Lower K- and L-theory (London Mathematical Society Lecture Note Series 178)

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London Mathematical Society Lecture Note Series, 178

Lower K- and L-Theory Andrew Ranicki Reader in Mathematics, Unversity of Edinburgh

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www. Cambridge. org Information on this title: www.cambridge.org/9780521438018 © Cambridge University Press 1992 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1992 A catalogue recordfor this publication is available from the British Library ISBN-13 978-0-521-43801-8 paperback ISBN-10 0-521-43801-2 paperback Transferred to digital printing 2006

For Gerda

Contents Introduction §1. Projective class and torsion §2. Graded and bounded categories §3. End invariants §4. Excision and transversality in A"-theory §5. Isomorphism torsion §6. Open cones §7. tf-theoryof Ci(A) §8. The Laurent polynomial extension category Afz,^"1] §9. Nilpotent class §10. if-theory of A^,*" 1 ] §11. Lower if-theory §12. Transfer in Ji-theory §13. Quadratic L-theory §14. Excision and transversality in L-theory §15. L-theory of Ci(A) §16. L-theory of Fk[z,z~l] §17. Lower L-theory §18. Transfer in X-theory §19. Symmetric L-theory §20. The algebraic fibering obstruction References Index

1 8 12 18 28 46 53 63 69 81 86 101 106 112 119 131 138 145 149 153 156 167 172

INTRODUCTION

1

Introduction The algebraic if-groups K*(A) and the algebraic L-groups L*(A) are the obstruction groups to the existence and uniqueness of geometric structures in homotopy theory, via Whitehead torsion and the Wall finiteness and surgery obstructions. In the topological applications the ground ring A is the group ring Z[?r] of the fundamental group ?r. For Ktheory a geometric structure is a finite CW complex, while for L-theory it is a compact manifold. The lower K- and L-groups are the obstruction groups to imposing such a geometric structure after stabilization by forming a product with the z'-fold torus T = S1 x S1 x . . . x S1 , arising algebraically as the codimension i summands of the K- and Lgroups of the z-fold Laurent polynomial extension of A

AWT)]

=

A[zu(z1)-1,z2,(z2)-\...,zi,(zi)-1).

The object of this text is to provide a unified algebraic framework for lower K- and L-theory using chain complexes, leading to new computations in algebra and to further applications in topology. The 'fundamental theorem of algebraic A"-theory' of Bass [7] relates the torsion group K\ of the Laurent polynomial extension A[z,z~x\ of a ring A to the projective class group KQ of A by a naturally split exact sequence 0 —» Ki(A) — • K!(A[z]) © K^Alz-1}) —> KtiAfaz-1]) — • K0(A) — • 0 . The lower iiT-groups K-i(A) of [7] were defined inductively for i > 1 to fit into natural split exact sequences 1 0 —> K-i

i+Mfo])

i() —+ 0 ,

generalizing the case i — 0. The quadratic L-groups of polynomial extensions were first studied by Wall [84], Shaneson [72], Novikov [48] and Ranicki [57]. The free quadratic i-groups L\ of the Laurent polynomial extension A[2, 2"1] (z = z~x) of a ring with involution A were related in [57] to the projective quadratic L-groups L* of A by natural direct sum decompositions The lower quadratic L-groups L*'(A) of [57] were defined inductively for i > 1 to fit into natural direct sum decompositions

