one is unbounded and contains p1(H1), the other, denoted by T1, is compact. The image p1(Z1) is compact. Since pi is an open map, p1(Zi) contains T1. Since p1 is an open map

J: P. OTAL- HYPERBOLIZATION OF 3-MANIFOLDS

III

and since M(GI) is not compact, pj(ZI) equals TI . It follows that pi is an embedding when restricted to Zi and therefore also when restricted to Hl . The surface Ea = 1(E1) cuts N2 into two components, one of which, denoted by H2, is contained in H2 . Certainly p2 restricts to H2 as an embedding since it does already on H2. Also, for some constant b' > 0, p1(D) and p2(D) are contained respectively in N5(GI) and Ns(G2). Therefore, up to replacing in the original description of N', p2(Ei) by pi (V) and b by b', we obtain a hyperbolic manifold N" (diffeomorphic

to N'). The genus of p1(El) is strictly smaller than the genus of pl (EI) since OD is not homotopic to 0 on pI (EI) . Hence this process ends up, after a finite number of steps, with the case when the gluing surface is incompressible, bringing a hyperbolic manifold diffeomorphic to N. 0

2.2 The skinning map The skinning map, introduced by Thurston, allows us to formulate the hypothesis of Maskit's combination theorem as the existence of a fixed point for a certain map defined on a Teichmiiller space. In this section, we define this map and we enunciate the Fixed point theorem. We keep the same notations and hypothesis as for the definition of the Ahifors-Bers map. For 1:5 i:5 k, g is a quasi-Fuchsian group by Maskit's theorem (cf. §1). Thus Sl(1 f) has two connected components, fli and l) . Let ri be a Fucbsian group such

that Si is conformally equivalent to HZ/ri. For each i, there is a quasi-conformal homeomorphism fi of C such that: (i)

fi(H2) = . li i

(ii)

fio -yo f,,,' =y', for all yE ri.

Set T(8M) = xi'(ri) and T(W = x7(T) i . Let s E T(8M) and let be the quasi-conformal deformation of G such that s = 8(p, (p-). Then p(I'N) is conjugated to ri by the quasi-conformal homeomorphism ip o fi . This defines a point in 9(l'). By the Abifors-Bers theorem again, ST(ri) is parametrized by the product T(ri) x T(I'i) . The first Ahlfors-Bers coordinate of (per;, ip o fi) is si . We denote the second coordinate by s;. Definition.- The skinning map associated to M is the map a : 7(8M) - 7(V) defined by

a(s) _ (4,...sk).

The Riemann surface s, can be interpreted as the "outside" structure on the component Si, whether s'i is the "inside" structure, i.e. the one which appears when one takes off the "Skin" of the manifold M. Another way is to consider the covering of M having fundamental group 7ri (Si) . This covering is homeomorphic to Si x R R. Any quasi-conformal deformation of G with Ahlfors-Bers parameters equals to (si, , si,) can be lifted to a quasi-conformal deformation of this covering. The parameters of this deformation are (si, s;) .

112


Notation.- Let s E 7(8M). We will denote o(s) by o(OMy), thinking of o(OM°) as a hyperbolic metric on aM (in the same way as we think of s as a hyperbolic metric on OM, cf. §1).

A reformulation of Maskit's combination theorem. We use now the skinning map to formulate in a different way the hypothesis of Maskit's combination theorem. We keep the same notations as in the statement of this theorem. Consider first the particular case when S separates N into two components.

Let o; be the skinning map associated to M,. Let (si, zi) E 7(8Mi) where si denotes the coordinate on the factor Si and zi denotes the coordinates on the

components of OMi others than Si. Fix zI and z2 and denote by vi'(si) the coordinate of aai(si,zi) on the factor Yi . Consider the quasi-conformal deformation M(Bjz') of Mi . Then Ni is deformed to the point with Ahlfors-Bers coordinates (si, o (si)) . The hypothesis of Maskit's combination means exactly that (1)

f(s1)=a21(s2) and f* oai(sI) = s2

When (1) is satisfied, Maskit's combination theorem asserts that M181,:') and MM'''E can be "glued together", yielding a hyperbolic manifold diffeomorphic to N. (1) can be stated in a more symmetric way. The diffeomorphism r from SI U S2 to itself defined by r(x, y) = (f -1(y), f (x)). induces a map r* : T(SI U 2 -+ T(SI U S2). It is straightforward to check that (i) means precisely that r*(Qi (sI), a2(s2)) = (SI, S2), or in other terms, that (sI, S2) is a fixed point of the composition of r* with (aj, 02) . This particular case extends similarly to the case when S is connected but does

not separate N. Consider now the situation of the Final gluing theorem, i.e.

when S is not

necesarily connected. Then M is the disjoint union of hyperbolic manifolds with incompressible boundary MI, Me, , M..

Definition. - The Teichmiiller space 7(8M) (resp. 7(W) ) is defined as the product of the Teichmiiller spaces T(OM) (rasp. 7(M) ). The skinning map a : T(OM) - T(M) is the product of the skinning maps ai associated to Mi . Let r be an orientation reversing diffeomorphism of OM which permutes the components by pairs. Then r induces a map r* : T(OM) - T(OM). With these notations, the following theorem is merely an extension to this general

case of the Maskit combination theorem in the formulation given above. Its proof follows from the original statement by induction on the number of components of S.

Theorem. - If r* o a has a fired point, then Mlr is hyperbolic. Using that, the Final gluing theorem becomes equivalent to the following:

Thurston's fixed point theorem. - Let M be a hyperbolic manifold with incompressible boundary which is not an interval bundle. Let r be an orientation reversing

J.-P. OTAL- HYPERBOLIZATION OF

113 involution of 8M which permutes the components by pairs. If M/T is atoroidal, then T' o a has a fixed point. We conclude this section by computing the skinning map associated to a connected hyperbolic manifold M which is an interval bundle. There are only two possibilities for M up to diffeomorphism. It is either the product of a closed orientable surface S with the interval [0,11 or the twisted interval bundle over a non-orientable surface T (cf. §7).

In the first case, we can identify S with the quotient 1}12/r for some Fuchsian group r . Then 7(8M) ='7(r) x T(P) and the skinning map is given by a((x, y)) = (y, x). In the second case, 8M is identified with the orientation cover S of T T. The deck transformation t of this cover reverses the orientation on S and induces therefore a map t' : 7(r) -+ T(11) . By definition, we have. a = t` .

The computation in these two cases shows that a is an isometry when ?(8M) and T(lif7l) are endowed with their respective Teichmiiller distances. Hence when M is diffeomorphic to an interval bundle, a is an isometry. However if M is connected and is not an interval bundle, then or is contracting. We will understand why it is so soon in this chapter, when we compute the derivative and the coderivative of a. There is another way to prove this contraction property, by using the Teichmiiller theorem which describes the extremal quasi-conformal map between two homeomorphic Riemann surfaces (cf. [Mor]).

2.3 The derivative and the coderivative of a McMullen's proof of Thurston's fixed point theorem entails a detailed analysis of the

derivative of r* oa. The derivative and coderivative of a at a point s E ?(8M) are expressed in terms of the geometry of the skinned surface o(8M'). Like a leopard skin, a(8M') is covered by spots.

The leopard spots. Same hypothesis and notations as for the definition of the skinning map. Let s E

T(am). Definition. - The image of a component U of n(Gs) n ip(rti) in the surface (5i)/p(r;) is called a spot. Notation. - Let U C a(OAP) be a spot covered by a component U of S2(G'). Let rU be the stabilizer of U in G-. Then U is conformally equivalent to the quotient U/rh np(I I). Hence U covers the component U/rU of 8M' (cf. Figure 2.2). We denote this component by Xu.

Remark. - The quasi-conformal homeomorphism ip projects to a homeomorphism between o(8M) and a(W) which maps the spots on o(8M) to the spots on a(BM'). In particular, the topological configuration of the spots on a(8M') does not depend on s. Since ip conjugates G with G", the topological types of the covers U -+ XI, associated to the spots do not depend on s either.

114


Figure 2.2

Since the components of 1(G) are simply connected, the inclusion of each spot in a(8M) induces an injection on the fundamental group.

The coderivative of a. Notation. - For a spot U contained in a(8M'), we denote by eu the Theta operator Au/x,, associated to the cover U --* XU.

Let 0 E Q(a(8M8)). Since the restriction 4u of ¢ to U is integrable, we can define a holomorphic quadratic differential eu4u . It is an element of Q(8M8) with norm less than or equal to Ij4'uII (cf. §1). Thus, we can sum the differentials euOU when U varies over all the spots contained in a(8M'). This defines an element of Q(8M') , denoted by EU eu4u The derivative of a.

The tangent space to T(8M) (resp. to T(a(8M))) at s is isomorphic to the quotient B(8M')/N(8M8) (resp. to B(a(8M'))/N(a(8M')). Let µ E B(8M'). Let µ(z)dz/dz be the pull-back of µ under the covering fl(G8) -+ W. By setting

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IL . 0 on L(11(GS)), we obtain an element µ E L°°(C) such that A(-f(z))7,(z) = µ(z), 7 (z)

for almost all z E C and for all ry E G-. In particular for all i, the restriction of Fa to ip(fli) is invariant under p(I;). It defines therefore a Beltrami form jai on (SZi)/p(r) . In this way, we obtain a Beltrami form µ = (µì) on a(8M5) . With the above notations, we have:

Proposition 2.1. (i)

µ -+ µ induces a map from B(W)IN(W) to B(a(i9M))1X(a(8M'))

which is the derivative of a at s, and (ii) the coderivative of a at s is given by

d9o¢ _ EevOuU

Proof.- Let 11 : B'(8M) -+ T(OM), and H : B'(a(8M)) -, T(a(8M)) be the projections defined in §1. Recall that lI (resp. H) induces an isomorphism between

B(8M)/N(8M) (resp. B(a(8M))/N(a(8M))) and the tangent space to 7(8M) (resp. 7(a(8M))) at OM (resp. a(OM) ). Choose a local differentiable section t of H, defined in an open neighborhood V of 8M in T(OM) (cf. §1). It follows from the definition of a that we have, for s E V: a(s) = Hot(s) . This implies that the map µ -+ µ projects to a map from B(OM)/N(8M) to B(a(8M))/N(a(8M)) which is the derivative of a at 8M. The differentiablity of a at an arbitrary point follows from this special case, by using the naturality of the differentiable structure on T(8M) (cf. §1).

Let p E B(OMS). Denote by iaU the restriction of ja to U, and by the pairing between holomorphic quadratic differentials and Beltrami forms on U (cf. §1). By definition, µ vanishes in the complement of the spots. Thus, for any 0 E Q(a(OMS)), we have (µ, 0) = F, (uU, 00U, U

where the sum carries over all the spots U C a(8M5). By definition, jiU is the pullback of µ under the covering U -+ XU. Since the pull-back operator on Beltrami forms is the adjoint of the Theta operator, we have (Fiu,0u)U = (µ,OU0U)xa

Therefore

(µ, 0) = (A, E eu0U) U

This proves Proposition 2.1 (ii).


116

Proposition 2.1 shows the relation between a and the Theta operators associated to the spots. It implies that, for s E T(OM) and b E Q(o(8M')) , we have 11d;a

5EIIeuIIIIOull. U

supu J1OulI I. In order to study more precisely the contraction In particular properties of a, we will first establish related contraction properties for the operators 6u . We describe now the topological type of the covers associated to the spots.

The leopard spots and the topology of M. If M is an interval bundle over a closed surface, each component of o(8M') is a spot and the cover associated to it is trivial: then d'a is clearly an isometry (in that case, we had already noticed that o was an isometry). Therefore, we assume in the rest of this section that M is not an interval bundle. Since this topological type of the covers associated to the spots does not depend on

s, we may suppose M' = M. Definition. - Let S be a compact surface contained in a connected surface X. We say that S is incompressible if it is connected and if ir1 (S) injects into nI (X). A cover Y -4 X between connected surfaces is geometric if irj(Y), viewed as a subgroup of vI (X) , is equal to the fundamental group of a proper incompressible surface S C X. In this case we say that the cover Y --- X is associated to S.

Examples. - The universal cover of any surface is geometric: it is associated to a disc. A non-trivial finite cover of a geometric cover is not geometric.

Proposition 2.2. (i)

Let U C a(8M) be a spot. Then the cover U -+ Xu is geometric.

(ii) All but finitely many spots in o(8M) are simply connected.

Proof. - We keep the same notations that we used to define the spots. Let S, be the component of 8M such that U is contained in a(Si). To simplify the notations

we denote Sli by '9, f1i lj h -and r, by r. Then r' is a quasi-Fuchsian group and 1(r') = Sl U Sl . Let U C Il be a component of the preimage of U. Denote by

r the stabilizer of U in G. Since u = U/rnr' and Xu = U/r, the cover U -+ X corresponds to the subgroup r n r' of iri(Xu) = r. Therefore, (i) is equivalent to say that r n r' is the fundamental group of a proper incompressible surface of XU .

Since r' (resp. r) is quasi-Fuclisian, the frontier of 1 (resp. U) in d.° equals the Jordan curve L(r') (resp. L(r) ). Consider the closed set F = L(r) n L(r'), which is invariant under r' n r. We note first that F is a proper subset of L(r) . For if F were equal to L(r), then 1l(G) would be the disjoint union of Sl and U. This would imply that r' is a subgroup of G of index at most 2, i.e. that M is an interval bundle (cf. §7).

Let yEr. Wehave 7(sa)n5=0 if yornI° and y(5)=5 if 'Y Ernrl. This implies that F and 7(F) are not linked on L(r), i.e. that no pair of points in

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F alternates on L(r) with a pair of points in 7(F) . We consider now distinct cases according to the cardinality of F.

1) IF? 2. Let C(F) denote the convex hull of F in U U L(r) for the hyperbolic metric on U .

Lemma 2.3. - Under the covering U -' XU , the frontier of C(F) in U maps to a disjoint union of embedded closed geodesics.

Proof. - Let g be a geodesic in the frontier of C(F) in U. Since, for 7 E r the endpoints of g do not alternate on L(r) with the endpoints of y(g), g maps to a geodesic g on XU without transverse self-intersections. Since two distincts translates

of F are not linked, two distincts translates of C(F) intersect atmost along their frontiers; therefore, 8C(F) maps to a disjoint union of embedded geodesics. Suppose for a contradiction that for some geodesic g C 8C(F) , g is not a closed curve. Then

g is not compact. Hence there is an infinite sequence (7k) of elements of I' such that the geodesics 7k(g) are distincts and that their endpoints accumulate on two distincts points of L(r). Since the endpoints of g are contained in the frontier of F in L(r), the components 7k(ft) of 11(G) are all distincts for sufficiently large k . Then fl(G) has an infinity of components with diameter bounded from below by a non-zero constant. This is impossible (cf. §1).

0

1a) IF = 2. Then C(F) is a geodesic and, by Lemma 2.3 its projection is an embedded closed curve. Let 7 E r be the element represented by this geodesic. Then 7 leaves also 11 invariant. For, if this is not true, the components (7)"(S2) of f2(G) are all distinct and their diameter is bigger than the diameter of F. This is impossible (cf. §1). Thus, 7 E F' and r n r, is equal to the cyclic group generated by 7. This proves Proposition 2.2 (i) in this case.

1b) IF > 2. In this case, C(F) has non-empty interior. Let E' be the projection of C(F) to XU. By Lemma 2.3 and since XU is compact, the projection of 8C(F) to XU is the disjoint union of a finite number of embedded closed geodesics. Therefore, E' is a compact connected surface with geodesic boundary (however, the projection

of some components of OC(F) is maybe contained in the interior of E'). The complement in E' of an open regular neighborhood of the projection of OC(F) defines an incompressible surface S. Let S be the component of the preimage of S that is contained in-C(F). Since S is contained in the interior of C(F), we have,

for 7Er: 7(S) =S if 7Ern1" and7(S)nS=O if 7¢rnr'. Therefore rnr' equals a1(S) (up to conjugacy). This proves Proposition 2.2 (i) in this case.

2) F=0. Then, for any non-zero 7 E r, we have 7(U) n U = 0. Hence U is homeomorphic to U which is therefore simply connected. Thus the cover 0 -> XU is geometric.

11 8


3) F={f}. We show that this cannot happen in our case - when G does not contain parabolic elements (if G contained parabolics elements, it could happen that F were reduced to one point; a slight modification of the next argument could prove however Proposition 2.2 (i) in this situation as well).

Let g C 5/r' be the projection of the Jordan are L(r) - (f I. Suppose that g is compact. Then g is a closed curve which is not homotopic to 0 since L(I') - If} is not compact. Thus g is homotopic to a closed geodesic, and any of its lifts to SZ accumulates to the two fixed points of some hyperbolic element of r' (since G does not contain parabolics elements). This contradicts the fact that L(r) - {f} accumulates to f .

Thus g is non-compact. Since Sl/T' is compact, there exists a sequence of distincts elements 1k E r such that ryk(L(I') - {f}) accumulates to a point p E Sl . Then the domains yk(U) are distinct components of O (G) and their diameter is bigger than the distance from p to L(r). This excludes the case 3) and finishes the proof of Proposition 2.2 (i). To prove (ii), recall that, for any spot u C f t/I'', xl (U) maps injectively into iv. In particular, each spot is homeomorphic to the interior of a compact surface with boundary. Only finitely many spots can have strictly negative Euler characteristic,

since such a spot contributes at least -1 to the Euler characteristic of Sl/I". To exclude the presence of an infinity of spots which are homeomorphic to annuli, we argue by contradiction. Then there are also infinitely many spots U; C St/I" which are homotopic to the same simple closed curve c. These spots can be lifted in SZ to distincts components of 11(G) which have the same endpoints as some lift c of c. Since their diameter is bigger than the distance between the two endpoints of c, we obtain a contradiction. This finishes the proof of (li). 0 The next result can be proven with the same arguments as Proposition 2.2.

Corollary 2.4.- Let St and U be components of 11(G) with stabilizers r' and r respectively. Let ry E r be a hyperbolic element which has one fixed point in L(r) n L(V) . Then ry E r' n P. 0

Acylindricity. The hypothesis that all spots are simply connected will introduce an important dichotomy in the proof of Thurston's fixed point theorem. We now show that this situation reflects a topological property of M, namely that M is acylindricaL

Definition. - Let A denote the annulus S' x [0,11. Let M be a compact 3manifold. A continuous map f : (A, 8A) -+ (M, 8M) is essential if it induces an injective map on 7rj(A) and on ir1(A, 8A) . The image f (A) is an essential annulus.

We say that M is acylindrical if it does not contain any essential annulus.

Fact 2.5. - The manifold M = M(G) is acylindrical if and only if all the spots contained in a(8M) are simply connected.

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119

Figure 2.3 Proof. - Suppose that M is not acylindrical. Let f : A -+ M be an essential map. Let f* be the map induced by f on the fundamental group. Then f * (7r, (A)) is a cyclic group generated by a hyperbolic element g. Let f : A -+ M be a lift of f to the universal cover. Since f induces an injection of aI (A, 8A) , the two components

of f(8A)) are contained in distincts components ft and a' of f1(G). Then, the intersection of the closures of f1 and f1' in U contains the two fixed points of a conjugate of g. It follows from the proof of Proposition 2.2 that the image of ft' on the component of a(8M) which is covered by f1= - S2 is a spot that is not simply connected.

Conversely let U C v(8M) be a spot which is not simply connected. Suppose that

U is contained in the component v(SS) of o(8M). Let U C f1E be a component of the preimage of U. Since -U is not simply connected, and since i1(U) maps injectively into aI (Si) = N , U is invariant by a non zero element of r . Then the intersection of U with L(1',) contains the two fixed points of this element and it is easy to construct an essential map from A into M (cf. Figure 2.3).

120

CHAPTER 3

Holomorphic quadratic differentials

In this chapter, we study triples (X, x, 0) where X is a connected Riemann surface, x E X and 0 E 0(X) . We won't describe the topology on the set of these triples in all its generality. The reader is referred to [McM1) for the general definition. We will rather explain, in an elementary way, what it means for a sequence of such triples to converge, so that the proof of the main theorem, Theorem 3.1, can be reduced to classical compactness theorems on holomorphic functions.

3.1 Compactness properties of holomorphic quadratic differentials Definition. - A pointed Riemann surface is a pair (X, x) where X is connected Riemann surface and x E X X.

We recall first how the space of pointed compact hyperbolic Riemann surfaces can be compactified ([Thul),[Mu)).

Limits of pointed R.iemann surfaces. Consider a sequence (Xi, xi) of pointed compact hyperbolic Riemann surfaces with fixed topological type. The behaviour of this sequence, viewed as a sequence of pointed metric spaces where X= is endowed with the hyperbolic metric depends on the injectivity radius of Xi at xi. This is the largest radius of an open embedded hyperbolic ball centered at xi. It is denoted by inj (x;) . Since the topological type of X; is fixed, the hyperbolic volume of X, is constant: in particular inj(xi) is bounded from above by a constant depending only on X(Xi). We distinguish two cases according to whether inj(x,) is bounded from below by a non-zero constant or not.

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a) inj(xi) is bounded from below away from 0.