2

LOWER K- AND L-THEORY

with LI = L*, generalizing the case i = 0 with L* = L*. An algebraic theory unifying the torsion of Whitehead [88], the finiteness obstruction of Wall [83] and the surgery obstruction of Wall [84] was developed in Ranicki [60] —[69] using chain complexes in any additive category A. This approach is used here in the K- and L-theory of polynomial extensions and the lower K- and L-groups. Chain complexes offer the usual advantage of a direct passage from topology to algebra, avoiding preliminary surgery below the middle dimension. A particular feature of the exposition is the insistence on relating the geometric transversality properties of manifolds to the algebraic transversality properties of chain complexes. The computation Wh{V) = 0 (i > 1) of Bass, Heller and Swan [8] gives the lower K- and L-groups of Z, which are used (more or less explicitly) in Novikov's proof of the topological invariance of the rational Pontrjagin classes, the work of Kirby and Siebenmann on high-dimensional topological manifolds, Chapman's proof of the topological invariance of Whitehead torsion and West's proof that compact ANRs have the homotopy type of finite CW complexes. A systematic treatment of homeomorphisms of compact manifolds requires the study of non-compact manifolds, using the controlled algebraic topology of spaces initiated by Chapman, Ferry and Quinn. In this theory topological spaces are equipped with maps to a metric space X, and the notions of maps, homotopy, cell exchange, surgery etc. are required to be small when measured in X. The original simple homotopy theory of Whitehead detects if a PL map is close to being a PL homeomorphism. The controlled simple homotopy theory detects if a continuous map is close to being a homeomorphism, by considering the size of the point inverses. After an initial lull, the lower K- and L-groups have found many applications in the controlled and bounded topology of non-compact manifolds, stratified spaces and group actions on manifolds. The following alphabetic list of references is representative: Anderson and Hsiang [3], Anderson and Munkholm [4], Anderson and Pedersen [5], Bryant and Pacheco [13], Carlsson [16], Chapman [18], Farrell and Jones [25], Ferry and Pedersen [28], Hambleton and Madsen [30], Hambleton and Pedersen [31], Hughes [35], Hughes and Ranicki [36], Lashof and Rothenberg [42], Madsen and Rothenberg [45], Pedersen [51], Pedersen and Weibel [53], [54], Quinn [56], Ranicki and Yamasaki [70], [71], Siebenmann [75], Svennson [80], Vogell [81], Weinberger [86], Weiss and Williams [87], Yamasaki [89]. Karoubi [38] and Farrell and Wagoner [26] were motivated by the

INTRODUCTION

3

proof of Bott periodicity using operators on Hilbert space and by the simple homotopy theory of infinite complexes (respectively) to describe the lower If-groups of a ring A as the ordinary if-groups of rings of infinite matrices which are locally finite K-i(A) = KoiS'A) = K!(Si+1A) (t > 0) , with 5* A the i-fold suspension ring: the suspension SA is the ring defined by the quotient of the ring of locally finite countable matrices with entries in A by the ideal of globally finite matrices. Gersten [29] used the ring suspension to define a non-connective spectrum K(A) with homotopy groups *,-(K(A)) = Ki(A) ( i e Z ) . The applications of the lower K- and L-groups to manifolds generalize the end invariant of Siebenmann [73], which interprets the Wall finiteness obstruction [W] G KQ(Z[7T]) of an open n-dimensional manifold W with one tame end as the obstruction to closing the end, assuming that n > 6 and that IT = TTI(W) is also the fundamental group of the end. The following conditions on W are equivalent: (i) [Wq = O€^o(Z[7r]), (ii) W is homotopy equivalent to a finite CW complex, (iii) W is homeomorphic to the interior of a closed n-dimensional man_ _ ifold M, (iv) the cellular chain complex C(W) of the universal cover W of W is chain equivalent to a finite f.g. free Z[?r]-module chain complex. The product W x S1 has end invariant [W x S1] = 0 e K0(Z[TT x Z]) ,

so that W x S1 is homeomorphic to the interior of a closed (n + 1)dimensional manifold N. However, if [W] / O G KO(1[7T}) then N is not of the form M x S1 for a closed n-dimensional manifold M with interior homeomorphic to W. Motivated by controlled and the closely related bounded topology, Pedersen [49], [50] expressed the lower if-group K-i(A) (i > 0) of a ring A both as the class group of the idempotent completion Pj(A) of the additive category Cj(A) of Z*-graded A-modules which are f.g. free in each grading, with bounded morphisms, and as the torsion group of =

K0(Pi(A))

= Kiid+^A))

(t>0).