Let us identify Xi with the quotient of 1)2 by a Fuchsian group ri in such a way that the origin 0 E 1)2 maps to xi. By definition, the Dirichlet domain of ri with respect to 0 is the set of points closer to 0 than to any of its translates by non-zero elements of ri. It is a finite sided convex polygon Ti. By an Euler characteristic argument, the number of sides of Ti is bounded independently of i . The hypothesis on inj(xi) implies that Di contains a ball centered at 0 of radius independent of i. Therefore, up to extracting a subsequence, (Di) converges to a finite sided polygon (having perhaps some vertices on the circle at infinity) which is the Dirichlet domain

of a discrete group F. This group has finite covolume and is called a geometric

limit of ri. Let X = D2/r and let x be the projection of 0 E 1)2. We say that (X2,xi) converges to (X,x). One can prove that this convergence is equivalent to the convergence of ri to r, for the Chabauty topology, i.e. for the Hausdorif topology on closed subsets of PSL2(R) [CEGI.

b) inj(xi) tends to 0. We use the formulas for the hyperbolic metric on an annulus (d §1). Let e:5 e(2). Suppose that xi belongs to a component of X$°'ì which is a Margulis tube around a geodesic gi. Consider the geometric cover of Xi with fundamental group isomorphic to the cyclic group generated by gi. This cover can be identified conformally with Ae-Ri eRi , for Ri = rr2/C(gi) . By the Margulis lemma, the a-thin part of Ae-Ri eR, embeds under the covering A,-R, eR, -i Xi, and in particular, the annulus A,-Pi,.,,, for 2Ri

Pi = IT

cos-1490) .

e

Let a"i be the lift of xi which is contained in Ae evi . Denote by Y, the image of A,-R, eR, by the homothety of ratio 1/ii. Up to extracting a subsequence, Yi converge to the annulus A°,,,, = C' and the hyperbolic metrics on Y , resealed by the factor 1/ inj(xi) , converge to a flat complete metric on C'. We say that (Xi, xi) converges to (C',1). This normalization allows us, by looking at an appropriate cover of Xi -which is either the universal cover 1)2 or the annulus Yi- to compare charts around xi E Xi when Xi varies.

Definition. - Let (Xi, xi) be a sequence of pointed compact hyperbolic Riemann surfaces which converges to (X, x). Let ql E Q(Xi) and let 0 be a holomorphic quadratic differential defined on X. If inj(xi) does not tend to 0, let ql (z)dz2

(resp. (z)dz2) be the pull-back of Oi (resp. q) to 1)2. If inj(xi) tends to 0, let Y be the covering of XLassociated to the fundamental group of that component and let i(z)dz2 (resp. O(z)dz2) be the pull-back of ii (resp. 0) to Y, . We say that ¢i converges uniformly to 0 if ci converges to 0 uniformly over compact sets (in 1)2 or in C*). Theorem 3.1. - Let (Xi, xi) be a sequence of pointed compact hyperbolic Riemann surfaces with a fixed topological type which converges to (X, x) . Let Oi E Q(Xi)

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3 HOLOMORPHIC QUADRATIC DIFFERENTIALS

with ¢i # 0. Then there exists constants c, and a non-zero holomorphic quadratic differential 0 on X such that (c,oi) converges uniformly to ¢ up to extracting a subsequence.

Remark. - One important feature of this theorem is to produce a non-zero limit, merely by applying a homothety to ¢i . The existence of a limit for a sequence (0i), when I 10iII is bounded can be shown by a more elementary argument. It follows from the precompacity of a sequence of holomorphic functions with bounded L' norm. However, this is not sufficient to guarantee that the limit is non-zero, even if II4.II = 1.

Remark. - The limit 0 produced by Theorem 3.1 is not necessarily integrable. The basic example of a non-integrable limit appears in Lemma 3.4. In order to prove Theorem 3.1, we construct first non-zero holomorphic quadratic

differentials Oi on X, and 0 on X such that (0i) converges uniformly to 0 as i tends to infinity.

Proposition 3.2. - Let (Xi, xi) be a sequence of pointed compact hyperbolic Riemann surfaces with a fixed topological type which converges to (X, x) . Then there exists non-zero 8, E 4(X,) and a non-zero holomorphic quadratic differential 0 on X, such that (0i) converges uniformly to 0, up to extracting a subsequence.

Proof. - We consider two cases according as X is a finite volume hyperbolic surface or is an annulus.

1) X is a finite volume hyperbolic surface. We begin with the following lemma.

Lemma 3.3. - Suppose that (Xi, xi) converge to (X, x) where X is a finite volume hyperbolic surface. Then %i/x. converge weakly to 6pz/X, i.e. for any P E Q(ID2), @D2/X,P converges uniformly to 6D2/XP.

Proof. - Assume that Xi and X are uniformized by Fuchsian groups r, and r acting on D2, in such a way that 0 E D2 projects to xi and x respectively. Let P E Q(D2) . Fix a compact set K C D2. For each r < 1, there is a compact neighborhood Cr of Id E PSL2(R) such that, for any g f Cr, g(K) is contained outside of the disc Or of radius r. We may assume that the frontier of Cr in PSL2(R) is disjoint from 1' so that ri n Cr -+ T tl Cr as i tends to co. We can choose r so that the IPA-mass of DD2 - D,. is arbitrarily small. Therefore, for i sufficiently large, the difference J6Mp1XP(z) - 6D2/XP(z)J can be made arbitrarily small over K : this follows directly from Cauchy's formula.

We use now the fact that 602/X is surjective when X has finite volume. The proof can be sketched as follows. For a general Riemann surface X, the image of the restrictions of the polynomials to 1D2 is dense in Q(X), when Q(X) is endowed with the Weil-Peterson scalar product (cf. [Kr]). If the hyperbolic volume of X is finite, Q(X) is a finite dimensional vector space (cf. §1). It follows that en2/X is surjective in this case. For a composition of covers of Riemann surfaces Z -* Y - X,

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123

we have: eZlX = Ay/x o ®Z/ y. Hence if X has finite volume, the operator ey/X is surjective.

Lemma 3.3 and the surjectivity of ®D'/x yield a proof of Proposition 3.2 as follows.

1a) X is different from the thrice punctured sphere. In this situation, the vector space Q(X) is non-trivial. By the above, it contains a non-zero element of the form BD2/XP. By Lemma 3.3, (®c/X.P) converges uniformly to 0 = AD2/XP. For all sufficiently large i, B; = OD2/X,P is non-zero. This proves Proposition 3.2.

1b) X is the thrice punctured sphere. Then the space of integrable holomorphic quadratic differentials is trivial and the reasonning above cannot be applied. The proof we will give could be extended with a minor modification to the case when X has finite volume but is not compact. However, we present it only when X is the thrice punctured sphere: any puncture on X corresponds to a closed geodesic g; C X, whose length tends to 0, and since X is a thrice punctured sphere, we can choose such a geodesic g; which gives rise to a single puncture on X.

The cover of X; associated to the cyclic subgroup of r; representing the curve g; can be conformally identified with A,-R, CRi (cf. §1). Let e1I I S 1, we have e(E) :5 1. Equality holds if and only if the restriction

of µ to E equals ¢/I¢I a. e. Proposition 3.10 [McM2). - Let X be a compact hyperbolic Riemann surface. Let ¢ E Q(X) with IIOII =1 and let p E B(X) with IIµll:51 Let E C X be a measurable subset of ¢ -mass bigger than m for some m > 0. Suppose that each point of E is the center of an embedded hyperbolic ball on which the efficiency of the pairing 1- cma, where between qS and µ is less than 1- a, for some a > 0. Then c > 0 depends only on x(X) . Proof. - By a Vitali type argument, we can extract from the family of balls provided by the hypothesis, a family of disjoint balls {B,) such that the balls 5B, cover E. Then by Proposition 3.8, we have.

fii_<J

(1)

5

Bu

Isc(5) fB; II.

Since the balls B, are disjoint we have (01U)CUB; +EIIOIIB; (

(0, µ) = II0IICUB, II#IICUB,

i

II0IIB,

Using (1) we obtain

(0,A):5 1- II0IIuB; + (1- a)I I0I IuB; :5 1- am/c(5).

0

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CHAPTER 4

The volume form on open Riemann surfaces

Definition. - An open Riemann surface is a connected hyperbolic Riemann surface of finite topological type but of infinite hyperbolic volume.

Equivalently an open Riemann surface is conformally equivalent to D2 , to an annulus, or to the quotient of D2 by a Fuchsian group r with 11(1') o @.

Let Y be a hyperbolic Riemann surface. A 1-form of type (1, 0) on Y is a smooth 1-form rl which can be written in any complex chart on the form ii(z)dz. In any complex chart, the hyperbolic metric can be written on the form X(z)ldzl. So, when rl is a 1-form of type (1,0) on Y, the quantity Irl(z)I X(z)-1 is independent of the chart.

Definition.- The function (17) : Y -+ R defined by (17)(z) = jr7(z)jA(z)-1

is

called the hyperbolic noun of rl.

Thus (ii) is the norm of the 1-form n when Y is endowed with its hyperbolic metric. The norm of any tensor on Y can be similarly defined. For instance, when w is a 2-form on Y , (w) equals the ratio w/dv where dv is the hyperbolic volume form on Y.

Theorem 4.1 [Di]. - Let e > 0. Let Y be an open hyperbolic Riemann surface whose systole is bigger than e . Then there is a 1 -form q of type (1, 0) such that (i)

o+i = dv, and

(ii)

1I(77)I1

is finite and bounded by a constant C which is a function of e and

X(Y)-

Remark. - This theorem applies to give an isoperimetric inequality for domains in Y. If 7C C Y is a compact domain with smooth boundary, a direct application of the Stokes formula gives

Area(X) S Ct(8lC).

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131

This means that the hyperbolic metric of Y satisfies a linear isoperimetric inequality with a constant depending only on the systole of Y and on x(Y) .

Proof. - We consider only the case when Y has no cusps. Suppose first that Y is conformally equivalent to a disk or to an annulus. On W, the hyperbolic metric is given by

ds=1-IZ12IdzI. Hence the 1-form

-2ix '1=1-IZI2 dz satisfies Srj = dv and (>1)(z) = IzI. Therefore II(n)II. = I and n satisfies the conclusions of Theorem 4.1.

Suppose that Y is conformally equivalent to Ae-R eR = {z E C,

e-R < IzI < eR}.

The hyperbolic metric on this annulus is ds = 2RIzI

(ir

I dzl.

)

Hence the 1-form

-irr 2Rz

tan(x

log II 2R )dz

satisfies A? = dv . Moreover for all z E Ae R eR , we have (q) (z) =

I) < 1.

Therefore rl is the required differential. We note also that I I (+l) I I =1 is independent of the systole of Y .

In the other cases, since Y has no cusps, the Nielsen core of Y is a compact surface Yo with geodesic boundary . The surface Y equals the union of Yo and a collection of half-infinite annuli. For an open Riemann surface such Y a lower bound on the systole does not guarantee that Y remains in a compact set of metrics, since the length of 0Yo could tend to infinity. For instance, imagine a pair of pants tending closer and closer to a bikini. The strategy to prove Theorem 4.1 is to use the explicit solution constructed above in the half-infinite annuli which are components of Y-Yo and then to extend it over Yo using the Green's function on Y.

The Green's function on a Rlemann surface X. Definition. - Let X be a Riemann surface. A Green's function of X is a positive function G(.,.) on X x X - diagonal which satisfies:

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4 THE VOLUME FORM ON OPEN RIEMANN SURFACES

AG(., y) = by , i.e. for x not equal to y, the function x --, G(x, y) is harmonic and in a holomorphic chart around y, the function (i)

x -+ G(x, y) + 27r

log Ix - yl

is harmonic, and

(ii) G is minimal among all positive functions satisfying (iii).

In D2, the Green's function equals

-

log tanh(d(2 y)

),

where d is the hyperbolic distance. But there does not always exist a Green's function on a given Riemann surface (cf. [Ah3J, [Nic], [Ts]). For instance on a hyperbolic Riemann surface of finite volume, it does not. But, when a Green's function exists,

it is unique. Let Y be an open Riemann surface isomorphic to the quotient of D2 by a Fuchsian group r. Consider the positive function on D2 x 1D2 - diagonal (1)

G(x, y) =

-2 E

logtanh(d(x,-f(y))).

7Er

This series is invariant under r and therefore induces a function G on Y x Y diagonal.

Lemma 4.2. - Let Y = 02/T be an open Riemann surface. Then for all a > 0, the series (1) converges uniformly for all pairs (x, y) such that d(x, l: (y)) ? a. The function G is the Green's function of Y. Proof. - For d(x, ry(y)) >- o t, we have: - log tanh( d(x,2 (y)))

C(a)e-d(xrr(V))

for some constant C(a). By applying the triangle inequality, we see that the convergence of E Er e d(o,7(O)) implies the convergence of the series (1). From the formula of the hyperbolic metric in D2, we obtain that the general term in the second series is equivalent to (1- iry(0)j2). A direct computation based on the fact that the Mobius transformations preserve the cross-ratio implies that for any 0 E 81192 we have (cf. [Nic])

(Y-1)'(0) _ (1Since Y is an open Riemann surface, 1)(I) : 0. Let I C (2(e) be a small interval which is disjoint from all its translates by non-trivial elements of r. The total euclidean length of the union of the translates of I is less than the length of the circle. Therefore

E J h7'(0)ldo 0.

Consider the 1-form 'i' on Y which vanishes on Yo and which is defined on Ak by ,/I Ak = Akylk, where Ak(z) =

A(logRkz!

The 1-form rl' is smooth and satisfies A,-R,,,-R,/2. Thus the 2-form

)-

q' = dv on the union of the annuli

dvo=dV-;+l'=(1-EAk)dv-EaAkAgk has compact support in Y. Therefore we can define a function h on Y by (2)

h(z) = 4 fy G(z,w)dvo(w).

The 2-form '08h equals dvo. Therefore, rl = 77'+ 8h is the required 1-form if we can show that its hyperbolic norm is bounded by a function of e and X(Y). Since the hyperbolic norm of r( on Y is easily seen to be bounded only in terms of A, it suffices to study the L°° -norm of (0h).

4 THE VOLUME FORM ON OPEN RIEMANN SURFACES

134

Lemma 4.3. - There exist positive constants CI and C2 such that if u is a C2 function on 1D2 , one has

II(8u)II. < CIIIuII, +C211(aOu)II00

Proof. - Using the invariance of (.) under conformal automorphisms of D2, it suffices to prove that, for appropriate constants CI and C2 , the right-hand side is bigger than (8u)(0). For z contained in the disc 1/21D2 we have

I°u1 `

(1- Iz12)21108U)II00

< CII(8Ou)II00

Green's formula on 1/2D2 gives

u(z) =

Ji/2D'

G(z, w)Du(w)dw +

8G (z, w)u(w)dw. I1w1=1/2 8v

where G denotes the Green's function on 1/21D2. By differentiating this expression twice, we obtain Lemma 4.3.

By Lemma 4.3 (applied to the lift of h to D2), in order to obtain the required bound for II(8h)II,,,, it suffices to bound 11(OOh)II,,,, and 11h1[.. In fact it is easy to bound II(58h)II,o. The computation of Oak shows 11(aak Arlk)II= 0 is chosen for the moment small with respect to g' and to the distance between Z and 8KI"',°OI, but will be fixed more precisely later. Consider the 1-form 77,,, that was constructed in Theorem 4.6 on the unbounded components of Yn ,OOt . Its hyperbolic norm is less than a constant C depending

only on p and on X(Y). Since Wk''°°I C Y?* (Fact 5.4), we obtain by imitating the argument used at the end of the proof of Theorem 5.1:

>1

(K* +r.ml_Zn(*)) 61n(IFnI) - C 8(K'µ,+r.eo[-Zr(r)) BnI Fnlln I = ClArea(Kn '+r,°°f - Zn(r))I If r0 is sufficiently small, then for all r 0 which depends only on X(X). Choose A:5 p/2 such that (8K,I,21".00I) < v/2 and such that OKI"-ODI n Z = 0. Then, for all r:5 p', we have Area(K,I," +r,°`() - f(8Kn'+x,001) > I' 2

With this choice of p' , the number of zeroes of 8'that are contained in Kn',0°I is bounded above independently of n because it converges (counting multiplicity) to the number of zeroes of 0 contained in Kf"-0°f.. Hence the area of Z,,(r) and the length of its boundary are bounded from above by c1r2 and c2r respectively for constants cl and c2 which are independent of n. Therefore, there is some r0:5 )i such that, for all r -

f8(.Kk"+"-1-Z.(r))

2C

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Therefore TO

1

m(r)(f

0' (JF Ids) -

Q(8(Kn`'+T,ool

- ZZ(r))))dr

a

is bounded away from 0 independently of n by Fact 5.5. This leads to a contradiction

as in la). 2) Kn '°°1 does not lift to Y for sufficiently large n. An important subcase to keep in mind occurs when Y,, is the universal cover of X, . However this situation could be handled with the same methods as above. The main difference between the first case and second case occurs when K;,µ ODD although not liftable to Y,,, contains some boundary components which are liftable. Because of such curves, certain components of Wk',-[ intersect Yno,µl for µ' 0 is a lower bound for (emmJ on K,, - Z,, (r) (cf. Lemma 5.2). Since Y,, covers X,,, its universal cover is naturally identified with HD2 . Let P.

be the lift of F,, to D2. Let Z (resp.

Z (resp. Z,, ) in 12 .

Lemma 5.6. - Up to extracting a subsequence, the functions P,, converge uniformly on compact subsets of D2 - Z to a holomorphic function P.

Proof. - Let X C ID2 be a compact set. Under the covering ID2 --+ K, the "degree" of the projection K -+ K is finite, i.e. the cardinality of the preimage of any point of K which is contained in K is bounded, independently of that point. Hence since (X,,, tend to (K, x) , the degree of the projections K - X is bounded independently of n. The same property holds a fortiori for the projections K Y . By the uniform convergence of @ O to 0, Z converges to Z and

Z (r) converges to 2(r) when n tends to oo. It follows that for any r > 0, the Ll-norm of F,JX - Z(r) is bounded independently of n. By Cauchy formula, this implies that F converges uniformly to a holomorphic function F on compact sets in X - Z, up to passing to a subsequence. 0

2a) F is not constant. Since P is holomorphic and not constant, there exists a point q E 12 - 2 such that F(q) has a non-zero imaginary part. By the uniform convergence of F to F, there exists positive numbers a, n and p so that for all q in the ball B(q, p) and for all sufficiently large n, we have

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5 CONTRACTION PROPERTIES OF THE THETA OPERATOR

IFP(q)I > a > O, and

(i) (ii)

0 < q < arg Pn(q) < a - i .

Suppose that q is near the origin in D2, so that the projection Bn of B(q, p) on Xn is contained in K?,-[. If p:5 A12, Bn is embedded for all sufficiently large

n. Since q 0 Z, the ball B(q, p) is at distance bigger than p from Zn for any sufficiently small p and for all sufficiently large n . In particular Bn is contained in Kn - Z,,(p). Let qn denote the image of q in Y,,. For sufficiently large n, B(qn, p) maps homeomorphically to Bn, under the covering Yn -+ Xn .

Let x and y be complex numbers such that x + y = 1, I xI ? a > 0 and 0 < ri _ c(rl+a)

Thus, for all sufficiently large n

Iennl f (OnI FnI - 1)(enOn)dv

f-3(Bo) I0ni -

J

>

(IF.(z)I + I1- Fn(z)I - 1)m(p)dv a(v,,.p)

?c(q ,a)m(p)

dv,

fB

where m(p) is a lower bound of (e,Qin) on Kn - Zn(p). This is impossible as tends to 1. 2b) F is a non-zero constant.

Let yn E Yn be the image of 0 E IDa under the covering map D2 -i Y,,. Since ii projects to xn, inj(yn) ? e. Let B(yn,R) denote the ball in Yn of radius R centered at yn.

Lemma 5.7.- For any sufficiently small positive r, the hyperbolic area of B(yn, R) - Z,,(r) tends to oo with R, uniformly in n.

Proof. - Since Kl'`'°°I does not lift to Y,,

,

yn is contained in an unbounded

component of Y?'OD . Denote B(yn, R)(µ'0°1= B(yn, R) n Yl"'°°l.

Suppose that %, belongs to the Nielsen core Y,, of Yn. By Lemma 4.7, there is a geodesic ryn C 8Yn whose length £(y,) is bigger than p, such that d(y,,,ryn) - p , the volume of B(yn, R)1µ'°°1 is (much) bigger

than (R-C(p,X(Y)))p, for large R and for all n. Suppose that y,, belongs to component of Yn - Y (which is an annulus). Since inj(y,,) >- p a short computation shows that the volume of B(yn, R)0`'°°1 is (much)

bigger than Rp. In both cases, the volume of B(y,,, R) [A,-[ tends to infinity with R, uniformly

in n.

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Any ball of radius µ in Y which is contained in the preimage of Kn'°°l embeds into X . Therefore the cardinality of the intersection of Z with any ball of radius

p is smaller than the number of zeroes of anon and so is smaller than 4g - 4. Thus, for all sufficiently small r > 0, the volume of B(y,,, R)("'°°1 - Zn(r) tends to oo with R uniformly in n. This implies Lemma 5.7.

Fix r > 0 so that the conclusions of Lemma 5.7 are satisfied. For any R, IFFI is bounded from below over B(yn,R) - g,,(r) by BFI/2 for all sufficiently large n tends uniformly on B(0,R)Z- 9n(r) to the non-zero constant JFl. because By Lemma 5.7 the volume of B(yn,R) - Z,,(r) is bigger than 3/(m(r)JFI), for a sufficiently large R independent of n. This is impossible since the Ll-norm of Fn

on Y - Zn(r) is less than 1/m(r).

2c) F = 0. We will apply essentially the same argument as in 1). We introduce first some notations. Let s' 1- 6, the 0 -mass of A(o(8M'))' is bigger than 1- 77.

Proof. - Let 0 < 6 < 1. Assume that 11011=1 and lld9aoll >- 1- 6. Then, since d,'ao _ E Au#u , the 0-mass of the union of the spots U for which I I Du ou I >_ (1v 6)11Ou 11 is bigger than 1-,13. For any spot U such that I I8u#u l1 ? (1- 6) I IMu I1,

Theorem 5.3 asserts that the 4u-mass of A(Xu)' is bigger than (1 - c(6))Iloull for a constant c(6) which tends to 0 with 6 and which depends only on a and X(Xu) . Since there are only finitely many possibilities for the topology of the covers

U -+ XU (Proposition 2.2), c(6) can be taken independent of U. Thus the ¢-mass of A(a(8M'))' is bigger than (1- f)(1-c(6)) . Proposition 6.2 directly from this.

0

6.2 Inefficiency over the thin part By Proposition 6.2, if IId`.agll is near 1 for a unit norm ¢, the 0-mass concentrates over the amenable part. In this section, we show that the 0-mass concentrates over the a -liftable part.

Definition. - The e-liftable part Z(a(8M'))' is the union over the spots U C a(8M') of the lifts of the (Bitable) components of X4'1 and of Xu`'OOI. Note that i;,(a(8M'))' is a compact surface that might be empty. We denote by S(a(8M'))' the union over all the spots U C a(W) of the simply connected components in the

preimage of XU°'`l. Hence A(a(8M'))' is the disjoint union of £(a(8M'))' and S(a(8M'))e . Proposition 6.3. - There exists a constant em > 0 depending only on X(8M) ,

such that for all e < em and for all rl > 0, there exists 6 > 0 such that. let s E T(8M), let ¢ E Q(a(8M')) with 11011 = 1 and 1- 6, then the 0 -mass of £(a(8M'))' is bigger than 1- ri .

6 MCMULLEN'S PROOF OF THE FIXED POINT THEOREM

154

Although this proposition looks very similar to Proposition 6.2, its proof is entirely different: it lies on the next result, whose proof is partially "3-dimensional".

Proposition 6.4. - There exists a constant em depending only on X(OM) such that, for all 0 < e:5 EM, we have. let s E T(8M), let ¢ E Q(a(8M')) with 11011 = 1, let p E B(8M') with IIµII = 1, then every x E 8(o(8M'))' is the center of an embedded hyperbolic ball on which the efficiency of the pairing between 0 and

d,ap is smaller than 1 - a, where the constant a > 0 depends only on e and on x(8M) . Proof. - We argue by contradiction. The contradiction will follow easily from Theorem 3.1 once we suitably normalize the limit set of G' in U. Suppose that there exists a sequence (ei) tending to 0 and a sequence (si) in 7(8M) such that the systole of si is smaller than ei, for which Proposition 6.4 fails. Denote to simplify

Mi = M'" and G' = G'i. Then for each i, there is a point xi E 8(Q(8Mi))e+ such that: there exists Oi E Q(a(8Mi)) and µi E B(8Mi) with 1Iiijl = 1f1ci1l = 1, such that on any embedded hyperbolic ball Bi C u(W) centered at xi, the efficiency of the pairing between Oi and d,.vµi tends to 1.

For each i, choose a component f1, of 1l(G'), with stabilizer I'i, such that xi belongs to the component a(1l,/I'i) of a(8Mi)._Then fl(I'i) is the disjoint union of 1l(ri) and another component denoted by f,. By definition, fti covers a(fti/ri) . Let ii E 5i be a point in the preimage of xi. Let Ui be the component of f2(Gi) which contains ii. The projection of Ui to a(fli/ri) is the spot denoted by Ui which contains x Since xi E 8(a(80))' , there is a non-zero element ryi E G' which stabilizes Ui and which moves ii a distance smaller than ei for the hyperbolic metric on Ui. Since the component of S(a(8Mi))e` which contains xi is simply connected, ryi f ri (cf. the proof of Proposition 2.2). Since G' has no parabolic elements, ryi is a hyperbolic isometry. Let ai and wi be its two fixed points in 8D3 _U. With these notations, we have:

Lemma 6.5.- Let 0 < eo 0. If co is smaller than a constant which depends only on X(M) and on R, then in the normalization provided by Lemma 6.5, Sit contains AR, for all sufficiently large i.

Proof. - Recall that yi 0 ri . The convex hull C(ri) of L(ri) is the universal cover of N(ri), the Nielsen core of M(ri) (cf. §1). With the induced path metric, ON(ri) is isometric to a hyperbolic surface (cf. §1). Since aM is compact, aN(ri) is compact and there are only a finite number of possibilities for its topological type. Thus the injectivity radius of the induced metric on ON(ri) is bounded from above by a constant d depending only on X(OM). Therefore any point of 8C(ri) is moved a distance smaller than 2d by a non-zero element of ri .

For v > 0, n"(yi) contains the neighborhood of radius 2d of ne(cf. §1). Recall that, by the Margulis lemma, for any v:5 e(3), the tubes n' (g) corresponding to elements g E Gi which are not contained in the same cyclic subroup are disjoint. Using this, we prove by contradiction that for any e:5 e(3), ne '(-Ii) is disjoint from C(ri). Suppose this is not the case. Then, '(yi) must intersect ac(ri), n'-2d d

since yi f ri. We saw above that any point in aC(ri) is translated a distance less than 2d by some non-trivial element gi E ri. Thus the tubes ne(yi) and n` (gi o'yy o g,-') have non-empty intersection. By Margulis lemma, gi o -ti o gi r and yi are contained in the same cyclic group. Then yi and gi must have the same fixed points. Therefore yi E ri. This provides the required contradiction.

In our normalization, yi moves 0 E D3 a distance equal to co for sufficiently large i. Thus for any K > 0, the hyperbolic ball of radius K centered at 0 E 143 is contained in ne"£o(yi) . By the last observation, if we choose co = eo(K) so that ezd+xeo 0, we can choose co = eo(R) such that AR C fli , for all i sufficiently large. If R > 1, then Or C fti for all i sufficiently large.

We show now by contradiction that Ar embeds in fti/ri, for all i sufficiently large (in the normalization of Lemma 6.6). If Or does not embed into Sii/ri , then

there is a non-zero bi E ri such that bi(Ar) n Ar # 0. The two fixed points of bi are contained in L(ri) and in particular they are outside from AR. Thus, for the natural metric on PSL2(C) bi is C(R)-close to a parabolic isometry b which fixes oo and moves 0 E C an euclidean distance less than 2. In particular, b moves

then 0 E D3 a hyperbolic distance smaller than 2. When R tends to oo, C(R) tends to 0. Thus we can choose R sufficiently large such that bi moves 0 E 9D3 a hyperbolic distance smaller than 3, for all i sufficiently large. But yi moves 0 E D3 a distance smaller than eo = eo(R) (depending only on X(8M) ). Therefore, if ce

156

6 MCMULLEN'S PROOF OF THE FIXED POINT THEOREM

also satisfies eoe3 < e(3), the tubes nepe3(yi) and nfpe3(biyi6i I) must intersect. Since biyi6i-l and -ti do not belong to the same cyclic subgroup of Gi, this is impossible by Margulis lemma. This ends the proof of Lemma 6.7.

In the sequel, we fix to such that Lemma 6.7 and Lemma 6.6 for R = 2 are satisfied. In order to obtain a contradiction, we need to find, for each i, a hyperbolic ball centered at xi on which the efficiency of the pairing between (pi and d,, au, is bounded away from 1, for any non-zero holomorphic quadratic differential ¢i and for any Beltrami form pi of unit norm.

Note that up to extracting a subsequence, (ryi) tends in PSL2(C) to a parabolic

isometry fixing 0. Since each yi moves 0 E D3 a distance exactly equal to to, some subsequence of (ryi) converges to a non-trivial element y E PSL2(C). Since the translation distance of ryi tends to 0, y is parabolic. By the normalization of Lemma 6.5, y fixes 0 E C . Fact 6.8. - The point ii tends to 0 as i tends to oo.

Proof.- The hyperbolic metric on Ui can be written Ai(z)ldzl. By the Koebe 1/4-lemma and since 0 E BUi, ai(z) ? 1/4de,,,(z,8Ui) ? 1/41x( The hyperbolic distance between ii and 'y,), for the hyperbolic metric on Ui, tends to 0 as i tends to oo. Therefore, if ii would remain a bounded euclidean distance away from 0, the spherical distance between ii and yi(ii) would tend to 0. Then any accumulation point of ii would be a fixed point of -t. Since y is a parabolic isometry fixing 0 E C, this is impossible.

Let Bi denote the largest ball for the hyperbolic metric on fli which is centered

at ii and contained in the disc A = DI . Fact 6.9. - Any limit of Bi for the Hausdorff topology on compact subsets of C contains a neighborhood of 0.

Proof. - Let Ri be the radius of the largest Euclidean ball centered at 0 and contained in Sli . In our normalization, Ri is bigger than 2. By the Schwarz lemma, the conformal factor ai(z) of the hyperbolic metric on ili is less than the one of the hyperbolic metric on AR, which equals 2Ri

R - Ix12, As we noticed above, Koebe's 1/4-theorem implies Ai(z)

1

euc(,

i)

Therefore on A we have supo Ai 0 a constant to be fixed later. Then there is s E 7(8M)L such that the coderivative d,(r' o a)K has norm bigger than 1/(I + b). For 0 _ 2, then #(8E n 9) < #(8S n 9). This contradicts our choice of S. 0

Proposition 7.4. - Let (M, 9) be an irreducible and atoroidal manifold with.corners having empty boundary. Let S be a good splitting surface for (M, 9) . Then there is a trivalent graph 9''s C 8MS such that (i) (MS, 9') is an irreducible and atoroidd manifold-with-corners having empty boundary, (ii)

S' is a union of mirrors of (MS, 9's), and

(iii) (Ms, 5s, S') is obtained from (MS, 9') by erasing the mirrors contained in S. The meaning of this lemma is that we can add to 9S edges which are contained

in S' to obtain a graph 9s such that (MS, 9s) is an irreducible and atoroidal manifold-with-corners having empty boundary .

Proof. - When 8S 34 0, we begin by adding finitely many points to the set of vertices of SS which sit on &S', so that this new set of points forms the 0-skeletton of a triangulation of 8S' (i.e. so that there are at least 3 points per components). Denote by V this set of vertices. Choose now a triangulation T of S' which extends

this triangulation of V. Refine then 7 to a new triangulation 7' by modifying it inside each triangle as described on Figure 7.3. Let £ be the 1-skeletton of the dual cellulation of 7. Observe that each edge of £ which is contained in OS' contains at most one point of V in its interior, and in particular at most one vertex of 9s. The union of the edges of £ which intersect the interior of S' form a graph 8. The vertices of £' which are contained in the interior of S' have valence 3 and those which are contained in 83' have valence 1. Define 9s = 9S U£'. By construction, (MS, 9') is a manifold-with-comers which satisfies Proposition 7.4 (ii), (iii). Observe also that each mirror of (MS, 5s) which is contained in S' intersects 8S' in at most one edge, and that this edge contains at most one point of 9S in its interior.

We show now that (MS, 5s) is irreducible and atoroidal. If A is an annulus properly embedded in MS such that 7r1(A) maps injectively into 7r1(MS), then 7r1(A) maps injectively into 7r1(M), by Van Kampen and since S is incompressible.

Therefore, there is a parallelism in M between A and an annulus contained in a mirror of (M, 9), since (M, 9) is irreducible and atoroidal. By incompressibility, S is disjoint from this parallelism. Hence, A is parallel in MS to an annulus contained in a mirror of (MS, 9S). Let y C 8MS be a closed curve which bounds a disc embedded in MS and which intersects 9s transversally in at most 4 points. We consider distinct cases, according to the cardinality of y n OS'.

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173

1) 7cS'. Then y gives rise to a closed path contained in the 1-skeletton of 1' which follows at most 4 edge. One deduces from the way the triangulation 7' was defined that 7 bounds a disc on 8MS which intersects 9's (i.e. E') like on Figure 7.2.

2) 7nS'=0. Then w'(-t) is a dosed curve which intersects 9 transversally in at most 4 points and which bounds a disc embedded in M. Since (M, 9) is irreducible and atoroidal, r'(y) bounds a disc on 8M which intersects 9 like on Figure 7.2. Since S is a splitting surface, this disc is disjoint from 8S. It follows that y bounds a disc on 8MS which intersects 9' (i.e. 5S) like on Figure 7.2 (cf. Proposition 7.2). 3) 0(7 n 8S') = 2.

Then y n S' is a single are that we denote by k. We distinguish two subcases.

3a) knE'=0. Then k is contained in a mirror f of (MS, 5s) . By the construction of E , f is simply connected and e = f n 0,51 is a single 1-cell of E. Therefore, k can be isotoped relatively to 8k to an arc rc C e. By construction, e contains at most one vertex of 9S. Therefore if y' denotes the curve obtained from y by replacing k by rz and slightly perturbed to be disjoint from S', then 7r'(7') is a closed curve which intersects SS transversally in at most 3 points and which bounds a disc embedded in M. Since (M, 9) is irreducible and atoroidal, 7r'(-y') bounds a disc on 8M which intersects 9 like on Figure 7.2. This disc is disjoint from S. Therefore y bounds also a disc contained in 8MS which intersects 9s like on Figure 7.2.

3b) knE'00. Then q(y n 9s) 0,

then we define a C-splitting to be a splitting B -+ Bt such that if i : N(Br) -a N(B*) is the associated singular immersion then there does not exist paths a, 3 : [0, t + e] -+ /V (B,*) such that a(0), 0(0) lie in the same I-fibre of N(Bl ), for all s E

[0, t], i(a(s)), i(6(s)) lie in the same I-fibre of .M(B*), and for s > t; i(a(s)), i(/3(s))

QUASI-MINIMAL SEMI-EUCLIDEAN LAMINATIONS IN 3-MANIFOLDS

201

lie in distinct I-fibres of Ar(B*). Finally length a(i(n([0, t]))) < C. Intuitively this means that the branched locus of B has been blasted open at least distance C.

Definition 1.5. By a 2-dimensional abstract lamination p we mean a topological

space covered by charts of the form U, = T; x 1P2, where T; is a closed subset of [0, 1]. Furthermore if Ui fl U; 54 0, then the coordinate transformations are of the form (x, y) - > (v(x), 0(x, y)) where x E T;, y E 1

2.

This is a specialization

of a more general definition given by [Ca]. There Alberto Candel initiates a deep investigation of the differential geometry of 2-dimensional abstract laminations.

Throughout this paper we will refer to abstract laminations simply as laminations. We say that the lamination µ is immersed in M3 if there is a continuous map

J : p -+ M3 whose restriction to each leaf of p is a smooth or PL immersion. We

say that W is carried by the branched surface B if the mapping J factors through

an immersion into N(B*) such that the leaves of µ are transverse to I-fibres of N(B'). If y intersects each I-fibre of N(B`), then p is fully carried by B. Definition 1.6. If A is carried by B, then B1 is said to be a A-splitting of B if there

exists a lift of A to N(B') which after normal homotopy lifts to A (B; ). Definition 1.7. Generalizing the similar notion of [GO] we say that an infinite sequence of branched surfaces B1, B2, -

-

is a full splitting if there exists C > 0 such

that for each i there exists a j (i) such that the splitting Bi -4 B1() is a C-splitting. If B is the first term of a full splitting, then we say that B is fully splittable.

The following result is the direct analogue of Lemma 4.3 [GO] to immersed laminations. See [MO] for a version for laminations in finite 2-complexes.

Theorem 1.8. B is fully splittable if and only if B fully carries a lamination. Proof. Suppose that B is fully splittable. After making the lengths of the I-fibres

of N(B;) go to 0 as i -a oo, then the inverse limit of the Bi's is a lamination fully carried by B. Conversely if B fully carries A, then A provides the clue to constructing a full splitting. The resulting lamination A' arising from the inverse

202

DAVID GABAI

limit construction will generally be distinct from A, however it will be normally

homotopic to a sublamination of A in N(B`). 0 Definition 1.9. Let M be a triangulated 3-manifold, with triangulation r. Let r" denote its i-skeleton. Following Haken (Ha], if J is a 1-manifold, then we say that

f : J -a r2 is normal if f is a PL-immersion transverse to r' and for each edge e of the induced triangulation on J, f le is an embedding which sends the vertices

of e to distinct 1-simplices of r'. A normal 2-disc is a properly embedded 2-cell D in a 3-simplex a such that 8D is a normal curve crossing a given 1-simplex of

a at most once. If S is a surface, then we say f : S -+ M is normal or S is a normal surface if f is a branched immersion transverse to r2 and each 2-cell 9 in

the induced cellulation x on S gets mapped to a normal disc. Thus, all branch points occur in rl. For v E K° define s(v), the spinning of v, to be the number of 2-cells of s; which meet v. If e is a 1-simplex of r, define V (e), the valence of e, to

be the number of 2-cells of r which meet e. Also for v E 0c°, define V(f(v)) to be

V (e) where f (v) E e. Thus s(v)/V (f (v)) measures the local branching of f at v. We say that f has fake boundary branching if there exists v E ic° f1 8S, such that f

is an immersion at v, but s(v)/V (f (v)) > 1.

Define S(D) = EQE,.onOD s(a) to be the total spinning of D. Define A(D) _ S(D)/(irc° f c9DI) to be the average spinning of D.

Proposition 1.10. Let r be a triangulation of the 3-manifold M. There exists a branched surface B, called the canonical normal branched surface, with the following property. If A is any immersed lamination in M whose leaves are normal with

respect to r, then after normal homotopy A is carried by B. In particular the conclusion holds if A is a compact normal surface, possibly with boundary.

Proof. Figure 1.4 shows how to construct a canonical branched surface B on a 3simplex a. It is symmetric under the symmetries of a and carries the isotopy class

of each normal disc in a unique way. Thus by putting this branched surface in each 3-simplex of r one obtains the desired B. To start with, the three distinct normal quadralaterals and the four distinct normal triangles in or can be normally


203

isotoped to appear in o as follows. The quadralaterals pairwise intersect in arcs whose endpoints he on edges of v. Also if rf is a triangle then 7) f1 (U quads ) = 877.

Thus the union of the quadralaterals and triangles is isomorphic to the cone on the 1-skeleton of an octahedron together with four faces of the octahedron which

pairwise meet in points. (I.e. one color of a two coloring of the octahedron.) See

Figure 1.4a. By squeezing the faces in the appropriate manner, we obtain the desired branched surface B. See Figure 1.4b.

The shaded regions are the triangles. Attach hemispheres to each circle to obtain the quadralerals.

a)

A

Attach discs to each of Abc, Ade, Abe, Adc.

Smoothing very near corner A of the octahedron creates the branched surface B b) Figure 1.4

DAVID GABAI

204

Definition 1.11. A measured branched surface is a branched surface B together

with a non negative valued function m : {Sectors of B) -r iR such that if sectors 81, s2 merge into sector 83, then m(si) + m(s2) = m(s3). A measured lam-

ination is a lamination A together with a transverse measure m. I.e. for each lamination chart T; x R2 there is a Bore! measure supported on Ti. Furthermore the transversal overlap functions are measure preserving. If the measured lam-

ination (A, m) is carried by the branched surface B, then m induces a measure also called m on B. Define the Euler characteristic of a measured branched surface as follows. If (B, m) is a branched surface, then put a cell structure on B

so that the branched locus is contained in the 1-skeleton. If t is a vertex or edge,

define m(t) to be the maximal m value of a sector which contains t and define

x(B,i)=

Everticesm'(v=)-Eedyesm(ej)+Efacesm(fk).

This formula is invari-

ant under splitting and choice of cell structure. If (A, m) is a measured lamination, carried by a branched surface (B, m), then define X(A, m) = X(B, m). Definition 1.12. We need the notion of diagram [LS] which is a slight generalization

of a disc. (The first time reader is strongly advised to pretend that all diagrams are discs.) A diagram D is a simply connected finite 2-complex that is embedded in the 0

0

plane. Define D to be the interior of D as a subset of R1. Define 8(D) = D - D and

Bd(D) to be a simplicial map of the circle to the curve which traverses around D. See Figure 1.5. Call a diagram map f : D -3 M admissible if 8D has a triangulation 0 such that f takes each 1-cell of ¢ to a normal arc and f is transverse to r2, except at isolated points where f appears as in Figure 1.6. We say that g : Sr -r r2 extends

to the diagram G : D -3 M or G spans g, if GoBd(D) is normally homotopic to g. If f : D --+ M is an admissible diagram map, then define length(8f) to be the number of 1-cells in the induced triangulation on Bd(D).


205

Figure 1.5

Figure 1.6

Definition I.M. If f : S --r M is a normal surface, then define "area (f)" = Z11 s(v;)/V (f (vi)), where {v1,

,

are the vertices in the induced cel-

lulation is on S. Thus if f : S -> M is a normal immersion of a closed surface, then

area(f) is the number of vertices in rc, i.e. the Haken weight. This definition of

DAVID GABAI

206

area takes into account local branching and is additive under union. In a similar manner we can define area(f) where f : D -4 M is an admissible diagram map, or

if f is a map of a compact surface such that f I8D is normal and f is transverse to r2 except as in Figure 1.6. In these cases one uses the induced stratification on D

to compute s(vi) and sum over points of f-'(r').

A diagram f : D -+ M, and in particular a normal map of a disc, is said to be least area (resp. k-quasi least area) if area(D) < area(E) (resp. area(D) < k area(E)) among all diagrams E spanning OD. A normal surface f : L -4 M is least area if the restriction off to embedded discs

is a least area diagram. L is strongly least area [MO] if the induced immersion of the universal covering of L into M is least area. In a similar manner define strongly

k-quasi least area for a normal map f : L -> M. The leaves of a lamination are quasi strongly least area if there exists a k such that for each leaf L, the induced immersion of L into M is strongly k-quasi least area. The following Proposition is a restatement of the main technical result of Mosher

and Oertel [MO]. We translate their work about PL maps of discs into finite 2complexes into a Proposition about maps of discs into 3-manifolds. We generalize slightly to allow for k-quasi least area maps.