These if-theory identifications are obtained here by direct chain complex constructions, and extended to corresponding identifications of the

4

LOWER K- AND L-THEORY

lower L-groups of a ring with involution A as the L-groups of additive categories with involution L 0) . For i = 0 this is an unpublished result of Faxrell and Wagoner (cf. Wall [84,p.251]). The open cone of a subspace X C Sk is the metric space

O(X) = {txeRk^\te

[O,OO),ZEX}

.

Open cones are especially important in the topological applications of bounded K- and L-theory, because (roughly speaking) the controlled algebraic topology of X is the bounded algebraic topology of O(X). Given a filtered additive category A and a metric space X let Cx(A) be the filtered additive category of X-graded objects in A and bounded morphisms defined by Pedersen and Weibel [53], and let Px(A) be the idempotent completion of Cx(A). Let Po(A) denote the idempotent completion of A itself, and let K(A) be the non-connective algebraic if-theory spectrum of A, with homotopy groups 7Ti(K(A)) = Ki(P0{f\))

(i e Z)

= Ki(f\) (i ? 0) . The main result of Pedersen and Weibel [54] shows that the algebraic 7\"-theory assembly map is an isomorphism for X = O(Y) an open cone on a compact polyhedron 7 C S * , s o that A'*(PO(Y)(A)) = HlJ(O(Y);K(f\))

= i f ^ F ; K(A)) .

In particular, the algebraic A"-groups A'*(Po(y)(A)) of the open cone of a union Y = Y^ \J Y~ C Sk of compact polyhedra fit into a MayerVietoris exact sequence

—* A^(P o( y +) (A))©A',(Po(y-)(A)) A-,(PO(Y)(A)) -L Ki AVi(Po(y-)(A)) Carlsson [16] extended the methods of [54] to metric spaces other than open cones, obtaining a Mayer-Vietoris exact sequence for the algebraic

INTRODUCTION

K-groups i*T*(Px(A)) of a union X = X+ U X~ of arbitrary metric spaces —» Ki(Px+((\))®Ki(Px-(l\))

with = { x G l |d(x,z + ),d(z,ar) < 6 for some x+ € X + , z~ G X~} the intersection of the 6-neighbourhoods of X + and X~ in X. This exact sequence will be obtained in §4 for i = 1 using elementary chain complex methods, with the connecting map 8 : ^(CjtfA)) —> limtfo(P M ( x + ) x-,x)(A)) 6

defined by sending the torsion r(E) of a contractible finite chain complex E in Cx(A) to the projective class [E+] of a Qv;(x+,x-,X)(A)-finitely dominated subcomplex E* C J5, corresponding to the end obstruction of Siebenmann [73], [75] of the part of E lying over a b-neighbourhood of X + in X. In the special case X = I

+

u r

= R , X+ = [0,oo) , X- = (-oo,0]

the algebraic if-groups are such that tf.(CB±(A))

= *u(P R± (A)) = 0 , ff,(CR(A)) = ^ ( C K A ) ) .

The connecting map in this case is the isomorphism of Pedersen and Weibel [53] 0 : tfi(C!(A)) £ Ko(Po(A)) . The algebraic properties of modules and quadratic forms over a Laurent polynomial extension ring A[z, z~l) are best studied using algebraic transversality techniques which mimic the geometric transversality technique for the construction of fundamental domains of infinite cyclic covers of compact manifolds. The linearization trick of Higman [34] was the first such algebraic transversality result, leading to the method of Mayer-Vietoris presentations developed by Waldhausen [80]. In §10 this method is used to obtain a split exact sequence 1 0 —> —> /^(Ab.z- 1 ]) —» Jfo(Po(A)) —» 0 for any filtered additive category A. The split projection K\(f\[z,z~~l\)

6

LOWER K- AND L-THEORY

>ifo(Po(A)) is induced by the embedding of A^,^" 1 ] in Ci(A) as a subcategory with homogenously Z-graded objects. Let Pi(A) denote the idempotent completion Po(Cj(A)) of the bounded Z'-graded category Cj(A). The above sequence is used in §11 to recover the expression due to Pedersen and Weibel [53] of the lower if-groups of A as

K-i(f\) = K0(Pi(f\)) = Kiid+iW)

(i > 1) ,

using polynomial extensions (in the spirit of Bass [7]) instead of the delooping machinery. The quadratic L-groups L*(A) are defined in Ranicki [68] for any additive category A with an involution, as the cobordism groups of quadratic Poincare complexes in A. The intermediate quadratic L-groups £*(A) are defined for a *-invariant subgroup J C KQ(F\) to be the cobordism groups of quadratic Poincare complexes in A with the projective class required to belong to J. The algebraic transversality method is applied in §14 to obtain a Mayer-Vietoris exact sequence for the quadratic Lgroups L*(Cx(A)) of a union X = X + U X~ ... —f UmZ*(P M ( J C + i J C - i X ) (A)) — > L n ( C x + ( A ) ) 0 l n ( C x - ( A ) ) b

with Jb

-

In particular, the quadratic i-groups i»(Co(y)(A)) of the open cone of a union Y — Y+ U Y~ C Sk fit into a Mayer-Vietoris exact sequence ft • £n(C O (y+)(A))eL B (C O (y-)(A))

—^ I n (C o(y) (A)) - i » ^ with J = ker(A'o(Po(y + ny-)(A))-^A'o(Po(y+)(A))©A'o(Po(y-)(A))). In the important special case Y±

= {±1} , Y = F + U Y~ = S°

the open cones are OiY*)

= R* , 0(Y)

= R

and X»(CR±(A)) = 0 , L.(CR(A)) = L.(Ca(A)), J = K0(P0(H)) , so that the connecting maps define isomorphisms ^ L._i(Po(A)) .

INTRODUCTION

In §17 the lower L-groups of a filtered additive category category A are defined inductively by

= £.(Po(A)) ,

The isomorphisms 9 are used to obtain the expressions £(A) = Ln+i(Pi(f\)) = i n + i + 1 (C i + 1 (A)) ( n > 0 , i > l ) . The lower L-groups of a ring with involution A are the lower L-groups of the additive category with involution of based f.g. free A-modules. The ultimate lower quadratic L-groups of A are defined by L(A) = lim 4 - > ( A ) (n € Z) , i

and as in Ranicki [69] there is defined an algebraic L-theory spectrum with homotopy groups The Mayer-Vietoris exact sequence of §14 shows that the algebraic Ltheory assembly map of [69, Appendix C] is an isomorphism for X = 0(Y) the open cone of a compact polyhedron Y C Sk, so that

In §20 the chain complex methods are used to provide an abstract treatment of the obstruction theory of Farrell [21], [22] and Siebenmann [76] for fibering a manifold over the circle S 1 . In Ranicki and Yamasaki [70] the lower if-theory algebra is applied to the controlled topology of Chapman-Ferry-Quinn, obtaining systematic proofs of the results of Chapman and West on the topological invariance of Whitehead torsion and the homotopy finiteness of compact ANRs. The bounded surgery theory of Ferry and Pedersen [28] is the topological context for which the lower L-theory algebra presented here is most directly suited. However, in Ranicki and Yamasaki [71] the algebra will be applied to the controlled surgery theory of Quinn [56] and Yamasaki [89]. An earlier version of this text was issued as Heft 25 (1990) of the Mathematica Gottingensis. I should like to take this opportunity of thanking the Sonderforschungsbereich 'Geometrie und Analysis' in Gottingen for its manifold hospitality on various occasions since its inception in 1984.

8

LOWER K- AND L-THEORY

§1. Projective class and torsion This section is a brief recollection from Ranicki [64], [65] of the algebraic theory of finiteness obstruction and torsion in an additive category A. Give A the split exact structure: a sequence in A 0

>A

*

>B

i

is exact if there exists a morphism A: : C (i) jk = 1 : C —* C , (ii) (i k) : A ® C

>C

>0

>B such that

> B is an isomorphism in A.