Proposition 1.14 [MO]. Let D1, D2, in

lim;,

the

triangulated

compact

be a sequence of immersed normal discs,

3-manifold

M,

such

that

length(8Di)/area(Di) -4 0. If no Di has fake boundary branching, then

after passing to a subsequence, Di converges to a normal Euler characteristic-0 measured lamination A. If each of the discs is k-quasi least area, then each leaf of

A is strongly k-quasi least area. If each Di is least area, then A has no 2-sphere leaves. Finally, the induced metric on each leaf is complete. Remarks. The proof of the existence of A in [MO] is greatly complicated by the fact

that it is happening in the setting of branched immersed discs into 2-complexes.

In the setting of immersed normal discs without fake boundary branching in 3manifolds the argument is considerably simpler. Mosher and Oertel do a competent


207

job of explaining their proof of strong area minimization. Indeed, their argument shows that if A had a 2-sphere leaf, then for i sufficiently large, some component of Di would be a 2-sphere, which gives a contradiction. By construction each leaf of A is complete. One should be aware that the discussion of diagrams is implicit, though suppressed in [MO].

Idea of the proof of existence of A. Each Di is carried by B, the canonical normal

branched surface. Fix a small C > 0. By the usual diagonal argument one passes to a subsequence of the Di's and finds a sequence B = Bo, B1,

such that Bi is

obtained from Bi_1 via C-splitting where each By carries every Di for i > j. (This

step would fail if there was uncontrolled fake boundary branching.) By counting the number of times a given Di crosses a sector of B5, we obtain a function mi3

on the sectors of By which fails to be a transverse measure only because of the boundary of Di. Let ai5 be the maximal value of mi3. By considering m,5/aij and using the length area hypothesis, then by passing to a subsequence of the Di's the measures mil/ail converge to a fixed transverse measure ml on B1. By passing to

another subsequence we can assume that the Di's induce a measure m2 on B2 so that as measured laminations (B2,m2) is obtained by splitting (B1, m1). Thus by applying Theorem 1.8 to the sequence {(Bi, mi) } we obtain a measured lamination

A'. Finally take A to be the sublamination which is the support of the measure m. This type of argument, in tamer form, goes back to Plante [PI]. Since X(Di) = 1, the Euler characteristic of the approximate measure mil/aij on

B5 is approximately 1/aij and mii/aid -1 m it follows that the Euler characteristic of A is zero.

Definition 1.15. Let A be a lamination with a piecewise Riemannian metric on each

leaf, which varies continuously in the transverse direction. (E.g. a metric induced from a 3-manifold) We say A is conformally Euclidean if each leaf is conformally equivalent to the Euclidean plane. A measured Riemannian lamination A is said to be semi-Euclidean if the set of leaves conformally equivalent to the Euclidean plane

are of full measure and dense in A.

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DAVID GABAI

The following result provides the crucial link between the topology of the mea-

sured laminations and their geometry.

Theorem (Ghys [Gh] 1.16). Let A be a measured 2-dimensional abstract Riemannian lamination with out spherical leaves. X(A) = 0 if and only if almost all leaves are conformally Euclidean.

0

Remark 1.17. This was proven originally for 2-dimensional foliations of manifolds,

but Candel [Ca] pointed out that the proof holds for 2-dimensional abstract Riemannian laminations. §2

IMMERSED LEAST AREA SEMI-EUCLIDEAN LAMINATIONS

Theorem 2.1. If M is a closed non negatively curved aspherical 3-manifold, then there exists an immersed strongly least area, Euler characteristic 0, measured semi-

Euclidean lamination A. The induced metric on each leaf is complete.

Remark 2.2. i) Here M has a triangulation r, and the various measurements of length and area are computed simplically. ii) Theorem 2.1 should be viewed as a refinement of [MO]. Their work translated to closed 3-manifolds yields the above theorem with immersed replace by branched immersed.

Conjecture 2.3. If Ko is a non negatively curved 2-complex, then Ko is simple homotopy equivalent to a 2-complex Kl such that Kl supports an immersed least area Euler characteristic zero semi-Euclidean lamination. The following is the main result of this section.

Proposition 2.4. Let M be a closed 3-manifold with triangulation r. ri(M) is not negatively curved if and only if there exists a sequence of least area normal

immersed discs f; : D, -+ M, such that length(8f;)/area(f;) -+ 0 and no ft has fake boundary branching.


209

Remark. This result is a serious technical advance beyond the obvious translation of Gromov's theorem, group negative curvature implies linear isoperimetric inequality,

to normal surfaces in 3-manifolds. The new wrinkle is that no fi has fake boundary branching."

Proof of Proposition 2.4. Step 0. rrl (M) is non negatively curved if and only if there exists a sequence of least

area normal immersed discs fi : D'- --> M, such that length(8fi)/area(f;) -r 0.

Proof of Step 0. Since 7r1(r2) = n(M) is not negatively curved, there exists by Gromov [Gr], [Bo] a sequence of simplicial maps of discs a, : Ai -4 -' such that length, (Ai) /area, (Ai) -> 0. Where length,(O.-1i) (resp. area,(.4i) is the number of 1-simplices (resp. 2-simplices) in 8Ai (resp. in Ai.) Furthermore area,(Ai) is

minimal among all such simplicial maps of diagrams that span aj 8A,.

Its routine to translate this statement about 2-complexes to the statement of Step 0.

0

From now on we will assume that rrl(M) is non negatively curved and that Proposition 2.4 is false. Thus there exists a constant KO < 1 such that if f : D -+ M is least area normal immersed disc with no fake boundary branching, then

length(8f)/area(f) > Ko. Here is how we derive a contradiction. As a result of Steps 0 and 1 we find a sequence satisfying the conclusion of Proposition 2.4, except that each disc has fake boundary branching at a single vertex. We then delete a small neighborhood

of the branching vertex and take (in Step 2) a weak limit of these trimmed discs to obtain an immersed least area measured lamination A' of Euler characteristic 0,

with boundary S' x K, K a closed subset of [0, 1]. By capping off the boundary circles with discs, we obtain a measured lamination A of positive Euler characteristic

and derive a contradiction to the least area property of A. Step 1. There exists a sequence of least area normal immersed discs fi : D2 -9 M,

such that length(8fi)/area(fi) -+ 0. Furthermore all the fake boundary branching

210

DAVID GABAI

of fi occurs at a single vertex wi of BDi. Proof of Step 1. Let fi be as in Step 0 and let Di denote the cellulation induced on

Di by r. If J is a subcomplex of a cell complex, then let St(J) denote the union of all open cells whose closures intersect J and St(J) to denote the closure of St(J).

Define D;'''"' = Di - St(8Di). Claim 0. We can assume that D!"' is connected, that Di has no separating 1-cells, and St(v) n Dirim is nonempty and connected for each vertex v E &Di.

Proof of Claim 0. Consider a maximal collection of properly embedded arcs d;

in Di such that 86; C Do, interiors of the 6 's are pairwise disjoint and the Sj's satisfy the following additional property. Either d; is a separating 1-cell of Di, O

or bj C Di - A? and each component of Di - dj contains at least one open 2cell of Di. The number of such paths between two given vertices is uniformly

bounded by a constant K1, since M is compact and Di is least area. Thus for Euler characteristic reasons there are at most Kllength(8Di), such bb's. If C;, is

a component of Di - uSj, then let C. be the smallest subcomplex of Di which contains C. Since E" length(8C") < 4K1(length(8Di)), some C. will have very small isoperimetric ratio, if Di does. Finally C,, satisfies the conclusions of Claim 0. Thus by cutting down the Di's we can assume that they satisfy the conclusions

of both Step 0 and Claim 0.

0

We will supress cumbersome language by writing as if each D;"'as well as other to be defined objects are discs rather than diagrams. If V E DonODi, then define D; = D;'i"USt(v). Da is an immersed disc with possible fake boundary branching only at vi. Since D;'i"' is an immersed least area disc with

no

fake

boundary

branching,

it

follows

that

for

all

i,

length(8D;'im)/area(Di> K. Claim 1. The average spinning A(Di) -+ oo.

Proof. Otherwise .4(Di)

< A all

i implies length (8V'im)/area(D;'im)
0 as


211

We now define Fanj(v), which more or less is the smallest simply connected subset of D; (v) which contains all sinlplices distance j from v. More precisely, if v

is a vertex of OA then define inductively define St,''im(c) = St(St;' 1'(v)) where St;ri"'(v) = gt(v), all these complexes being computed in D. Let Fans (v) be the minimal simply connected subcomplex of D; which contains Stfri r.(v) Claim 2. Given p, there exists B(p) independent of i. such that for any v,., v8 E OD j,

area(Stt

n St-"'(v8)) < B(p). (Recall Stp`m(v,) C

and Stir"(vs) C

Dr .) a

Proof. If x is a vertex in Stp'i'(vr)nSty im(v3). then there exists a simplicial path ax of length < 4p from Vr to Vs which passes through x. The union of two such paths

a,, ap gives rise to a least area diagram Dx5 C Di, with OD,, C ax u a, c Dx,, and hence length 5Dxy < 8p. Since Al is compact, the number of such diagrams is bounded and hence if x, y E Stp im(vr) Cl Sty; i'"(v8), then the distance between x and yin Dgri' is uniformly bounded, independent of i. Since the valence of vertices

of D,ri"' are uniformly bounded, independent of i, Claim 2 follows.

Claim 3. Given p, there exists a constant K3 independent of i such that at most K3(length(8Di) distinct pairs (Fanp(vr), Fanp(v8)) intersect.

Proof. If Fanp(vr)n Fanp(vs) # 0, then define a,, to be a shortest simplicial path in 1-cells of D°' U D; ° from yr to vs. Thus ar8 is embedded of length < 4p. By rechoosing ar8 we can assume that D,rim n a,8 is connected, and using Claim 0

that either ar8 Cl D;r""' # 0 or ar8 C 8Di. Furthermore, if ar8 n D;ri'" 0 0, then after possibly deleting the first and/or last 1-cells of a,., the remainder is properly

embedded in Di. Also if ar8 C 0Di, then length(ar8) < 4. Thus there are at most 4 length(8Di) peripheral acs's and at most 5N non peripheral ar8's, where N is the number of properly embedded ar8's. Since at most finitely many paths ar8 can pass through a given vertex of Diri'n, it follows that there exists a uniform bound K2 on

the number of aye's which can intersect a given a,,,, inside of D,". The collection

212

DAVID GABAI

of properly embedded arc's can be partitioned into K2+1 subsets, with the property

that if a,.,, au are distinct elements of a given subset, then a n auv = 0. For Euler characteristic reasons each subset has at most length(8Di) - 1 elements, so N < (K2 + 1)(length(8Di) - 1) and Claim 3 follows with K3 = 5(K2 + 1) + 4.

Claim 4. For every k > 0, there is a N > 0 such that i > N implies that there exists a v E AO n Di and a j such that length(8Fanj(v))/area(Fanj(v)) < 1/k. Proof. Fix k and i. If v is a vertex in 8Di, then let SextFanj(v) denote 8 Fanj(v) n

Di.

If there exists j such that length(BextFanj(v)) < s(v)/2k, then

length(BFan,,(v))/area(Fanj(v)) < (s(v)/2k+ length( Fanj(v) n 8(D°)))/s(v) and hence

either

length(8Fanj(v))/area(Fanj(v))

s(v)/2k. In the latter case, for

all p > j, length(Styim(v) n 8D;) > s(v)/2k. Now choose p so that p/(6k) > 8/Ko. Therefore for all vertices v of 8Di

i) length(8Fanj(v))/area(Fan1(v)) < 1/k for some j < p or ii) length(Styim(v) n 8D°) > s(v) /2k or

iii) area(Styi"t(v)) > 8s(v)/Ko. Indeed if neither i) nor ii) held for v, then for all j < p, length(8e,tFanj(v)) > s(v)/2k and hence area(Styim(v)) > (1/3) Ej=1length(BextFani(v)) > ps(v)/6k > 8s(v)/Ko. This argument was inspired by [Pa]. Let v1,

, v.,,t denote the vertices of 8Di. It follows that either

a) length(8Fanj (v8))/area(Fani(v,)) < 1/k for some vertex v8 of 8Di and some

j <por

b) En

n 8D;) > S(Di)/4k or

c) En larea(St1 tm(vn)) > 4S(Di)/Ko We show that b) does not hold if i is sufficiently large. Claims 2,3 imply that the sum of the pairwise area overlaps of the Sty im's is bounded above by B(p)K3length(8Di). Possibility b) implies that the sum of the total pairwise area

overlaps of the Sty im s is > S(Di)/12k = A(Dj)length(aDj)/12k. Now apply Claim 1.


We show that c) does not hold if i is sufficiently large.

213

If c) holds then we

obtain area(D;rim) > area((U;L1Styrim(u,;))- St(3D1)) > 4S(D,)/Ko - B(p)K3

length(0Di) - S(D1) >

2S(Di)/Ko + S(D2) - B(p)K3length(5Di)

length(OD1")/Ko + (.4(Di) - B(p)K3)length (aD1) > area(D rig").

>

BY Claim

1, the latter inequality holds if i is sufficiently large. Thus for fixed k, a) holds for i sufficienity large.

Thus we can assume that each Di has fake branching except at the vertex w1. Furthermore by the compactness of Al. we can assume that for all i, f (w1) = w E r'. O

Let Ei = Di - St(vi) and hi = fil (Di - St(wi)). Let -i be the 1-simplex of -r which contains w and let of = 0 St(wi) -3D,. Since hi has no branching or fake boundary

branching we can assume that for all i, length(0h2)/area(h2) > K0. Thus there

exists K7 > 0 so that for i sufficiently large length (ai)/area(E1) > K which in particular implies that length(3EE - ai)/length(ai) -i 0. Since the restriction of fi to St(wi) is an immersion, hi(ai) is an embedded arc which spirals around 83t(-;). Step 2. Either Proposition 2.4 is true or there exists an immersed normal measured lamination (A, m) by strongly least area leaves such that X(A) = 0. Furthermore aA

is a lamination of the form S' x K C 09t(y) where K is compact and m(K) = 1.

Proof of Step 2. Let ni be the number of times h(ai) spirals around St(y). Here we abuse notation by letting St(-y) denote the union of all 3-simplices of r which nontrivially intersect y. As in the proof of Proposition 1.14 normalize the various weights associated to the discs Ei by dividing by n1. Since area(Di)/ni is uniformly bounded, thus we can apply the proof of Proposition 1.14 to let the normalized discs

Ei/ni weakly converges to the measured lamination (A, m). Since aA is the weak

limit of the sequence 0Ei/ni and length(0Ei-ai)/ni = 0, any weak limit of 0Ei/ni is a weak limit of hi(ai)/ni. Because each Di is immersed in M, ai is an embedded spiral in 9St(ry), so it follows that any weak limit is of the form Si x K where K is

compact and m(K) = 1.

214

DAVID GABAI

Step 3. Obtain the contradiction. Proof of Step 3. Let (A',m') be the immersed measured lamination obtained from

(A,m) by capping off each component of S' x K by a disc. X(A') = 1 and so by the Connes sphere theorem [Co] (or [MO1] for an elementary proof) there exists a

measured sublamination S of A' consisting only of 2-spheres and X(S) > 1.

Since A is strongly area minimizing it has no 2-sphere leaves. The rest of the argument follows implicitly from the proof of "strong area minimization of leaves

theorem" [MO]. See the next paragraph for a hint. If some leaf S of S intersects

D2 x K in a single disc F, then for i sufficiently large, there exists a subdisc Hi O

of Di which is normally parallel to S - F, and 8H2 C ai. This implies that Di is a sphere. Now suppose that some leaf S of S intersects D2 x K in exactly 2-discs 0

0

F1, F2. Given n E N, and an essential are 0 in the annulus A - F1 U F2, then for i sufficiently large, there exists a subdics Hi of Di of the form [0, 1] x [0, n] where

(8[0,1]) x [0, n] C a, and Hi naturally projects to A as part of an n-fold cyclic cover. Finally each component of [0, 1] x (0, n} naturally projects isomorphically

onto /3. Thus by coning off Hi f1 ai one obtains a subdisc of Di whose boundary

length is constant, independent of i, but whose area a oo as i -* oo. This thereby contradicts the least area property of Di. A similar argument works when SnD2 xK

has r > 2 components. However the cases r < 1, together with m(K) = 1, implies X(S) = 1 and that the generic leaf of S-D2 x K is an annulus thereby contradicting

the r = 2 argument. Here is the idea behind what was used in the previous paragraph. Recall that we had, by Proposition 1.14, a sequence of measured branched surfaces (Bi, mi) which

converged to a measured lamination with measure m, and that A was the minimal

sublamination which supported m. Also j > i implies that (BB, mj) is obtained by splitting the measured branched surface (Bi, mi). Therefore if T is a compact leaf

of A, then for i sufficiently large T embeds in B2. If r = 1, then T is a disc so for C sufficiently large, any C-splitting either isolates or destroys T. Since the latter does not occur, T must appear as an isolated sector in B for j sufficiently large. If

QUASI-MINIMAL SEMI-EUCLIDEAN LANIINATiONS IN 3-::1ANIFOLDS

215

r = 2, then T is an annulus and a holonomy argument shows that for i sufficiently

large near T C Bi, part of each D j > i, spirals around T. The spiraling must get arbitrarily large to obtain the annulus T in the limit. Our [0. 1' x of that spiral. This completes the proof of Proposition 2.4

ni is a piece

0

Proof of Theorem 2.1. Combine Propositions 2.4 and 1.14 with Theorem 1.16. O

§3

HYPERBOLIC CELLULATIONS

Definition 3.1. Let 0 be a piecewise linear cellulation of the 3-manifold W. This means that M is obtained by taking a disjoint union of 3-dimensional polyhedra v1,

, on and pairwise identifying 2-dimensional faces in a PL fashion. Thus to

each cell C; of 0, there is a piecewise linear mapping f= : C, -r ,If which restricts to

an embedding on C;. We will usually suppress mentioning the f is and will denote

the i-skeleton of 0 by ,`. As in the setting of triangulations define a normal curve (resp. local

embedding)

to be an immersion f : a -a 02 transverse to 01, where a is a compact 1-manifold

and for each 1-simplex 17 in the induced triangulation on a, f la = f1 o t, where t : tl -3 Cs is an embedding into the 2-cell Ci with 8q going to distinct edges (resp. t) not necessarily going to distinct edges.).

Similary define a normal disc to be a map f : D2 -r M, which factors through

an embedding into a 3-cello of 0, so that f1BD -> d! is normal and (the lift into o) crosses a given edge of or at most once. If S is a surface, define a map f : (S, 8S) -a (M, s12) to be normal if f is transverse to y,2, f l8S is a normal curve,

and for each 2-cell rl in the induced cellulation on S, f IS is a normal disc. If S

is a surface, define a map f : (S, 8S) -a (M, 02) to be a-transverse to 02 if f is transverse to 02 except at isolated points of 8S where f appears as in Figure 1.6 and furthermore f JOS is a local 7/,-embedding. We say that f : S -+ N is normal, or S is a normal surface, if f is a branched immersion a-transverse to ,i,2, inducing a cellulation rc on S such that each 2-cell of n gets mapped to a normal disc in Vi.

DAVID GABAI

216

In particular f IOS is a normal curve. In order to suppress notation, we will often view S as lying in M. E.g. if Y is a

submanifold of M we may refer to S n Y rather than f'1(Y). If a is a curve in '2 transverse to 01, then define length(s) to be the number of 1-

cells crossed by a. If f : S -4 N is a-transverse to'02, then f induces a stratification

rc on S, where f -' (i") = rc' the i-th strata. We'll abuse notation by calling a com-

ponent of rc' - X''1 an i-cell. Finally define area(S) = E 1 s(vi)/valence(f (vi)). Here v1,

, v are the vertices of rc, s(vi) is the local number of 2-cells of r

touching vi, and valence(f (vi)) is the local number of 2-cells of 0 which come into f (vi). For example if f : S -+ M is an immersion of a closed surface, then area(S) is simply the intersection number of S with ik1. This definition of area keeps track of the local branching at vertices and is additive under unions, e.g. if S = Sl U S2, SI n S2 C OS1 n OS2, and f IS is normal, then area(SI) + area(S2) _

area(s). View the mass of M as being concentrated near ip'. Think of a 1-cell 0 as a D2 X I. If k 3-cells of 0 touch g5, then pretend the D2 x I is subdivided into k wedges, one for each 3-cell. Finally imagine that each D22 x t, t E I is subdivided

into k pie slices, the area of each slice being 1/k.

We may use the notation length,r(a) or area,G(D) to make clear that we are measuring with respect the cellulation 0.

Lemma 3.2. If f : S --> M is a mapping of a closed surface into the aspherical manifold M with cellulation Vi, then f can be homotoped to a normal surface, pro-

vided that for each essential simple closed curve in S, 10 [f.(a)] E irl(M) (i.e. f injects on simple loops).

Proof. First homotope f to a generic least area immersion transverse to rG2. After a further homotopy we can assume that the induced statification ic is a cellulation. Among all such maps choose one that minimizes double points of f If -I (VJ2). If n is a 2-cell of rc which maps into the 3-cell o, then f 1877 is an embedding (when lifted)

into a. Otherwise a homotopy of f supported near n reduces the number of double


337

points. Thus we can assume that the restriction of f to each 2-cell is an embedding (when lifted) into its associated 3-cell. Such an f is normal.