The class group A"o(A) is the abelian group with one generator [A] for each object A in A, subject to the relations (i) [A] = [A1] if A is isomorphic to A', (ii) [A © B] = [A] + [B] for any objects A, B in A. A chain complex C in A is n-dimensional if Cr = 0 for r < 0 and r >n d d C : ... > 0 • Cn —> Cn-i • . . . — > C\ > Co . A chain complex C in A is finite if it is n-dimensional for some n > 0. The class of a finite chain complex C in A is the chain homotopy invariant defined by

[C] = E ( - ) r [ C r ] € A'o(A) . r=0

The k-fold suspension of a chain complex C is the chain complex SkC defined for any k £ Z by a dimension shift — k dSkc

= dc : (5 C ) r = Cr_fc —> (5 C ) r - i = Cr-fc-i .

If C is n-dimensional and n + k > 0 thefc-foldsuspension 5feC is (n + fc)dimensional, with class

[SkC] = ( - ) * [ q G tfo(A) . The idempotent completion Po(A) of A is the additive category with objects (A,p) defined by the objects A of A together with a projection p = p2 : A >A A morphism / : (A,p) >(B,q) in Po(A) is a morphism / : A >B in A such that qfp = f : A >B. The full embedding

A—+P0(A);

A—>(A,1)

will be used to identify A with a subcategory of Po(A). The reduced class group of Po(A) is defined by A'o(Po(A)) =

1. PROJECTIVE CLASS AND TORSION

9

Given a ring A let B-^(A) be the additive category of based f.g. free A-modules. The idempotent completion Po(B^(A)) is isomorphic to the additive category P(A) of f.g. projective A-modules, and K0(P0(Bf(A))) f

Ko(Po(B (A)))

= K0(P(A))

= KQ(A) ,

= K0(A) .

Let (B,A C B) be a pair of additive categories, with A full in B. A chain complex in B is homotopy A-finite if it is chain equivalent to a finite chain complex in A. An f\-finite domination (D, / , g, h) of a chain complex C in B is a finite chain complex D in A together with chain maps / :C >D, g : D >C and a chain homotopy h : gf ~ 1 : C >C. The projective class of an A-finitely dominated chain complex C in B is the class of any finite chain complex (D,p) in Po(A) which is chain equivalent to (C, 1) in Po(B)

[C] = [D,p] € AVPo(A)) . See Ranicki [64] for an explicit construction of such a (D,p) from an A-finite domination of C. The reduced projective class is such that [C] = 0 € K0(P0(f\)) if and only if C is homotopy A-finite. A chain homotopy projection (D,p) is a chain complex D together with a chain map p : D >D such that there exists a chain homotopy p~p2

: D

> D.

A splitting (C,f,g) of (D,p) is a chain complex C together with chain maps / : C >D, g : D >C such that gf ~ 1 : C >C, f g ~ p : D >D. The projective class of a chain homotopy projection (D,p) with D a finite chain complex in A was defined in Luck and Ranicki [44] by

[D,p] = [C] € tfo(Po(A)) for any splitting (C, / , g) of (D,p) in Po(A). See [44] for an explicit construction of an object (DutPu) in Po(A) such that [D,p]

= [Du,Pu]

- [Dodd] e #o(Po(A)) ,

with Deven = A) 0 D2 © D4 © . . . , Dodd

= Dt © D3 © D5 © . . . ,

Du = Deven

© Dodd

= Do © Di © D2 © . . . .

The reduced projective class is such that [-D,p] = 0 G ifo(Po(A)) if and only if (D,p) has a splitting (C, / , g) with C a finite chain complex in A.

10

LOWER K- AND L-THEORY

A finite chain complex C in A is round if

[C] = o e

K0(f\).

The projective class of an A-finitely dominated chain complex C is such that [C] = 0 G Ko(Po(f\)) if and only if C is chain equivalent to a round finite chain complex in A. The reduced projective class is such that [C] = 0 G K o (Po(A)) if and only if C is homotopy A-finite. PROPOSITION 1.1 If in an exact sequence of chain complexes in B 0

> C

> D

> E

> 0

any two ofC,D,E are Fk-finitely dominated then so is the third, and the projective classes are related by the sum formula [C] - [D] + [E] = 0 G tfo(Po(A)) . D

The torsion group of an additive category A is the abelian group K\(f\) with one generator r ( / ) for each automorphism / : M *Af, subject to the relations (i) r(gf : M >M >M) = r(f : M >M) + r(g : M >M) , (ii) r^fi

:L

>M

>M

(iii) r(f © / ' : M © M'

>L) = r(f : M

>M) ,

>M © M')

=

r(/ :

>M) + r(/ ; : M'

M

>M') .

A stable isomorphism [f] : L >M between objects in A is an equivalence class of isomorphisms / : L © X >M © X in A, under the equivalence relation (f:L®X

>M®X) - (f':L®X'

>M © X')

if the automorphism a = (/'1 has torsion r(a) = 0 G The composite of stable isomorphisms [/]

: L

— M , [y] : M —» N

represented by / : L@X

v M®X

, g : M ©F

is the stable isomorphism [gf] : L

>N represented by the composite

( N @Y

>N®X®Y.

>L has a well-defined torsion

= r(f:L®X

1. PROJECTIVE CLASS AND TORSION

11

with The stable automorphism group of any object M in A is isomorphic to An additive functor F : A phisms of algebraic K-groups

>B of additive categories induces mor-

Fo : i^o(A) —> K0(B) ; [M] —> [F(M)] , Fj : ^ (A) — ff,(B) ; r(g : M — M ) —+ r(F(g): F(M)—»F(M)) . The relative K\ -group K\ (F) is the abelian group of equivalence classes [M, JV, ^f] of triples (M, TV, ^) consisting of objects M, JV in A and a stable isomorphism [g] : F(M) >F(N) in B, subject to the equivalence relation (M, iV, ^f) ~ (Mf ,N',g') if there exists a stable isomorphism [h]:M®

N1

>M' 0 JV in A such that

TWH^M'1

- F(M

® M')

>F(M ® M')) = 0 e A^(B) ,

with addition by [M1,N1,g1] + [M2,N2,g2]

=

[M1@M2,N1®N

The relative K\ -group fits into an exact sequence A'i(A) —U U Ki(B) K(B) —> Ki(F) K(F) —> K K(l\) 0(l\) - ^^ K0(B) with Kl(B)

— . Ki(F) ; r(g : M^M)

— • [0,0,5] ,

a : Ki(F) —» A'o(A) ; [M,iV,(,] — The full embedding A

>P0(A) ; M

[M] - [iV] .

>(M,1)

induces an isomorphism of torsion groups ifx(A) —> ifi(Po(A)) ; T ( / : M— + M) —* r ( / : (M,l) which is used to identify The following additivity property of torsion will be used in §4. LEMMA 1.4 Let f : M >M be an automorphism in A. Given a resolution of M by an exact sequence in A 0

> K - ^

L - ^

M

>0

12

LOWER K- AND L-THEORY

and a resolution of f by an isomorphism of exact sequences in A .0

0

0©j

•B®K

the stable isomorphisms [g] : A

>B, [h] : A

1

r ( / ) = r([/i]~ [^f] : A

>B are such that

>A) £ ifi(A) . 1

PROOF The stable automorphism [ft]" ^] : A the isomorphism

>A is represented by

/ L /T\ 1 _ \ (n /^T\ 1 r , \ . A /T\ Tf /T\ 7" v D /T\ 1^" /T\ 7" I 11> KXs •*• L/J \y M7 -*• l\. ) • •**• vi' •**• ^37 •*-' ' •*-' vi/ •**• \ I / -*-/