Lemma 3.3. (Epstein - Penner (EP)). If N is a complete noncompact hyperbolic 3-manifold of finite volume, then N has a cellulation by ideal polyhedra.

0

Question 3.4. (Igor Rivin) Does every complete noncompact hyperbolic 3-manifold of finite volume have a cellulation by ideal triangles?

Definition 3.5. If the complete noncompact finite volume orientable hyperbolic 3-

manifold N has a cellulation by ideal polyhedra, then removing a neighborhood

of the ends of N one obtains a compact manifold .11 whose boundary is a non empty union of tori. The induced cellulation on r1f is a cellulation by truncated ideal polyhedra. We call such a cellulation a relative hyperbolic cellulation. Call the newly created 2-cells (on DM) the facets. Definition 3.6. Define a combinatorial geometry (n, angle) on the compact surface S to be a cellulation rc together with a function angle: {vertices of 2-cells of '} -r [0, 7r]

such that if v E rc°, w1,

-

, w are the vertices of 2-cells of n which are identified

with the vertex v of rc and Angle(v) d`=I °

" angle(w;), then Angle(v) = 2ir for i=t

v E S.

If C is a 2-cell with vertices v1, -

, v,,, then define fc K = E 1 angle(vi) -

(n - 2)rr = 2ir - E', (7r- angle(vi)) and fs K = E', ff, K, where C1,

-

, Cm

are the 2-cells of rc. If v is a vertex of rc, v E 8S, then define y9 (v) = ir-Angle(v)

and fas y9 = E.,EK°n8Sy9(vi). Call a 2-cell C respectively negatively curved, flat or positively curved if fC K is respectively < 0, 0, or > 0.

PL Gauss Bonnet theorem (We] 3.7. If S has a combinatorial geometry (b, angle), then fS K + fas y9 = 2rrx(S). Remark 3.8. The proof is an elementary combinatorial exercise. See [We] for the case of a closed surface.

Remark 3.9. The following result is the obvious generalization of an observation of

Andrew Casson which was stated for normal closed surfaces with respect to ideal

218

DAVID GABAI

triangulations on hyperbolic 3-manifolds. Lemma 3.10. Let 7P be a relative hyperbolic cellulation on Y. If S is an immersed surface in Y which is normal with respect to ii. Then f induces on S a combinatorial O

geometry (K, angle) where h is the induced cellulation on S and if v E Y, then angle(v) is the dihedral angle at the edge of the associated 3-cell of b. Otherwise angle(v) _ 7r/2.

Corollary 3.11. (Casson [Ca]) Let 4' be a relative hyperbolic cellulation on Y. If

S is an immersed closed surface in Y which is is normal with respect to 0, then

X(S) < 0. Equality holds if and only if S is a normally parallel to a peripheral torus.

Proof. Combine the the following result with Lemma 3.10 and Theorem 3.7.

Rivin's Lemma 3.12 [Ri]. If a is a normal closed curve on the boundary of an ideal hyperbolic polygon which crosses (not necessarily distinct) edges e1,

, e,,,

then 2ir < E 1(ir - d(ei)), where d(ei) is the dihedral angle at ei. Equality holds if and only if a is a simple closed curve which separates off a single facet from the rest.

Proof. Use the fact that any ideal polygon is the union of ideal tetrahedra and use the fact that for an ideal tetrahedron, dihedral angles of opposite edges are equal.

Remark 3.13. Its a famous theorem of Igor Rivin [Ri], that the converse is also true. That is any combinatorial geometry on a polygon satisfying the conclusion of Lemma 3.12 arises from a unique ideal hyperbolic polygon.

Remark 3.14. A closed surface of genus g which is (Riemannian) least area in a hyperbolic 3-manifold has area bounded above by (2g - 2)ir. The rest of §3 is devoted to establishing a similar result for least area surfaces with respect to relative hyperbolic cellulations. Unlike the Riemannian setting, it may happen that fc K > 0, where C is a normal disc. However, we show that if S is a normal surface

in the relative hyperbolic cellulation, then after a small homotopy and redefinition


219

of the combinatorial geometry, all such "positively curved" discs can be eliminated.

It will then follow that the total integral over the negatively curved normal discs is

bounded below by 2aX(S) - a length(OS). Therefore by the finiteness of normal disc types the negative curvature is concentrated in a finite area subsurface. After

shoving, the flat part of S out of Y, we are left with a surface Sy whose area is bounded above by Co( length(8S) - X(S)) where Co is a constant depending only on 7r.

Definition 3.15. A compact immersed surface S in the 3-manifold Y with relative hyperbolic cellulation i/i is h-normal if S is a normal surface with respect to

and

satisfies the following additional properties.

i)8Sn8Y=0, ii) S has no fake boundary branching, and iii) for each 1-cell Q in w (the induced cellulation on S) which lies in the 2-cell

n of rb, if Q separates in il, a single vertex from the rest, then either Q C 8Y, or (3 n 8S, j4 0. Call an arc of the latter type bad. See Figure 3.1.

Figure 3.1

220

DAVID GABAI

The next lemma is a version of Lemma 3.2 for surfaces in relative hyperbolic cellulations. It says that at the cost of pushing part of S out of Y, S can be

homotoped to be h-normal, without decreasing X(S).

Lemma 3.16. Let Y be a codimension-0 submanifold of the compact aspherical 3-manifold M with Y fl 8M = 0. Let ip be a relative hyperbolic cellulation on Y 0

and f : S -a M satisfy OS C Y, f IOS is normal, f injects on simple loops, f is transverse to 02 near 8Y (and in particular transverse to 8Y) and f is boundary incompressible in the following sense. There is no properly embedded arc a c SnY

with endpoints in distinct 1-cells of the induced triangulation on 8S, such that f la

can be homotoped rel 8a into a 2-cell of r(i. Assume also that no component of f -1(Y) is a 2-disc disjoint from 8S. Then S can be homotoped rel OS to S1 such that Sl fl Y is h-normal respect to

and X(Si fl Y) > X(S fl Y). Furthermore no

component of Si fl Y is a disc disjoint from 8S1.

Proof. Using the boundary incompressibility of f, first homotope f rel OS u (S n

8Y) to an immersion a-transverse to 02. Homotope f near OS to eliminate fake boundary branching at the expense of introducing intersections as in Figure 1.6.

If some component of f'1(Y) - f-1(02) is not a disc, then f can be homotoped rel OS to reduce IS n 011. If for some 2-cell C of lc, f (8C) crossed a 1-cell e of 1i more than once, then a homotopy of f reduces IS n,01 1. (But X(S fl Y) may rise if

e C 8Y.) If S had a bad arc ,Q, then a small homotopy eliminates this arc thereby reducing IS fl Y fl V51 I and possibly increasing IS fl 01 1. Argue as in the proof of

Lemma 3.2 to homotope f so that the restriction of f to each 2-cell of ,c is an embedding into its associated 3-cell. All of these homotopies can be accomplished without decreasing X(S fl Y) or introducing disc components of S fl Y. The result 0

follows by induction on (IS fl Y fl'iP11, IS fl,01 ) ordered lexicographically.

Proposition 3.17. Let 0 be a relative hyperbolic triangulation on the compact 3manifold Y which is contained in the interior of the compact 3-manifold M. Let f : 0

S -+ M be a map of a compact surface transverse to 8Y such that f 18S C Y. There

exists a constant Co such that if SnY is an h-normal surface, then S is homotopic


221

relOS to a surface S1, such that area(S1 fl Y) < Co(lengthl8S) - :HIS: l 1-)). Corollary 3.18. Under the hypothesis of Proposition 3.17, there exists a C1. such that length(8(S1 fl Y)) < C, (length(8S) - y(S fl Y).). Proof. Apply the following Lemma 3.19 taking C, = CCo.

Remark. If S is a Riemannian least area surface in Y which goes far out the cusps,

then the region out in the cusp accounts for little Riemannian area, but large PL area. The passage from S to S, amounts to shoving the cusp stuff out of Y.

Lemma 3.19. Let v be the maximal valence of an edge of

'.

If T is a-transverse

to z/,, then area(T) > length(8T)/v. 0 Proof of Proposition 3.17. Let r be the induced cellulation on S fl Y. Define a combinatorial geometry (rc,angle) on S fl Y as follows. Let v be a vertex of the 2-cell rl of ,c.

If f (v) E 8Y, define angle(v) = a/2. If v E 8S fl V51, then define angle(v) = 0.

If v E 8S - ail, then define angle(v) = 7r/2 (Such vertices arise as in Figure 1.6.)

Otherwise angle(v) is the associated dihedral angle of the 3-cell a of ' which contains rl.

Claim. If C is a 2-cell of Ic, then C is nonpositively curved. If C is flat, then either

a) f (C) is parallel to a facet and C is disjoint from 8S, or b) f (8C) encircles a vertex of 0, or

c) f (C) encircles a single edge e on the boundary of the 3-cell which contains f (C), e ¢ 8Y and C fl 8S = 0. See Figure 3.2.

DAVID GABAI

222

Type a)

Type b)

Type c)

Flat 2-cells Figure 3.2

Proof of Claim. If a is the 3-cell which contains f (C), then f IC lifts to an embedding (also called f) of C into a. First suppose that f(8C) is disjoint from the facets. Let e denote C with the following combinatorial geometry. The 1-skeleton of C equals 8C, f-1(+(11) f1 C are the vertices of 8C, and each point x of f-1(?P1)

is assigned the dihedral angle of a at f(x). Thus by Rivin's lemma, e is either negatively curved or is flat and parallel to a facet. If c f1 8S = 0, then C and C will have the same combinatorial geometry. If C f1 8S 0 0, then by comparing the

combinatorial geometry of C with that of C it follows that fC K < ft K < 0. If 8C crosses exactly one facet and has no bad subarcs, then by pushing 8C off

of the facet, in at least one of the two possible ways, one obtains a new normal curve bounding a disc C' which is not more curved than C. Any two facets of a truncated polyhedron are connected by at most one edge. Thus if C' is flat, then C is negatively curved. If 8C has a bad subarc (3, then by decree angle(v) = 0 for the non facet vertex v of 8J3. Another application of Rivin's Lemma implies that such a C is flat if and only if 8C crosses exactly 3 edges of ?,.

If 8C crosses exactly two facets, and C C a is a 3-cell of 0, then a similar argument to the one above shows that either C is negatively curved or 8C separates

off a single edge of a disjoint from M.


223

If 8C crosses more than two facets, then C is negatively curved. 0 Even accounting for the possibility that part of a 2-cell lies in OS. there are only a bounded number of combinatorial possibilities for negatively curved h-normal 2cells in l'. Thus there exists cl > 0 which depends only on t such that -cl is the

maximal possible value of fc K where C is a negatively curved h-normal 2-cell.

By definition for v E 0 ° n OS either f (v) E e 1 in which case -t,, (v) = r. or

f(v) f v/i' in which case -ys(v) = 0. Also if o is a component of f-'(8Y). then fQ yy = 0. Therefore fs K = 2rX(S) - rlength(8S) and hence there are at most (rlength(8S) - 27rX(S)) negatively curved 2-cells of x.

Let c2 be the maximal valence of an edge of V1. The Claim and the fact that there is no fake boundary branching implies that there are at most c2 length(8S) flat 2-cells of type b). Since a fiat 2-cell of type c) can only share an edge with either another type c) flat 2-cell or a negatively curved 2-cell, and that a string of c2 type c)

flat 2-cells leads to an area reducing homotopy of S, it follows that after homotopy

there can be at most %i' (rlength(8S) - 2rX(S)) type c) flat 2-cells. Here c3 is the maximal length of the boundary of an h-normal 2-cell. Thus for some constant c4,

there are at most c4(length(8S) - X(S)) 2-cells of re which are not flat of type a). Now let W C S be the union of type a) flat 2-cells. Homotope S to St by a homotopy 0

supported in W to push most of W out of Y. So if T denotes IT' with a small collar of

OW removed, then S1nY = ((Sf1Y)-T)UA, where A is a union of annuli connecting O

OT straight to 8Y. Therefore area(SinY) < area((S-T )n Y)+2c_length(814') < 0

c5area((S - W) n Y) < c5c3c4(length(8S) - X(S)) = Co(length (8S) - X(S)). Proposition 3.20. Let zli be a relative hyperbolic triangulation on the compact 3manifold Y C M. Suppose that f : S ->' Al is a map of a compact surface such that for each component b of OS, either d nY = 0 or f (d is an immersed curve in 6-2 n Y

which is transverse to 01, f injects on simple loops, f is transverse to 8Y', f is a-transverse to 0' and no component of f-I(I') is a disc disjoint from 8S. Then S can be homotoped rel 8S to S1 such that area(Si n Y) <Max(O, Co(length((OS) n

Y)-X(SnY))).

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DAVID GABAI

Proof of Proposition 3.20. The main improvement of this Proposition over the last

is that f : S -> M is not h-normal. We argue by induction on c(S) = length((OS)n

Y) - X(S n Y) and can assume that S is connected. If c(S) = -2, then S is a 2-sphere in Y and can be homotoped out of Y. Now assume that the Proposition

is true for c(S) < k. Using the induction hypothesis, its routine to verify the Proposition if (8S) n Y is not a union of normal curves.

Suppose that there is an arc a C f -1(Y) with endpoints in distinct 1-cells of r such that f Ice is homotopic into a 2-cell of *, via a homotopy fixing 8a and supported in Y. Boundary compress along a to get a new map fl : T -> M which

satisfies the hypothesis of the Proposition, and c(f1) < c(f). By induction, f1 is homotopic rel OT to gl which satisfies the conclusions of the Proposition. If Ti is

the resulting surface, then attach a little 1-handle to T1 (missing u") to obtain a surface S1 which is homotopic to S rel 8S and satisfies the conclusion.

Therefore we can assume that f satisfies the hypothesis of Lemma 3.16. Now apply Lemma 3.16 and Proposition 3.17.

Corollary 3.21. If Sl is as in Proposition 3.20, then length(8(Sl n Y)) < Max(0,C1(length((8S)nY) -X(SnY))). Remark 3.22. The main results of §1-2 can all be stated in terms of cellulations rather than triangulations. In particular Proposition 1.14, Theorem 2.1 and Propo-

sition 2.4 hold by substituting triangulation with cellulation. The proofs parallel those of §1-2. Of course a version of Proposition 1.10 also holds for cellulations,

but there is not an obvious canonical branched surface which carries all normal laminations. For the sake of reference we single out

Proposition 3.23. Let M be a closed 3-manifold with cellulation ib. iri(M) is not negatively curved if and only if there exists a sequence of least area normal immersed discs fi : Di -+ M, such that length(8fi)/area(fi) -> 0 and no fi has fake boundary branching.


225

§4

THE UBIQUITY THEOREM

Definition 4.1. If a is a homotopically trivial curve in the 3-manifold 3f and k C M - a is a smooth simple closed curve, then define the wrapping number wr(a, k) _

min{lEflkj where E C Al is an immersed 2-disc transverse to k such that 8E = a}.

Theorem 4.2. Let k be a smooth simple closed curve in the closed, irreducible 3manifold M with cellulation zr such that k is transverse to (,2 , disjoint from chl and

for each 2-cell 77 C 0, Ik f1 t)I < 1. Then either Al is toroidal, or k is contained in a

3-cell or there exists a constant C such that for any least area disc D a-transverse to 02 with 8D fl k = 0 we have area(D) < C(wr(8D, k)+length(8D)). Remark. Here length and area are measured as in Definition 3.1. Recall that normal immersed discs are examples of a-transverse discs. The following result is an immediate consequence of Theorem 4.2.

Ubiquity Theorem 4.3. Let k Q B3 be a smooth simple closed curve in the closed, atoroidal, irreducible 3-manifold M with cellulation .0 such that k is transverse to V52, disjoint from V51 and for each 2-cell 71 C /i, lk fl 771 < 1. There exists constants

K and L such that if D is a least area disc a-transverse to ,2 with OD (1 k = 0 and

length(8D)/area(D) < L, then wr(8D, k)/area(D) > K. 0 Corollary 4.4. Let A be a quasi-least area semi-Euclidean lamination in the closed atoroidal irreducible 3-manifold M. If k C M - A is a smooth simple closed curve,

then k lies in a 3-cell. Proof of Corollary 4.4. Suppose that A is normal with respect to the cellulation'0

and all measurements are taken with respect to 1i. After passing to a subdivision of zG and isotopy of k we can assume that for each 2-cell ti of 0 that lk fl rll < 1. Note that the property of a leaf being conformally Euclidean or quasi-least area is

preserved after passing to subdivision. If L is a conformally Euclidean leaf of A, then by the Ahlfors Lemma, [Ca] p. 499, there exists a sequence of embedded discs

ti C E2 C

C L the universal covering of L, such that length (8Ei)/area(Ei)
0, then Figure 4.1 shows how to reembed Y U 14`0 U ... U TV,-, into Al in such a way that BTV, bounds a solid torus V, in M - Y' U GT o U

U I'VV_1, where IV; (resp. OlVr, Y') denotes the

reembedded TVt (resp. 81F, Y). There are an infinite number of such reembeddings,

by "twisting the neck". Now fix a graph structure on Y. This structure induces a Seifert fibering on 8W,.. At most one reembedding corresponds to spanning Seifert

fibres of 8Wr by meridinal discs of V,. After any other reembedding, A1- II 0 U

DAVID GABAI

228

U W;_1 is a graph manifold. By repeating this operation r - 1 more times, we conclude that M is obtained by attaching a solid torus to the boundary of a graph manifold. Since such a manifold is either reducible, has finite 1r1 or has a

lrt-injective immersed torus, Claim 1 is established. Note that if iri(M) is finite,

then M has a linear isoperimetric inequality. 0 k

k

k

k

k

W, bounds a solid torus

aw does not bound a solid torus

Figure 4.1

Given a'-least area disc D, define Wy, (D, k) =

wr(8D, k) + lengthy (8D) area,y (D)

We may suppress the subscript 0 when the cellulation is understood. Claim 2. We can assume that -0 satisfies

a) t,b Y is a relative hyperbolic cellulation and in particular for each i < r, i restricts to a cellulation on W2.

b) There exists a D2 X S1 C Wo(k) called N1(k) such that k c N1(k) is a core

of Ni(k), ,2 n N1(k) consists of 8N1(k) and a finite number of pairwise disjoint meridinal discs. Furthermore if a is a 3-cell of i/ilNi (k), then k n a is a properly O

embedded unknotted arc. Note that 01 n N1(k) = 0. See Figure 4.2

QUASI-MINIMAL SEMI-EUCLIDEAN LA.".MINATIONS IN 3-MANIFOLDS

229

A schematic view of IN, (k)

Figure 4.2

Proof of Claim 2. If ¢ is a subdivision of ii and Theorem 4.2 holds for Q, then Theorem 4.2 holds for t(,. Use the fact that there exists a K1 > 0 such that if

D is a '-least area disc, then there is a 0 least area disc, also called D, with Wy,(D,k) unchanged such that D is a Kl-quasi ¢-least area disc. Furthermore lengtho(8D) < lengtho(8D) < K1lengthJ,(8D). The last inequality uses the fact

that an embedded arc a in a 2-cell of 0 is isotopic rel 8a to an arc crosing 01 a uniformly bounded number of times, via an isotopy disjoint from k. (This is where we use the hypothesis that for each 2-cell 77 of ip, lkfl I < 1.) Thus a sequence (Di}

of t/i-least area discs with WO(D;) -+ 0 gives rise to a sequence {E;} of q-least area discs with WW(Ei) -+ 0.

Conversely suppose that q5 is a subdivision of such that if a is a 3-cell of LY'

and a f1 k $ 0, then cla = 1iJa. Then standard arguments show that there exists a K2 > 1 such that a ¢-least area disc D give rise to a V,-least area disc E where length,y(OR) < K2Ien the(8D) and OE is homotopic to 8D via a homotopy disjoint area.. (E) area*

from k. Thus if Theorem 4.2 holds for 0, then it holds for 0. Similarly this result follows in the following case. Here if a is a 3-cell of 0, then either k no, = 0 or k 11 a

DAVID GABAI

230

is an unknotted arc and p is a subdivision of t' such that 02 fl k = 02 fl k.

Given ip, we first construct a subdivision V)1 with the property that for each 3-cell a of V51, k n a = 0 or k fl a is an unknotted arc. The subdivision is created

on each 3-cell B of

as follows. Let t = B fl k. Think of B as D2 x I with

8t C D2 xO and tin generic position with respect to the height function coming from

the I-factor. Now attach a finite number of pairwise disjoint properly embedded 0

2-cells C1,

, C. C B, one for each local minimum of t C B, with the property

that for each i, C= separates off a 3-cell B; with Bz fl t is an unknotted arc and to (B -U" 1B;) is a trivial B3-link on s components. Now subdivide each C2 along

a properly embedded arc such that each "half' of of C; hits t exactly once. See Figure 4.3. After subdividing along another s - 1 2-cells we obtain a cell division

on B such that each 3-cell intersects tin an unknotted arc. 0 is the result of these n + s - 1 subdivisions of y. One readily checks in n + $ - 1 steps that Theorem 4.2 holds for v if and only if it holds for 'P1.

a)

Figure 4.3

b)

Let N1(k) be a D2 x S1 in Wo whose core is k such that N1(k) fl V)i = 0 and N1(k) fl vi is a disjoint union of meridinal discs. Subdivide V51 to 7P2 so that N1 (k) nVj = N1(k) n 02 U oN1(k). By the second paragraph of this proof, Theorem 4.2 holds for 02 if and only if it holds for V51.