Choosing a splitting morphism fc : M defined an isomorphism s = (ik) : K®M

v

/I /T\ If'" /T\ 7" ' •**• vl/ •**• vI7 •*-' •

>L for ^ : L

>M there is

> L

such that M for some morphism e : M

K. Thus

: (A © K) 0 M

.,(ffeljr)(ael

> (A © K) © M ,

e

}=

/J "7 o^ lo

o

i

: (A@K)@M®K—>(A®K)@M@K

,

and

§2, Graded and bounded categories The bounded X-graded category Cx(A) of Pedersen and Weibel [53] is defined for a metric space X and a filtered additive category A, as a subcategory of the following (unbounded) X-graded category. DEFINITION 2.1 Given a set X and an additive category A define the X-graded category Gx(A) to be the additive category in which an object

2. GRADED AND BOUNDED CATEGORIES

13

is a collection {M(x) \x £ X} of objects in A indexed by X, written as a direct sum

M = J^ M(x) , xex and a morphism / = {/(y,*)} : L = is a collection {/(y,x) : L(x) >M(y)\x,y £ X} of morphisms in A such that for each x £ X the set {y £ X \f(y,x) ^ 0} is finite. It is convenient to regard / as a matrix with one column {/(y,x) \y £ X} for each x £ X (containing only a finite number of non-zero entries) and one row {/(y, #) \x £ X} for each y £ X. The composite of morphisms / :L >M, g : M >N in Gx(A) is the morphism gf : L >N defined by

where the sum is actually finite. Note that if X is a finite set the functor GX(A) —•> A ; M —> J ] M(x) is an equivalence of additive categories. EXAMPLE 2.2 Let A = Bf(A) be the additive category of based f.g. free A-modules for a ring A. For any set X write An object in Gx(A) is a based free A-module which is a direct sum

M = J^M(x) xex of based f.g. free A-modules M(x) (x £ X). The morphisms in Gx(A) are the A-module morphisms. D

An additive category A is filtered if to each morphism / : L >M there is assigned a filtration degree 8(f), a non-negative integer such that (i) Kf + 9)< max(S(/), 6(g)) for any / , g : L >M, (ii) % / ) < «(/) + % ) for any / : L^M, g : M^iV, (iii) S(-l : X >L) = S((l 0) : I 0 M >L) L—>L®M) -
= 0,

14

LOWER K- AND L-THEORY

(iv) 6(f) = 0 for any coherence isomorphism / . The morphisms / : L >M of filtration degree < b define a subgroup Fh(L, M) of HomA(£, M) such that oo

F0(L, M ) C F 1 ( I , M ) C . . . C ( J Fb(L, M) = HomA(L, M) . 6=0

DEFINITION 2.3 Let X be a metric space, and let A be a filtered additive category. (i) An object M in Gx(A) is bounded if for each x € X and each number r > 0 there is only a finite number oi y £ X with c?(x, y) < r and M(y) non-zero. (ii) A morphism / : L >M in Gx(A) is bounded if there exists a number b > 0 such that (a)«(/(y,i)) 6 . (iii) The bounded X-graded category Cx(A) is the subcategory of Gx(A) consisting of bounded objects and bounded morphisms. The idempotent completion of Cx(A) is written

PX(A) = Po(Cx(A)) . • An X-graded CW complex (K,px) is a finite-dimensional CW complex K together with a continuous map px ' K >X. For each r-cell e = er C K let Xe ' Dr >K be the characteristic map, and let Pe = PKXe : Dr

> X , 7 (e) = pe(0) G X .

The X-graded CW complex (K,px) proper, and the maps

is bounded if PK • K

>X is

are uniformly bounded, i.e. there exists an integer b > 0 such that d(pe(x), pe(y)) < b for each cell er C K and all x,y € i? r . Given a ring A let B-^(A) be the additive category of based f.g. free A-modules with the trivial filtered structure F0Hom=Hom, and write Cx(B'(A)) = Cx(A) . EXAMPLE 2.4 Given an X-graded CW complex (K,PK) and a regular covering K of K with group of covering translations n consider the cellular based free Z[7r]-module chain complex C(K) as a finite chain complex in the X-graded category GX(Z[TT]) with C(K)r(x) the based