By (EP] there exists a hyperbolic cellulation on Y and hence there exists a

QUASI-MINIMAL SEMI-EUCLIDEAN I.AMIN AT1ON IN

.`.!A.IFOI.DS

23!

cellulation i3 on Al satisfying a) and b) and which agrees with t _ near A 'I

By

the uniqueness of PL structures for 3-manifolds

there exists a cellulation V,

on Al isomorphic to 03 via an isomorphism which restricts to the identity on Nj (k

such that 04 and 02 have a common subdivision

and

(k; = t.:fl-Vt(k) =

t21Nr(k). By the first two paragraphs Theorem 4.2 holds for &4 if and only if it holds for tPo if and only if it holds for 02. Therefore Theorem 4.2 holds for va if and only if it holds for P. Since 0.1 satisfies the conclusions of C lain; 2. that result

follows. 0 Claim 3. It suffices to prove Theorem 4.2 for discs D such that DD C 1'.

Proof of Claim 3. It suffices to prove Theorem 4.2 for discs D such that OD N1(k) = 0. Indeed by construction of N1(k), a small hornotopy disjoint from k.

pushes aD off of N1(k) and so D gives rise to a least area disc E a-transverse

to 0 with En N1(k) = 0,wr(aE,k) = wr(aD,k) and D. E have approximate isoperimetric inequalities.

Since balls and solid tori have linear isoperimetric inequalities it follows that there exists a K3 such that if y is a locally f-embedded homotopically trivial curve

in ill - Nr (k) then there exists a hornotopy H : Sr x I -r Al of , to -.r such that 0

yr is locally sb-embedded, yt c Y and area(H) < K3length(,). This implies that if D (resp. E) is a least area disc spanning -y, (resp. -f I), then

area(E) > area(D) - area(H) length(yr) < length(y) + v(area(H)) < (1 + vK3)length(y) wr(yt, k) < wr(y, k) + 3area(H) < wr(y, k) + 3K3length(y)

The first and third inequalities are immediate. The second follows from Lemma 3.19. To obtain the fourth, note that H can be chosen so that if n is a 2-cell in the 0

induced cellulation on Sr x I, and H(t) n Nt (k) # 0, then HIt is an embedding and H(t) fl ik' ; 0. (This uses the fact -y fl N1(k) = 0.) Since H can be chosen so it follows that IH'r(k)1 < 3area(H).

that 177 f1 kf < 1 and area(HIri) >

s

DAVID GABAI

232

Thus a sequence of discs {Di} such that W (Di, k) -> 0 gives rise to a sequence 0

{Ei} with W(Ei, k) -> 0 and 0E; C Y. 0 Let E be an immersed disc in RI a-transverse to ,2 such that aE = OD and JE fl kI = wr(OD, k). Among all such discs with these properties choose one that

minimizes IaY fl El. Let Ey = E n Y. Assume that length(OE) > 0.

Claim 4. -X(Ey) < wr(aE, k) Proof of Claim 4. Since Y has no essential annuli, and I4Y fl El is minimal, each component of Ey disjoint from aE is a disc with at least two open discs removed. Define a partial ordering on the components Fl,

, F. of Ey, by Fi < Fj if Fi is

separated from OE by Fj. Let Ft = UFj 5F; Fj and Fi be the subdisc of E bounded

by the outermost component of aFi. Since for some i, Ey = Pi and E = Pi it suffices to show that for each i, -x(Fi) < JPi n kI - 1. To prove this let Fi be a minimal component of Ey. All but one component of aFi bounds a disc in E whose

interior is disjoint from F;. Let G be one such disc. The incompressibility of OWi in W,, i > 0 implies that G C WO. The incompressibility of OWo in Wo - k implies

that G fl k ,{ 0. Hence -X(F,) < JFi n kI. Now inductively assume that for all Fi < F -X(Fi) < iFi n ki - 1. Again let G C E be the disc spanning an innermost 0

component of aFj. By either the induction hypothesis, if GnEy # 0 or the previous 0

argument, if G fl Ey = 0 it disjoint from OE follows that -X(G fl Ey) < IG fl kI -1

and therefore -x(Fj) < JFj fl ki - 1. 0 Claim 5. E can be homotoped to E' rel aE such that if F is the component of Ey which contains OE, then

1) area(F) < Co(wr(OD,k)+ length(OD)) and

2) length(OF) < CI (wr(aD, k)+ length(OD)). 0 Proof. Let S be the compact codimension-0 submanifold of E such that s n Y is exactly the component of EnY which contains OE. By Proposition 3.20 and Claim

4, S can be homotoped rel aS to Sl such that area(Sj n Y) < Co(length((aS) n Y) - x(S n Y)) < Co(length(aE) + wr(OE, k)). Since the homotopy of S to SI

QUASI-MINIMAL SEMI-EUCLIDEAN LAMINATIONS i:`: 3-}.t AN!FOLDS

233

extends trivially to a homotopy of E and DE = DD conciuwion 1 . follows. (Here E'

is the homotoped E and F is the component of St ' Y containing DE.; Conclusion 2) follows from Proposition 3.21.

Claim 6. There exists a constant

such that if a is a closed curve in l1';. which is

homotopically trivial in Ill, then a bounds a disc

-4

such that area(A)

< k2(length(a)).

Proof. Since balls and solid tori have linear isoperimetric inequalities and each

W,i > 1, lies in a 3-cell, Claim 6 holds (using the same constant k3) for all

ac

1, and for all a C l4'o which are homotopicalh. trivial in 110.

Let P C Wo be a normal curve which generates rl(IF0). There exists a k4 such that if a c Wo is not null homotopic in loo, then a is homotopic in W o to 31, n ¢ 0

via a homotopy L : S' x I -* Al such that area(L) + Ink < k;length(a). Since a is homotopically trivial in M,3" and hence 3 (since r1(M) is torsion free) are homotopically trivial in M. If B (resp..4) is a least area disc bounded by .3 (resp.

a), then area(A) < Injarea(B) + area(L) < (1 + area(B))(Inj + area(L))

< 2(1 + area(B))k4length(a) = kslength(a),

where k5 = (1 + area(B))k4. Finally, take k2 =max(k3, ks).

To complete the proof of Theorem 4.2 observe that OD bounds a disc J which is the union of F (which was defined in Claim 5) and least area discs bounded by

OF - 8D. Thus by Claims 5 and 6 we obtain area(D) < area(J) < area(F) + k2length(OF) < (Co + Crk2)(wr(OD, k)+ length(DD)).

§5

UBIQUITY IN THE SMOOTH CATEGORY

Theorem 5.1. Let k be a smooth simple closed curve in the closed irreducible Riemannian 3-manifold M. Then either M is toroidal or k is contained in a 3-cell

DAVID GABAI

234

or there exists a constant C > 0 such that if D is a least area disc with 8D n k = 0, then area(D) < C(wr((9D, k) + length(8D)).

Proof- Step 1. Either k C B3 or M is toroidal or M = Y U Wo U . . U W, where 6'i o is a D2 x S', k C W'o, 8Wo is 7r1-injective in WO - k, W n W1 = 0 if i $ j, and if i > 1, then W; lies in a 3-cell and 8WW is a torus which is 7r1-injective in 0

W. Furthermore Y # 0 and Y has a hyperbolic structure of finite volume. Finally

YIi(WoU...UI'Vr) =81'. Proof of Step 1. For the proof of step 1, read (in §4) from the beginning of the proof of case 2, through the end of the proof of Claim 1.

0

Step 2. It suffices to establish Theorem 5.1 for any Riemannian metric on M.

Proof of Step 2. Use the fact that changes in the Riemannian metric changes arc

length as well as area by uniformly bounded multiplicative amounts. 0 0

Let X be a space diffeomorphic to Y and give X a complete hyperbolic metric.

Let T x [0, oo) cut off the ends of X and be parametrized so that each T x i is a union of r + 1 horospherical tori and for z E T, z x [0, oo) is a geodesic parametrized

by arc length. Let N(OY) C (Wou...UIV,.)-k be a closed regular neighborhood of 8Y. Let g be a Riemannian metric on M such that g1Y is isometric to X -T x (0, oo) and gIN(8Y)

is isometric to T x [0,1]. I.e. gIY U N(Y) is the pullback metric induced from a diffeomorphism f : Y U N(Y) -a X - T x (1, oo) such that f (Y) = X - T x (0, oo)

and f (N(8Y)) = T x [0,1].

Step 3. a) There exists y > 0 with the following property. It suffices to prove Theorem 5.1 for discs D where the geodesic curvature at each point of 8D n Y is bounded above by -y.

b) We can assume that OD n N(8Y) is a union of curves of the form z x [0,1].

c) We can assume the 8D is embedded in M. 0 Let D be a least area disc in M such that 8D n k = 0. Let E be a smooth disc which spans 8D, is transverse to k U N(OY), IE n kj = wr(8D, k) and satisfies the

QUASI-MINIMAL SEMI-EUCLIDEAN LAMINATIONS IN 3-:M1ANICOLDS

235

following two conditions.

i) No component of E n Y disjoint from 8E is either a disc or an annulus

ii) If a C E fl Y is a simple closed curve, then either a bounds a disc in E ^ Y or a is homotopically nontrivial in Y. Remark. The reader may have noticed that we have abused notation by identifying

the disc E with its image in M thereby suppressing the mapping f : E -3 ,If which

defines that image. Thus it would have been more precise to have first defined f : E -a M and then rephrased i) by "no component of f't(1') is a disc or annulus

disjoint from 8E" and ii) by "If a is a simple closed curve in f `(Y), then either a bounds a disc in f'1(Y) or f Ice is homotopically nontrivial as a map into 1'. In the following statement E n Y denotes a subset of the disc E.

Step 4. -X(E fl Y) < wr(8E, k) Proof of Step 4. By hypothesis wr(OE, k) = IE fl kI. Define a partial ordering on

the components Fr,

, F,,, of E fl Y, by Fi < FF if Fi is separated from OE by

Fj. Let F2 = Up,- r1/2(a, [E]).

Thus if g is at least two we have (a, [E])/(2g - 2) < sup 4lrIIaIIh/IIShIIh,

and in the case that E is a torus we see that the right-hand side is infinite. This is the desired result, in view of the characterization of the dual Thurston norm at (2).

Combining this lemma with the previous one, we obtain:

Corollary 2.3. If c is a Spin` structure on Y and the dual Thurston norm of c1(c) is bigger than 1, then there exists a Riemannian metric on Y for which the monopole equations have no solution.

We shall see later that the inequality in Lemma 2.2 is actually an equality for many 3-manifolds Y (Proposition 3.8).

255

3

The monopole invariants

Obtaining invariants from the rreonopole cquutions

There is a well-understood procedure by which we can extract some ruetric-

independent data from the set of solutions to the n:onopole equations. to obtain an invariant S14"(I', C) E Z

depending only on Y (a closed, oriented 3-manifuld) and the Spin structure c. A careful account of the most general case is given in :4. 27'. and there is a model for this construction in [35], where Casson's invariant is given a gauge-theory interpretation. In this subsection, we review this construction, restricting ourselves sometimes to the simpler cases. The monopole equations on a Riemannian 3-manifold Y are the variational equations for a functional

CSD(A, ) = Z

- .4o) n (F+ F) - 2 j(. Dt ) dvol.

the Chern-Simons-Dirac functional of .4 and (D. (A reference connection .4o is chosen to define the first term, but a change of reference connection only changes the functional by addition of a constant.) The equations are invariant under the natural symmetry group of the Spin` bundle II which is the group G of maps u : Y -> S' acting on IT', by scalar endomorphisms. This group acts on A and (D by

A -a A - (u-'du)1

H U. The functional CSD is invariant only under the identity component of G. The

component group of G is the group of homotopy classes of maps Y -+ S'. which is isomorphic to H' (Y; Z): the change in the functional under a general element of G is given by

CSD(A - (u-'du)1, u-1)) - CSD(A, (D) = -4-,,'([u] '- ct(c)) [Y].

So CSD does not descend to a real-valued function on the quotient space C = {(A, 4))}/G, but does descend to a circle-valued function whose periods are multiples of 47rz.

3.

256


The orbit space C is an infinite-dimensional manifold except at points where the action has non-trivial stablilizer. These are the configurations with

= 0 (the reducible configurations), whose stabilizer is the circle group

of constant maps u. If cl (c) is zero or a torsion class, then there are spin connections for which .4 is flat, and these are reducible solutions to the equations. To define SW (Y; c), one first chooses a perturbation of the functional CSD so as to make the critical points non-degenerate. To do this, one may

add an extra term, setting CSD,(A, ) = CSD(A, ) + J(A

- Ao) A (ip),

where ,4 is an exact 2-form. The variational equations are now

p(F4 + ip) - {$ ®,D'} = 0 and for a suitable choice of p (such p are dense), the Hessian will be nondegenerate at all irreducible solutions in C. A proof is given in [12). (The condition that p be exact means, among other things, that the function CSD, still descends to a circle-valued function on C and that its periods are the same as those of CSD.) Critical points of CSD,, on C have infinite index: the Hessian is a selfadjoint operator which, like the Dirac operator, has a discrete spectrum which is infinite in both the positive and negative directions, so one cannot define an index i(a) at a critical point a as the dimension of the sum of the negative eigenspaces. However, there is a way to define a relative index between any two critical points. If a and b are two non-degenerate critical points of CSD, and -y(t) is a path in C which joins them, then the Hessians H7 a family of operators for which one can define a spectral flow - the number of eigenvalues which move from negative to positive in the family. One can defines the relative index i(a, b) as the spectral flow. There is a further point to attend to, which is the possible dependence of the spectral flow on the choice of path. The fundamental group of C is again isomorphic to Hl (Y; Z), and for a closed path y, the spectral flow is given by SF(1) = ([u] - cl(c)) [Y],

where u is the corresponding element of Hl. As noted above, cl(c) is alway divisible by 2, so at least the parity of i(a, b) is well-defined. After settling

Basic classes

25 7

on a convention as to which is to be which, we can divide the critical points into even and odd, using the relative index. The invariant SW (Y. c) can now be defined as. roughly speaking. the euler number of the vector field CSDA on C. We restrict our attention to the case

that cl(c) is not a torsion class, so that there are no reducible solutions to the equations, and after choosing u so that the solutions are non-degenerate we set

SW(c) = #(even) - #(odd).

!; 11;

The main technical point here is that the set of critical points in C is compact. and hence finite under our non-degeneracy assumption. The proof of the compactness property of solutions to the monopole equations is a straightforward application of standard techniques, starting from the Co bound obtained by applying the maximum principle to (4). The quantity (11) is independent of the choice of Riemannian metric on Y and the choice of p. Our' ur failure to fix a convention about which is even and which is odd leaves an overall sign ambiguity in the invariant.

If cl(c) is torsion, a similar definition leads to an invariant in the case that bt (Y) is non-zero: one can perturb by a small, non-exact, closed 2-form

p to remove the reducible solutions. The case of bt(Y) = 0 is a little more subtle, but an invariant can be defined: see [4, 271. Basic classes

We call a Spinc structure c on a closed 3-manifold Y basic if the monopole invariant SW(Y, c) is non-zero. In this case, we also refer to the first Chern class cl (c) as a basic class if Y. For our present purposes, these definitions are only interesting when bt (Y) is non-zero and ct (c) is not torsion. The definition of the invariant SW means that, if c is basic, then the corresponding perturbed monopole equations (10) have at least one solution, for every Riemannian metric and a dense set of exact 2-forms p. The compactness

properties of the equations imply that the non-emptiness of the solution space is an open condition, so it is also true that the original equations (3) have solutions for every Riemannian metric h. This observation, together with Corollary 2.3, yields a relationship between basic classes and the genus of embedded surfaces:

258

3.


Proposition 3.1. If Y is a closed, oriented 3-manifold with bl # 0 and a is a basic class on Y, then the dual Thurston norm of a satisfies jal. < 1. In other words, for any oriented embedded surface E in Y representing a class a, we have

x-(E) ? (a,o')

Aonopole invariants and the Alexander invariant

Whether the above proposition is useful depends on what else one knows about the invariant SW and the basic classes. Meng and Taubes [30] showed that when bl (Y) is non-zero, SW (Y, c) is completely determined by a classical invariant, the MIilnor torsion. While it is interesting that torsion is calculated

by the gauge theory route, this result does make Proposition 3.1 look less interesting.

The situation is easiest to describe when the 3-manifold Y is the result of zero surgery on a knot K in S3: that is, Y is obtained by removing a solid torus neighborhood of K and replacing it while interchanging the longitude and meridian curves on its boundary. In this case, the information contained in the Milnor torsion is the Alexander polynomial of K. We shall use the symmetrized Alexander polynomial, whose shape is

alt

with a, = a_i. We refer to r as the degree of the polynomial. The betti number of i = Y(K) is 1, and the second cohomology is Z. There is therefore exactly on Spin` structure ck with c1(ck) = 2k. Its monopole invariant can be expressed in terms of the Alexander polynomial:

Theorem 3.2 (Meng-Taubes [30]). On the manifold Y(K) obtained by zero-surgery on K, the monopole invariants are given by SW (I ; Ck) _ E jai+lkl j>0

In particular, SW (z, Ck) = 0 fork > r - 1.

(12)

.Monopole invariants and the Alexander invariant

239

(The symmetry SW(Y,ck) = is a general property o the monopole invariants, and follows from a symmetry of the equations.; With this interpretation of the monopole invariants. we call reformulate Proposition 3.1 as Statement 3.3. If E is a connected, oriented surface in Y(Ii

rrp;ri.5E+rziirig

the generator of H2, then g(E) > r. where r is the degree of the Alexander polynorraial.

Indeed, the theorem tells us that the extreme basic- classes in H2 2 arise from the Spin` structures ct(,_l), with cl(c)[2] = =(2r (The restriction to connected surfaces has no effect on our statement when b: = 1.) The statement above, however, is an elementary consequence of the definition of the Alexander polynomial, in the formulation which expresses .-l:; ;a) in terms of the homology of the infinite cyclic cover of Y(K). Furthermore. the inequality between the genus of 2 and the degree of the Alexander poly-

nomial is not a sharp one in general. Here is a relevant result:

Theorem 3.4 (Gabai [15]). If 2 is an embedded surface in Y(li) representing the generator of H2, then g(E) is no smaller than the genus of the knot.

This theorem is not self-evident. The proof of the result comes from refinement of the existence theorem for taut foliations quoted earlier: a minimal-

genus spanning surface for K can be filled out to a taut foliation F of the complement of a neighborhood N(K) in S3, and this can be done so that the boundaries of the leaves are a family of longitudinal circles. The solid torus N(K) has a trivial foliation by ineridianal disks, and in the surgered manifold Y(K), this foliation joins with F to give a taut foliation of the closed manifold.

The genus of a knot K and the degree of Al are in general different. The untwisted Whitehead double of any non-trivial knot has genus 1 and Alexander polynomial 1. Figure 1 shows a Whitehead double of a trefoil knot; a spanning surface is formed from a ribbon which follows the course of the trefoil, together with a small hand with one full twist at the clasp. Under the connected sum operation, the genus of a knot is additive. while the Alexander polynomial is multiplicative, so one can easily obtain knots of large genus whose polynomial has small degree.

260

3.


Figure 1: A doubled trefoil. Monopole classes

The result of the previous subsection is disappointing. The lower bound for the genus of embedded surfaces in terms of basic classes is no better than the lower bound for the genus of a knot which the Alexander polynomial provides: it does not capture the topology. However, the reason for the disappointing result is that too much of the content of the gauge theory has been disposed of in passing from the solution set of the monopole equations (3) to the integer invariant SW(Y, c) which counts these solutions. Rather than count the solutions, let us take a step backwards and simply make the following definition: Definition 3.5. A class a E HZ(Y; Z) is a monopole class if it arises as cl(c) for some Spin` structure c on Y for which the equations (3) admit a solution for every choice of Riemannian metric h on Y. This definition ensures that the basic classes are monopole classes. Also, Proposition 3.1 applies to monopole classes, just as it applies to basic classes;

so a monopole class a has dual Thurston norm at most 1. (This is simply another step backwards, to Corollary 2.3, from which Proposition 3.1 was deduced.)

However, the monopole classes are in general a larger set than the basic classes. The following theorem is our central result. We shall give an outline of the proof in section 5.

Monopole classes

261

Theorem 3.6. If Y is a closed, irreducible, orienled smooth, taut foliation - ' by oriented leaves, then

manifold with a a monopole class.

Combining this statement with Corollary 1.6, one obtains

Corollary 3.7. If Y is a closed, irreducible oriented 3-manifold, then the -unit ball B(Y) C H2(Y; R) for the dual Thurston norm is the conre:c hull of the monopole classes (reduced to real coefficients). In other words. the Thurston norm on H2 is given by a

where the maximum is taken over all monopole classes.

(The extra hypothesis in Corollary 1.6, that there is not a basis for HZ consisting of tori, was there to ensure that Y had at least one smooth. taut foliation. The hypothesis is unnecessary in Corollary 3.7, for the accidental reason that 0 is always a monopole class, on account of the reducible solution with 4) = 0.) Thus the monopole classes give us sharp information about the genus of embedded surfaces, while the basic classes (in general) do not. For example. in the 3-manifold Y(K) obtained by zero-surgery on a knot K of genus g7 the classes ±(2g - 2) in H2 = Z are monopole classes. This means that. for the Spin` structure which we called c9_1i solutions of the monopole equations always exist, even though the algebraic count of the solutions will be zero if the Alexander polynomial has small degree. We can also return to our discussion of the scalar curvature, and see that

our inequalities there were sharp also. Our previous results stated that the unit ball for the norm 41rsup IIa-Ih h

IIshIIh

was sandwiched between the convex hull of the monopole classes and the unit ball for the dual Thurston norm (Lemmas 2.1 and 2.2). Knowing that these last two coincide, we can replace the inequality of Lemma 2.2 with an equality:

Proposition 3.8. If Y is a closed, irreducible oriented 3-manifold, then the dual Thurston norm on H2(Y;IR) is given by

IaI = 47r sup IISa-,

262

4.

,where the

Detecting monopole classes

is taken over all Riemannian metrics on Y.

Remark. Up until this point. it was by no means clear that the supremum on the right hand side was ever finite. Our results now say that the supremum is reached. in the limit, by stretching the metric along a cylinder [-R, R] x where 7- is a minimum-geniis representative for a class a with (a, a)

4


How can one detect that a given class is a monopole class, without it being a basic class' An existence theorem is needed for solutions to the equations. A simple scenario in which one can see that solutions must exist arises when the 3-manifold in question is embedded in a suitable 4-manifold. We therefore turn to dimension four. The 4-dirnensiorcal equations

The equations we have been discussing are a 3-dimensional version of the monopole equations which were first introduced, by Witten [42], in dimension 4. On an oriented Riemannian 4-manifold X, a Spin` structure c consists of a hermitian vector bundle 11' of rank 4, together with a Clifford multiplication

p : T*X -+ End(W)

with the property that, if e', ... , e4 are an orthonormal coframe at a point in X, then the endomorphisms p(ei) are skew-adjoint and satisfy the Clifford relations p(e')p(e') + p(el)p(e') = -2bi,j.

Clifford multiplication is extended to forms of higher degree as before. It is a consequence of this definition that the spin bundle W decomposes into two bundles of rank 2, T4 "+ e IT , whose determinants are equal. The action of 1-forms maps 117. -p Ii'-, the action of 2-forms preserves the decomposition, and one can characterize I['- as the subspace annihilated by p(w) for all selfdual 2-forms w (forms satisfying *w = w). A spin connection is defined as before, and given a spin connection A, one has a Dirac operator DA acting on sections of W. We write D+i for the restriction of DA to W+, which is an operator D : I'(W+) -+ r(w-).

The 4-dimensional equations

2G3

The connections on A21.1'+ and induced by a spin cui:nection A are equal, and we write .4 for either. NV e write c; (ci for the first Cl:ern class of W+, which is the class represented in de Rhauri cohomoloay by the form (i/27r)FA.

Once again, the set of Spin` structures is acted on transitively by H2(X; Z), and we have the same rule, cl(c +e) = cl(c) + 2e.

which shows that cl (c) determines c to within a finite ambiguity measured by the 2-torsion subgroup of H2(X; Z). The 4-dimensional monopole equations are the following pair of equations for a section 4i of and a spin connection .4:

p(Fj)-{4®4i'}=0 Dad = 0

(13)

The first equation is to be interpreted as an equality between endomorphisms of W+. The curly brackets denote the trace-free part on 11'+, not on I1', and F+ denotes the projection of the curvature onto the self-dual forms, as usual. The moduli space All, is the space of solutions (.4.4i) modulo the action

of the gauge group, G = Map(X, St). N e can also perturb the equations, rather as in the 3-dimensional case, by an arbitrary self-dual 2-form q:

p(Fa+in)-{4i(9V}=0 D,+.t(D = 0.

(14)

We write M,,,, for the solution space. In dimension 3, for a generic perturbation, the irreducible solutions are

isolated. In dimension 4, the equations have an index. We suppose X is compact and write d(c) = 4 (cl(c)2[X] - 2X(X) - 3o(X)), which one can also recognize as the second Chern number, c2 (F1'+) [ X j. The

basic facts about the moduli space are these:

Proposition 4.1. The moduli space M n is compact. For an open, dense set of perturbations 17, the irreducible part of the moduli space (the locus of solutions with 0 0) is a smooth manifold of dimension d(c), cut out transversely by the equations.

264

4.


Remark. The proof of compactness runs much as in the 3-dimensional case, beginning with an essentially identical calculation leading to (4).

For the unperturbed equations, a solution with = 0 means a connection -4 in :12tj'' with anti-self-dual curvature, and hence an anti-self-dual representative for c1(c). An anti-self-dual form has negative square, so there can be no solutions if c1(c)2[X] is positive. If cl(c)'-[X] is zero, there can be reducible solutions only if cl(c) is a torsion class, so that there is a flat connection. The same is true of the perturbed equations if i is small. Even if c, (c)2[X ] is negative, however, there can only be a reducible solution if 2irc1(c) - is represented by an anti-self-dual form. The real cohomology H2(X ; i) is the direct sum of the self-dual and anti-self-dual harmonic spaces, so the space of q for which such a representative exists is an affine subspace of codimension b+(X ), the dimension of the space of self-dual forms. If b+(X) is non-zero, there is no reducible solution for generic 77, and if b+(X) is at least 2. there is no solution for all i in a generic path. Now let c be a Spin` structure with d(c) = 0, so that cl(c)2[X] = 2X+3Q.

W e shall suppose that X either has b+ > 2 or has 2X + 3a non-negative, and if 2X + 3a is zero we shall also ask that let cl (c) is not a torsion class. For a generic rl, and also for a generic path of 17, the moduli space k1,7 then consists of finitely many points which are transverse, irreducible solutions of the equations. The number of solutions, counted with suitable signs, is independent of the choice of perturbation and the choice of metric. We write SW (X, c) E Z

for this number. This is the Seiberg-Witten monopole invariant for X with the Spin` structure c [42]. As before, if SW (X, c) is well-defined and nonzero, we call ci (c) a basic class of X. Note that if SW (X, c) is non-zero, then the moduli space M, of solutions to (13) is non-empty for every choice of Riemannian metric. The first significant result about basic classes was proved by Witten in [42]. This was the statement that, for a smooth algebraic surface with b+ > 1 (e.g. a hypersurface in QF3 of degree 4 of more), the first Chern class and its negative (the canonical class) are basic classes. This was soon generalized by Taubes in [36, 37] to symplectic manifolds, in the following form. A symplectic structure w on a manifold determines an almost-complex structure uniquely up to deformation, and hence has Chern classes ci(w). The canonical class K, is -c1(w).

Stretching 4-manifolds

Theorem 4.2 (Taubes, [36, 37]). Let (X,-;;) be a compact sginplectw .fmanifold. Suppose either that b+ > 1. or that b : = 1 and K.. and A are both positive. Then the canonical class is a basic class.

Remarks. Note that the hypotheses rule out C?2. (On a Kdhier r:anisuld, the sign of K - [w] is opposite to that of the mean scalar curvature.} The statement for the case b+ = 1 can be sharpened, but not without refining our treatment of the monopole invariants. A full treatment of the nionopole invariants in the case b+ = 1 is given in [26].

This version of the theorem is a little careless. As well. as determining a canonical class K, a symplectic structure gives rise to a canonical Spin` structure c,,, with cl(c,) = K. Taubes result asserts that this Spin` structure is basic, and in fact t15j

under the hypotheses of the Theorem. Stretching 4-manifolds

A simple relationship between the 3- and 4-dimensional equations gives the following criterion for a class cx on a 3-manifold to be a monopole class.

Proposition 4.3. Let Y be a closed, oriented S-manifold embedded in a compact, oriented 4-manifold X. Let a be a basic class on X. Then the restriction a]y is a monopole class on Y. Proof. Let c be the Spin` structure with cl (c) = a and SW (-Y, c) non-zero, so that solutions to (13) exist for all metrics h on X. Let by be a Riemannian on Y, and let hl be any metric on X such that a collar neighborhood [-1, 11 x Y carries a product metric dt2 + hy. Let hR be obtained from hl by replacing

this short cylinder by a longer cylinder, [-R, R] x Y, for R > 1. For each hR, there exists a solution (AR, (DR) on X. The idea of the proof is to show that, as R approaches oo, we can find a subsequence such that the corresponding solutions converge to a translation-

invariant solution on some portion of the cylindrical piece. A translationinvariant solution can be interpreted as a solution of the 3-dimensional equations on Y, for the metric hy, so showing that a solution exists. Since by is arbitrary, the class c1(c)]y is a monopole class.

4.

266


To make this work. we need first to understand the relationship between the equations in dimensions three and four. On a cylinder [-R, R] X Y with a product metric, the action of p(dt) gives an isomorphism between W+ and U . Using this isomorphism, Clifford multiplication by 1-forms orthogonal to dt become endomorphisms of TI'+. In this way, Y acquires a Spin` structure (with spin bundle TV 3 = TT" 11.) which one can call the restriction of c. Given

a solution of the unperturbed 4-dimensional equations on the cylinder, one can apply a gauge transformation to make the dt component of A zero. If A is in such a temporal gauge, it can be recovered from the path A(t) in space of spin connections on 1', obtained by restricting A to the slices {t} x Y. The spinor on the 4-manifold gives a path 4)(t) in the space of spinors on the 3-manifold. In a temporal gauge, the 4-dimensional equations (13), become the following equations for the paths A(t) and (P(t),

P(A) = -P(FA) +

®V}

4=-DA),

(16)

in which D.4 now stands for the 3-dimensional Dirac operator, and the dot is differentiation with respect to t. These equations can be recognized as the downward gradient-flow equations for the functional CSD(A, 4b). Having understood this relationship, we can complete the proof. The solution (AR. 4DR) on the cylinder [-R, R] x Y can now be interpreted as a gradient trajectory for the Chern-Simons-Dirac functional. The compactness properties of the equations can be used to show that the change of CSD along these trajectories is bounded, by a constant independent of R. It follows that, when R is large, there is at least some portion of the cylinder in which the change in the functional is small. Passing to a subsequence, one obtains in the limit a translation-invariant solution to the equations on the cylinder, in a temporal gauge. This is a critical point of CSD, or in other words a solution of the 3-dimensional equations. This outline is filled out in [23]. 0

Remark. It is worth commenting that, if c restricted to Y is trivial, the solution whose existence is established by this argument may only be the trivial solution. With some additional hypotheses however, one can establish a stronger conclusion. Using Theorem 4.2, we can draw the following simple corollary.

Floer homology

Corollary 4.4. Let Y be a closed oriented ?d em,i,edrled in a closed. syrnplectic 4-manifold (-1_o). If bri.V: = 1. ;appo e steer. tie l:;putr. es of Theorem 4.2 hold. Then li_-11c' H'(1') i.. a rno;rc:pufe clr:, s an 1'. (In fact, solutions on 1F exist for the Spin` structure This obse_ation. with a little ingenuity, is already enough to show that the set cAr morropole

classes can have larger convex hull than the set of basic classes on a 3manifold. We will need to adapt the corollary. however. before it becornes very useful. It is a puzzling question to characterize the classes a H'which arise in this way, for a general 1'. Floer homology

There is a well-understood framework in which to place the ideas just discussed, namely the framework of 'Floes horology'. The model in the literature that is closest to what we need is Floer's construction in i11' of an invariant of 3-manifolds, using the gradient-flow of the Chern-Simons functional (of an SU(2) connection). There appears to be no serious obstacle to adapting [11 to the ChernSimons-Dirac functional, particularly in the case that cj (c) is not torsion. but there is not yet a complete account of such a construction in the literature. Nevertheless, it is clear how to proceed, and we shall content ourselves with some remarks. The starting point of [11) is the basic observation that one can calculate the homology of a compact manifold 11 by the following recipe. Choose a Morse function f on _1I whose gradient How satisfies the additional `Morse-Smale' condition, that the stable and unstable manifolds of all the critical points meet transversely. This means in particular that the trajectories which run from a critical point a at t = -oc to a critical point b at t = +oo form a family of dimension equal to the difference of the indices of a and b. Of these degrees of freedom, one is the freedom to repararnetrize the trajectory y(t) as y(t + c). If a and b have index differing by 1, the trajectories are isolated once one forgets the parameterization. Now form the vector space C with a basis ea indexed by the critical points a, and define a linear transformation 8 by a(ea) = 1: nabeb,

where nab counts the number of trajectories from a to b in the case that i(b) = i(a) - 1 and is zero otherwise. To avoid questions of orientation, one

268

4.


can take Z/2 as the field of coefficients for C. Then one shows that 82 = 0 and the ker 8/im8 is the homology of M. In particular it is independent of the choice of f and the choice of Riemannian metric used to define the gradient. Without having an alternative definition of the homology however, one can verify this independence directly. Floer applied this construction in an infinite-dimensional setting, taking the Chern-Simons functional as f. In the monopole setting, one should use the functional CSD on the space C. The situation is simplest in the case that cl (c) is not torsion, so that the monopole equations on Y have no solution with 1) = 0. As mentioned above, for a suitable exact 2-form µ, the perturbed functional CSD, has non-degenerate critical points, and these are a finite set. Although there is no well-defined difference of indices between a pair of critical points, we can measure the index difference between a and b along a given trajectory ;, as the spectral flow of the Hessian, as before. If a Morse-Smale condition is satisfied, we can then construct C and 0 as before, defining nab as the number of trajectories whose spectral flow is 1. unfortunately, we cannot expect to achieve the stronger Morse-Smale condition by such a restricted class of perturbations as the addition of an exact 2-form p. One must seek a larger class of perturbations. At the same time, it is a particular property of the equations involved that the spaces of trajectories have any reasonable compactness properties (as noted in the previous subsection, the gradient trajectories can be interpreted as solutions of the 4-dimensional monopole equations), and one must choose the perturbations of CSD, so as not to upset this feature. We make some remarks in the following subsection about how one might define a suitable larger class of perturbations, and for the moment we shall pass over this point. It is precisely here that more work needs to be done to carry through the Floer program for the monopole equations. After taking care of perturbations and compactness, one should arrive by this construction at a vector space HF(Y, c) with Z/2 coefficients and an even-odd grading. It should depend only on Y and c, not on the choice of Riemannian metric or the perturbation chosen for the equations. The construction makes clear that HF(Y, c) is zero if the original monopole equations, or their perturbation, have no solution. Also, the euler characteristic of HF(Y, c) (the difference of the odd and even betti numbers) is equal to the integer invariant SW (Y, c). The usefulness of Floer homology in the present context is that Proposition 4.3 and Corollary 4.4 can be strengthened, so as to conclude that the

Perturbing the gradient flow

egg

Floer homology is non-zero. (For the 4-dimensional Yang-Mills invariants. this role for the instanton Floer homology of !ll was first noted by Donaldson.) For example, if Y is embedded in a closed sylnplectic 4-manifold (X, w) (satisfying the hypotheses of Corollary 4.4 in the case i ; X = 1;. and if K, lr is not torsion, then one would conclude that HF(Y. C, ' is non-zero. once the definition of this Floer homology was in place.

Perturbing the gradient flow

It may be worth noting that perturbing the Chern-Simons-Dirac functional so as to achieve a Morse-Smale condition for the trajectories of the gradient flow may not be particularly difficult. Let Y be a closed Riemannian 3manifold with Spin` structure c, and let S be the space of all pairs '.-1. (D ). where A is a spin connection and D is a section of W. The space C above is

the quotient S/G. Let AO be a fixed spin connection, so that we can write the general spin connection as

A=Ao+a1, so identifying the space of spin connections with the space of imaginaryvalued 1-forms a, as before. We have already considered adding to CSD a function of the form

Tr(a)=i f aAP for an exact 2-form µ. Let us now relax the requirement slightly, and suppose only that p is closed. Take a collection µ1, ... , PN of closed 2-forms, which included a basis for HZ(Y), and let -r1, ... , Ttti be the corresponding functions

of a. As noted earlier, the functions TZ on S are invariant only under the identity component Ge c G. The map

(T1...... N):S-+ RN commutes with the G action, however, when G acts on R` through a discrete

action of the quotient, G/Ge = H'(Y; Z), by translations. So to obtain a G-invariant function on S, we should take a function

F:RN - R

4.

270


which is invariant under these translations, and define

f(a,f) = F(ri(a),...

7TLr(a)).

(Functions of this shape include the 'smoothed-out' holonomy maps used in (111 for non-abelian connections.) Our function f is not yet sufficiently general, for it does not depend on $. To incorporate c, we can proceed as follows. Let L be the Greens operator for the ordinary Laplacian on C'0(Y). Thus L inverts the Laplacian when restricted to functions with zero mean, and the kernel and cokernel of L are the constant functions. Let H C Ge be the subgroup

H={eitI :Y -IIR, with The quotient Ge/H is the circle, represented by the constant maps, and G/H is represented by the harmonic maps Y -4 S', a group isomorphic to S' x HI(Y) Z). For any fixed spinor E r(w), let o be the complex function on S defined by the hermitian inner product

ao(a, ) = f (0a, 4,), where 'Ua denotes the expression ,ba =

The definition of 'a is such that it transforms as'' does under the action of H: if u = ei£, where has mean zero, then = e-Ld'aeiLd'dt , = Upa. Thus a,;, : S -# C is invariant under H. Under the circle Ge/H, however, it transforms with weight 1.

(N K

Now choose a collection of spinors vi (i = 1.... K), and let of be the corresponding complex functions on S. We now have a collection of functions T1, ... , TN> al) ... , aK) S -+ 1R

X

27-1

These functions are equivariant for G when G .:H is Blade to act suitably on

R x Cl'. The quotient of R,' x Ci" by this action is Ch /Si over the base T" x

a hoist+lc

itli fii:>er

where hi is the Betti number of 1'. Now choose a smooth function F:

x lC'

which is invariant under the action of G /H. and define

f(a,P) = F(r1....;;.y.a...... ul, ). Now consider perturbing CSD by the addition of such a function f. With terms of this sort we can Ci-approximate, for example, any smooth function on a compact submanifold of C (lying in the locus where 4) is non-zero), and quite formally the class of functions is large enough to give the necessary transversality. If the partial derivatives of F are hounded, it seems that the compactness theorems for spaces of trajectories hold up too. Thus. in the crucial calculation (4), one finds amongst other things an additional cubic term in D, which is local in the t coordinate, but non-local on Y, involving an expression of the shape

But such a term does not break the argument.

5

Monopoles and contact structures

We now turn to the proof of Theorem 3.6. In view of Corollary 4.4. one might hope to prove this by showing that if Y had a taut foliation F. then one could always embed Y in a closed symplectic manifold (.V,:o) in such a way that ci(w) restricted to Y was ±e(.F). Perhaps this can be done: such a result would be very interesting, and presumably very hard. A slight modification of this tactic leads to a proof, however. Applying a theorem of Eliashberg and Thurston [8], we shall embed Y in an open symplectic manifold whose ends have a cone-like geometry. We shall then extend the 4-dimensional gauge-theory techniques to this setting.

272

5.


Using the theorem of Eliashberg and Thurston

The following material can be found in [8]. To begin, the following proposition sheds a geometric light on the meaning of taut. (The converse to the proposition is true also, but is rather deeper.)

Proposition 5.1. If Y is a taut foliation of Y by oriented leaves, then there is a closed 2-form S2 on Y whose restriction to the leaves is positive. Proof. Let y be a closed curve, transverse to the leaves and meeting every leaf. (The existence of -y was our chosen definition of taut.) A small tubular neighborhood N(y) meets the leaves of .F in a foliation by disks, to give a product structure. Using the product structure, pull back to N(y) a 2-form w supported in the interior of the disk and non-negative there. The result is a closed form Q(y) which is non-negative on the leaves of F. By pushing y along the leaves, one obtains transverse curves running through any point of Y. By taking a suitable finite collection of such curves yy and adding up the corresponding forms S2(yi), one obtains a suitable Q. Now let a be a non-vanishing 1-form on Y whose kernel at each point is the tangent plane to F and whose orientation is such that a A S2 is positive. The integrability of the tangents to the foliation means that a A da is zero. On the cylinder [-1,1] x Y, consider the closed 2-form

w=d(tAa)+Q. In w2, the only term to survive is the terms dt A a A S2, which is positive. So w is a symplectic form.

We have succeeded in embedding Y in a symplectic 4-manifold with boundary, [-1,1] x Y, and it is not hard to see that the first Chern class cl (w) restricts to e(.F) on the 3-manifold. But this elementary step is insufficient for our needs. A contact structure on a 3-manifold is a field of 2-planes 1;' which strictly

fails to be integrable at every point of Y. This means that, if 1; is defined locally as the kernel of a 1-form ,6, then a A df3 is nowhere zero. If the 3manifold and the 2-plane field are oriented (as will always be the case for us), then a suitable form f3 exists globally. Note however that, as a non-vanishing 3-form, the product f3 A d,0 itself determines an orientation of Y. We shall say that the contact structure is compatible with the orientation of Y if the form f3 A df3 is positive, for some and hence for all choices of 0 with kernel

Using the theorem of Eliashberg and Thurston

273

;. The theorem of Eliashberg and Thurston which we need sates that the tangent planes to a foliation` can be defor':tted to gye a cc:uacr compatible with either orientations:

Theorem 5.2 (Eliashberg Thurston [8j). Lei . be. a Ji

ootlc.

foliation of an oriented 3-manifold, other than the foliationt of S

,r:e led S=

spheres. Then the 2-plane field TY can be C" approt'1'Inutect by contact ..tra:ctures S compatible with either the given o'rientatio'n of I" or ;t6 oppo,-ite.

-

(An example of this phenomenon arises when Y is a circle bundle over a surface arising as a compact left quotient of SL(2. =y}. A left-invariant 2If - is plane field is determined by a 2-plane - in the Lie algebra sl 2. tangent to the null cone of the Killing form, then the corresponding 2-piane field is a foliation. If the Killing form is either definite or hyperbolic on -. then the 2-plane field is a foliation, compatible with one or other orientation of the 3-manifold.)

Now let us return to the manifold X _ [-1.11 x Y with the svrnplectic form w constructed above. The oriented boundary of X is ]' Y (the bar denotes the opposite orientation), and r foliates both components. Using the Theorem, one obtains foliations l;_ and f . on I' and F. compatible with their respective orientations, and at a small angle from the tangents to Y. If the angle is made small enough, we can arrange that these contact structures are compatible with w, in the weak sense that w is positive on the 2-planes at the boundary:

wj > 0

at a-t.

(17)

approximate. (To Indeed, w is positive on the tangent planes to F. which clarify the signs involved, this sort of compatibility holds for the standard. Kahler, symplectic form on a pseudo-convex domain in C2, such as a ball. when the boundary is given the contact structure defined by the complex tangent directions.) To summarize, starting from a 3-manifold Y with a foliation F, we have constructed a symplectic 4-manifold (X, w) in which Y is embedded, with K,,Iy equal to the euler class e(Y). The 4-manifold has a contact structure i; = _ U i;+ on its boundary (compatible with the boundary orientation, in a `convex' direction), and the symplectic form is compatible with s, in the sense described by (17).

274


5.

Four-manifolds with contact boundary

Although we have not embedded I' in a closed manifold, the convex contact structure on the boundary of X is all we shall need, because we can extend the monopole invariants SW(X, c) for closed manifolds so as to define similar invariants for 4-manifolds with contact boundary. We give an account of the construction from [21]. Let X be a compact, connected, oriented 4-manifold with non-empty boundary 9X, and let f be an oriented contact structure on 8X, compatible with the boundary orientation. In the presence of a metric, any oriented 2-plane field such as f determines a Spin` structure on 8X, and hence a 4-dimensional Spin` structure on a collar of the boundary. One can think of this in various ways. For example, define the spin bundle bl' on aX can be defined as the sum C®i;`, where the second summand means that the oriented 2-planes of t; are being regarded as complex lines: then define Clifford multiplication at a point y by picking a basis el, e2i e3 of tangent vectors at y, with er the positive normal to f and declaring that these act on (C ®t;°)y by the Pauli matrices

i (0

Oi)

'

(0

01)

,

(0i 0)

using the basis vector e2 to trivialize 1;. Alternatively, one can think of this as a special case of the way in which an almost-complex structure determines a Spin` structure in even dimensions. Note that the spin bundle comes with a canonical section 4% = (1, 0).

Now let c be any extension of ct to the interior of X. Given such an extension, we shall define a monopole invariant SW (X, t;, c) E Z,

which is a diffeomorphism invariant of the triple (no condition on b+(X) is needed).

The invariant SW (X, , c) is defined as follows. First we enlarge the manifold X by adding expanding cones to the boundary components. In more detail, if Y is a 3-manifold with a positive contact structure defined as the kernel of a 1-form 0, then there is a symplectic form on the cone (0, oo) x Y, given as w = d(f (t)a)

Four-manifolds with contact boundary

2

'J

Figure 2: The geometry of Z.

for any monotone increasing function f of t c (0. cc). To reproduce the way in which R4 with its standard symplectic structure arises from S3, we prefer to set w = (1/2)d(t23.).

We apply this standard construction to the components of 8X. and attach conical pieces [1, oo) x 8X to the boundary, to obtain an open 4-manifold Z (diffeomorphic to the interior of X):

Z = X U [1, oo) x 8X. On Z we choose a Riemannian metric h compatible with on the conical pieces. This means that there are local orthonormal coframes in which w can be expressed as el A e2 + e3 A e4. Figure 2 shows an illustration of Z in the case that X is the manifold [-1,11 x Y from the previous subsection. In this case, the contact structures c_ and T give symplectic forms w= on the two conical ends.

The symplectic structure w on the conical pieces determines a canonical Spin` structure c,, there, essentially the same as the Spin` structure determined by e on the boundary of X. The choice of c on X gives an extension

5.

276


of Comega to all of Z. The spin bundle W+ = W+ has a canonical section 4)o of unit length on the conical pieces of Z, and there is a unique spin connection there, with the property that DA04o = 0. (Such a connection is determined by any non-vanishing spinor on a 4-manifold.) We extend 4o and AO arbitrarily over the remained of Z. Motivated by the constructions of [36, 371, we now consider a modified version of the Seiberg-Witten monopole equations on the Riemannian manifold Z. The equations are

p(Fa) - {k

p(Ft DA4i=0.

f 4lo ®,Do} (18)

We can also consider, as before, perturbing these equations by the addition of a self-dual 2-form 71, which should decay on the ends of Z:

p(F +i7)) - {4)®4)'} =p(FAO) - {4)o®4o} D,+g4)=0.

The unperturbed equations are set up so that (Ao, 4)o) satisfies the equations on the conical ends, though not necessarily in the interior, since (Do may not satisfy the Dirac equation there. We seek solutions (A, OD) in general which are asymptotic to (AO, (Do) at infinity. It turns out that, even if only mild decay is required of A - AO and 4) - 4o, any solution of the unperturbed equations will approach the canonical pair exponentially fast after adjustment by a suitable gauge transformation. We again write M, (or M,,,, in the perturbed case) for the set of such solutions, considered up to the equivalence relation defined by the gauge transformations. Note that there can be no `reducible' solutions with 4) = 0, because 4) is required to approach the unit-length spinor 4o at infinity. The main facts about the moduli space are these:

Proposition 5.3 ([21)). The space MM,,1 is compact, and for generic rl it is a smooth manifold cut out transversely by the equations. In this case, the dimension of the moduli space is given by

d(c) =c2(l'ti', ,4)o)[Z,Z\X],

(which is the relative euler class of the bundle W+ on Z, relative to the non-vanishing section (1o on the ends).

Symplectic filling

(The formula for the dimension coincides with the alto""nzttivi

,,r r:u a

c2(l47+(XJ which we gave in the closed case.)

In the case d(c) = 0, we can again count the number of solutions. either with suitable signs, or just modulo 2, to obtain a definition of the invariant SW(X, l;, c). It is again independent of the choices trade. such as the 1-form ,13, the metric h, and the perturbation q. Symplectic filling

In the definition of SW (X, , c) as just described. no svntplectic form on X is involved and none is needed in the construction. When one has a sx n:piectic structure on X compatible with s at the boundary. then there is a nonvanishing theorem for the invariant. The symplectic form w deter:nines a

canonical Spin` structure c;, on X, and the compatibility condition with means in particular that c;, and c are the same at the boundary. so and invariant SW (X, , c,,) is defined. The theorem is then:

Theorem 5.4 ([21]). If X is a compact 4-manifold with contact structure on the boundary, and w is a symplectic form on X compatible with c- then SW (X, e, c

Once the analytic framework of the previous subsection is in place. the proof of this result is very much as the same as the proof of (1-0) from '371. There is one point to note, however. The proof begins by constructing a symplectic

form wz on Z, the manifold with conical ends. Although Z is a union of pieces each of which carries a symplectic form (as illustrated in Figure 2 for the case of [-1, 11 x Y), these forms do not necessarily agree at the joins, and even their cohomology classes may be different. Nevertheless, one can patch the forms together, in that one can find a symplectic form wz which is asymptotic to the conical form on the ends, and agrees with the given form w on X except in a small neighborhood of a?i. See [21], for example. The stretching argument used in Theorem 4.3 and Corollary 4.4 works just as well in the present setting of 4-manifolds with contact boundary. From the above theorem, we can therefore deduce: Corollary 5.5. Let (X, w) be a symplectic 4-manifold with a compatible contact structure a on the boundary. If Y is an oriented 3-manifold embedded G in X, then K,,Iy is a monopole class.

278

5.


using the construction of Eliashberg and Thurston, with X = [-1, 1] x Y, we deduce Theorem 3.6:

Corollary 5.6. If Y is a closed, irreducible, oriented 3-manifold with a smooth, taut foliation F by oriented leaves, then e(Y) is a monopole class.

0 (Again, the hypothesis of irreducibility excludes S' X S2.) Slightly more generally, one can apply Corollary 5.5 to the case that Y is parallel to 8X or a component of &Y. In Eliashberg's terminology, [8], a contact 3-manifold Y is symplectically fillable if Y arises as the correctly oriented boundary of a 4-manifold X carrying a compatible symplectic form w. If Y arises as a union of components of such a boundary, then it is symplectically semi-fillable. Corollary 5.7. If a contact 3-manifold (Y, 1;) is symplectically semifillable, is a monopole class. then

One useful feature of this corollary is one can form a connected sum of semifillable contact structures, and so obtain results about monopole classes on reducible 3-manifolds also.

All these statements can be sharpened a little, because one knows which Spin` structure is involved, not just the Chern class. More significantly perhaps, one should draw the stronger conclusion that the Floer homology is non-zero in these cases. For example, if (Y, £) is symplectically semifillable and et is not torsion, one would conclude that HF(Y, cf) is non-zero. (A more refined statement could be made in the case of a torsion class.) In particular then, one should say:

5.8. If Y carries a taut foliation, then the Seiberg-Witten Floer homology of Y is non-zero for the corresponding Spin` structure. The obstruction to proving such statements is no larger than the problem of verifying a suitable construction of Floer homology. Invariants of contact structures

There is a way to rephrase part of the construction just described. Given an oriented 3-manifold Y and a contact structure f compatible with the orientation, one can form a symplectic cone [1, oo) x Y with symplectic form w.. as before, and attach to it a cylinder (-oo, 1] x Y, as shown in Figure 3.

Invariants of contact structures

Figure 3: Defining an invariant of ( Y'. ;) .

Using the Spin` structure given by , one can then write down a version of the monopole equations on this 4-manifold which resemble the deformed equations (18) on the conical piece and resemble the usual equations (13) on cylindrical end, where they can be interpreted as the gradient flow equations for a trajectory of CSD. For each critical point a of the functional CSD on Y" (or of the perturbed functional CSDL considered before), one can look at the moduli space of solutions (A, (P) which are asymptotic to (.4o, 4)o) on the cone. and which descend from the critical point a on the cylinder. After perturbation, these moduli spaces are smooth manifolds. For each a, let n(a. ) be the number of solutions belonging to zero-dimensional moduli spaces, counted with signs as usual, and consider the expression E naea a

If we as an element of the chain group C which defines Floer homology. reducible solutions, then is not torsion, to eliminate the suppose that in other a (by arguments that are familiar the above sum is closed under

applications of Floer homology), and the resulting homology class

17, naea] E HF(Y,C) a

is an invariant of the contact 3-manifold.

(19)

280

6.

Potential applications

In this way, one can define an invariant of contact structures, once Floer homology is in hand. These seems the most natural setting in which to place the constructions of [28], where collections of contact structures are exhibited which are homotopic as 2-plane fields but not isotopic as contact structures. (The manifolds Y in [28] are homology spheres, so one needs to tackle Floer homology for the case that cl(c) is zero.) The non-vanishing result, Corollary 5.7, should be rephrased so as to say

that the invariant (19) of (Y, ) is a non-zero element of HF if the contact structure is semi-fillable.

6


When the Seiberg-Witten equations were introduced as an alternative gauge theory tool to replace the self-dual Yang-Mills equations exploited by Don-

aldson, an important link with topology was temporarily lost. The 3dimensional companions of Donaldson's Yang-Mills invariants are, as we have

mentioned, the instanton Floer homology I (Y) introduced in [11], and the Casson invariant A(Y) [1]. These play the roles of HF(Y, c) and SW(Y, c) from the monopole theory. Floer defined I(Y) for homology 3-spheres Y by studying the gradient-flow of the SU(2) Chern-Simons functional, whose critical points correspond to flat SU(2) connections, or representations of 7rl(Y) in SU(2). There is also a version I,,,(Y) for the case that Y is a homology S' x S2, where the chain group is built from flat SO(3) connections with non-zero Stiefel-Whitney class [3]. Defining I(Y) in other situations presents technical difficulties related to reducible connections. In the definition of I(Y), there is an immediate connection with the fundamental group. Thus, for example, Floer's instanton homology vanishes if Y is a homotopy sphere (Floer's definition did not use the trivial representation of 7rl). No such statement can be made very easily for the monopole Floer homology (our tentatively defined HF(Y, c)). On the other hand, in the Seiberg-Witten version, we have a handle on anon-vanishing result: irreducible 3-manifolds with bl 0 0 admit taut foliar tions by Gabal's results, and we have argued that a foliation forces HF(Y, ) to be non-zero. If one could establish even a weak relationship between the monopole HF(Y, c) and instanton Floer homology, then there would be a useful payoff.

Surgery and property 'P'

281

Surgery and property 'P'

To elaborate on the last remark, the application we have in mind is in line with Casson's application of the invariant \(Y) to the question of 'property F. This is the question of whether one can make a simply connected 3manifold by non-trivial surgery on a non-trivial knot K. ((If one cannot. then K is said to have property P.) Work of Gordon an Luecke '18, showed that one cannot make S3 this way. So if one did manufacture a homotopy-sphere by surgery on a knot, it would be a fake 3-sphere (a counterexample to the Poincare conjecture). The basic example to consider is +1 surgery, in which a neighborhood N(K) is removed and sewn back in so that the meridian on N(K) is attached to a curve in the class of the meridian plus longitude. Let us call this manifold Y1, or Y, (K). We also have the manifold ho obtained by zero-surgery, which is a homology St x S2. To prove property P, we would like to know that Y; has

non-trivial fundamental group, and to this end we could seek to show that 7rl(Yr) had non-trivial representations in SU(2). or that I(I i) is non-zero. There is a powerful tool at hand, in the following theorem of Floer. (This is a special case of his èxact triangle' in instanton homology [311).

Theorem 6.1 (Floer, [3]). In the above situation the instanton Floer homology I(Y1) of the homology sphere Yi is isomorphic to the instanton Floer D homology II(Yo) of the homology S' x S2 obtained by zero-surgery.

In this form, the theorem is already hard to prove. But the special case we are interested is actually very easy to prove: Corollary 6.2. If the instanton Floer homology 1,,,(Yo) is non-trivial, then 0 Yt is not a homotopy 3-sphere. (In fact, if Yt is a homotopy 3-sphere, then one can easily use the holonomy perturbations introduced by Floer to deform the equations FA = 0 on Yo so that they have no solutions for SO(3) connections A on Y with non-zero W2.) Suppose now that we could prove:

Conjecture 6.3. If the monopole Floer homology HF(Y; c) of a homology St X S2 is non-trivial for some Spine structure with ct(c) not torsion, then the instanton Floer homology I,,(Y) is non-trivial also.

Then we would be home, at least in the case that the genus of K is 2 or more. Indeed, we have learned from the foliation theory that in this case, Yo

282

6.


admits a smooth, taut foliation having a genus 2 surface has a compact leaf. The euler class of this foliation is non-trivial, and modulo the verification of the definitions, the monopole Floer homology of Yo for the corresponding Spin` structure has been shown to be non-zero. The conjecture would imply that instanton Floer homology is non-trivial also, and it would follow that Yl was not a homotopy sphere, by Floer's result. (In the case of genus 1, there are two additional difficulties. The first is that Gabai's foliation may not be smooth. The second is that the relevant Spin` structure on Yo has cl = 0, so one must treat reducibles with respect. The second point is moot, perhaps, because one would need to consider all Spin` structures together, most likely, to prove the conjecture above.) Note that, without using any connection between the instanton and monopole Floer homologies, one could try and reprove the theorem of Cordon and Luecke by establishing an exact triangle for the monopole Floer homology.

The Piidstrigatch-Tyurin program

Conjecture 6.3 does not stand unsupported. Indeed, there is evidence for a closer relationship. In introducing the monopole invariants of 4-manifolds, Witten [42] conjectured a very specific relationship between the monopole invariants and the older instanton invariants, for a large class of 4-manifolds. (This conjecture is extended in [32].) A mathematical approach to proving Witten's conjecture was proposed by Pidstrigatch and Tyurin [33], and although their program does raise some technical challenges at the time of writing, it seems most likely that a proof will be along the lines suggested. There is work in this direction in [9]. The proposed method is to study a larger moduli space of solutions to the PU(2) monopole equations. These equations have the same shape as the equations (13) but involve a non-abelian connection A. The equations contain both the instanton equations and the usual monopole equations. If one pursues this line in three dimensions instead of four, one can arrive at an elegant proof that the 3-manifold invariants we have called SW (Y, c) are related to the òdd' Casson invariant. For example, if Y is obtained by zero surgery on a knot K, there is a Casson-type invariant .(Y) which is the euler characteristic of the instanton Floer homology I,,(Y), and without encountering any of the technical difficulties of the 4-dimensional case, one

283

can prove that

A(Y) _

SW(l : c). Y

Of course, one can verify this relationship externally. because both sides can be reduced to the Alexander invariant. (In the case of the left-hand side.

this is due to Casson.) But the ìnternal' proof using the PU(2) equations seems to show that the relationship extends beyond the euler characteristics. It may well be that one can relate the Floer homologies using this approach.

7

Surfaces in 4-manifolds

Where the theory succeeds We turn now to the question of representing a 2-dimensional homology class a in a smooth, oriented 4-manifold X by a smoothly embedded. oriented surface E of minimal complexity. There are many statements here that can be made

to look very much like their 3-manifold counterparts, particularly when E has trivial normal bundle. (This condition implies that the self-intersection number o or is zero, but the converse is false, because we still allow that E may be disconnected.) For example, if o o- is zero; then PD;v) E H2(X) is the pull-back of the generator of H2(S2) by some map f : X -+ S2. and one can find a representative surface E as the inverse image of a regular value of f , so establishing that representative surface exists. Every embedded surface with trivial normal bundle arises this way, just as a surface in a 3-manifold Y arises from a map to S'. We have already introduced the 4-dimensional monopole equations and the basic classes, at least in the case that b+(X) > 1. For surfaces with trivial normal bundle, the basic classes provide a lower bound for the genus, just as in dimension three (compare Proposition 3.1):

Proposition 7.1. Let X be a smooth, oriented, closed 4-manifold with b+(X) > 1, let a be a basic class and E an embedded surface representing a class a. Suppose that the normal bundle of E is trivial. Then

x-(-r) > (a,o'). Proof. Being a basic class means for a that there is a Spin structure c with ct (c) = a for which the invariant SW (X, c) is non-zero. All we need, however,

7.

284


is that the moduli space Af, of solutions to the equations (13) is non-empty, for every choice of Riemannian metric on X. Just as in the 3-dimensional case (see (5)) ; if (.4. (D) is a solution for a given metric, then the Weitzenbock formula leads to

f

f s2dvol. x

In dimension 4, the first of the two Seiberg-Witten equations leads to the relationship IFf 12 = (1/8)14> 14, so there is a relationship between the L2 norms, IISI12/8.

(20)

Now let IIaII stand for the L2 norm of the harmonic representative, as before. We have IIail < (1/21r)IIFAII

On the other hand, alpha is the orthogonal sum of its self-dual and anti-self dual parts, and

IIa+II2 -IIa-II2=(a_a)[XJ, so from the inequality (20) we obtain the bound Ilallh < IIShIIh/(4r)2 + (1/2)a2[X],

in which we have once again adjusted our notation to indicate the dependence on a Riemannian metric h. We can write IIaikh

(a, a) One can pass from the proposition above to a statement about surfaces with non-negative normal bundle (that is, surfaces E such that each component has non-negative self-intersection number):

Proposition 7.2. Let X be a smooth, oriented, closed 4-manifold with b+(X) > 1, let a be a basic class and E an embedded surface representing a class o. Suppose that the normal bundle of E is non-negative. Then X_(E) > o o + (a, o). Proof. The statement can be deduced from the case of trivial normal bundle by using the `blow-up formula' for the 4-dimensional monopole invariants. It is enough once more to consider the case that E is connected. Let k = o or #kC?2 and et Xk = X be the connected sum of X and k copies of C2 with reversed orientation. Put Cek = a + el +

+ ek,

(22)

where ei is the generator of H2 in the ith copy of l 2. The blow-up formula ([19], Proposition 2) says that ak is a basic class on Xk if a is a basic class on X, because the monopole invariants of corresponding Spin` structures are equal. Let t be the embedded surface in Xk formed by an internal connected sum of E with the spheres representing the generators of homology in the Cp2

summands. (We orient these spheres so that the class ei evaluates as +1 on the ith sphere, and we form the sum respecting these orientations.)

286

7.


The surface ` has trivia! normal bundle and the same genus as E. So the previous proposition gives

(ak, [S'])

= k + (a, a) which is the desired result.

Remarks. In some applications of this proposition, a proof of the blow-up formula is not needed. One may know that a is a basic class by an application of Taubes' result, Theorem 15, when a arises as K, for some symplectic form, in which case one knows that ak is a basic class also, because there is a symplectic structure on X. with this class as its canonical class. Without using the blow-up formula, one can therefore deduce

X_ (E) > a a + (K,,, o).

(23)

This inequality is an equality for symplectic submanifolds, or smooth algebraic curves in a complex surface, where it is usually referred to as the adjunction formula. The inequality in the proposition is often called the 'adjunction inequality'. Unlike the 3-dimensional version of this statement, which led only to such basic facts as the lower bound for the genus of a knot in terms of the degree of its Alexander polynomial, the 4-dimensional version provides information which we can reach in no other way at present. Even with the 4-torus, where the Spin° structure c with cl = 0 has a monopole invariant SW (T4, c) = 1, we learn that embedded surfaces in T4 satisfy

which is a significant result. For example, it leads to a proof of Milnor's conjecture [31J on the unknotting number of torus knots [22]. (The absolute value appears on the right-hand side because T4 looks the same with either orientation.) By contrast, the 3-dimensional result, applied to the 3-torus, is entirely contentless. The result for T4 above is sharp: every class a can be represented by a surface of complexity More generally, if X carries a symplectic form a,

Where the theory hesitates

then Proposition 7.2 is sharp for at !east a signlfecar;t ran;. of cia se :s

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