2. GRADED AND BOUNDED CATEGORIES

15

f.g. free Z[7r]-module with one base element for each r-cell e C. K such that 7(e) = x. If (K,PK) is bounded then C(K) is defined in CX(Z[TT]). • A proper eventually Lipschitz map f : X >Y of metric spaces is a function (not necessarily continuous) satisfying (i) the inverse image of a bounded set is a bounded set, (ii) there exist numbers r, k > 0 depending only on / such that for all s > r and all x,y £ X with d(x,y) < s it is the case that d(f(x),f(y))Y let UM =

{f.M(y)\y£Y}

be the object in Cy(A) defined by UM{y) = If / : X >Y is a homotopy equivalence of metric spaces in the proper eventually Lipschitz category then /* : Cx(A) ^Cy(A) is an equivalence of filtered additive categories. A subobject N C M of an object M in Gx(A) is an object N in Gx(A) such that for each x £ X either N(x) = M(x) or N(x) = 0. Given an object M in Gx(A) and a subset Y C X let M(Y) be the subobject of M defined by yev The evident projection is denoted by :M

PY

> M(Y) .

Given also a subset Z C X and a morphism / : L

>M in Gx(A) write

f(L(Y)) C M{Z) to signify that f(x,y)

= 0 : L(y)

> M(x) for all x £ X - Z , y £ F ,

or equivalently P x - z / p y = 0 : L —-> M(X - Z) . Let (X, Y C I ) be a pair of metric spaces. For any integer b > 0 define the b-neighbourhood of Y in X to be ATb(Y,X) =

{xeX\d(x,y).A/&(Y, X) is a homotopy equivalence in the proper eventually Lipschitz category, with a homotopy inverse defined by sending x G Afb(Y,X) to any y G X with d(x,y) < 6. The induced isomorphisms of algebraic iiT-groups

will be used as identifications. DEFINITION 2.5 (i) The germ away from Y [/] of a morphism / : L >M in Gx(A) is the equivalence class of / with respect to equivalence relation ~ defined on the morphisms L >M by / ~ g if ( / - g){L(X\Mb(Y,X))) C M{Afc(Y,X)) for some b, c > 0 . (ii) The (X,Y)-graded category Gx,y(A) is the additive category with: (a) the objects of Gx,y(A) are the bounded objects M = ^ M(x) of (b) the morphisms [/] : L Y of morphisms / : L A morphism / : L

>M in Gx,y(A) are the germs away from >M in Gx(A). •

>M in Gx(A) with a factorization f :L

> N

> M

through an object N in Gjy/-6(y>x)(A) has germ [/] = [0] : L — , M . DEFINITION 2.6 The bounded (X,Y)-graded category Cx,y(A) is the subcategory of Gx,y(A) with the same objects, and morphisms the germs [/] away from Y of morphisms / : L >M in Cx(A). D y

Cx y(A) is the category denoted C^ (A) by Hambleton and Pedersen [31]. The germ [/] away from Y of a morphism / : L >M in Cx(A) consists of the morphisms g : L >M in Cx(A) such that (f-g)(L)CM(Afb(Y,X)) for some b > 0. Given a chain complex C in Cx(A) let [C] be the chain complex

2. GRADED AND BOUNDED CATEGORIES

17

defined in Cx,y(A) by d[C] — [dc] '- [C]r = Cr

> [C] r -1 = Cr-\ .

PROPOSITION 2.7 For every finite chain complex C in Cx,y(A) there exists a finite chain complex D in Cx(A) such that C = [D] . PROOF Let C be n-dimensional. For any representatives in Cx(A) > Cr-i

dc : Cr

(1 < r < n)

there exists a sequence b = (&o> b\,..., 6n) of integers br > 0 such that

(i)d c (c r (M f (y,x)))cc r .i (ii) (dc)2(Cr) C Cr_2(Mr_2(^ (Start with 6n = 0 and work downwards, 6 n _i,6 n _2,..., &o)« For any such sequence b let Z2 be the chain complex defined in Cx(A) by '0 0 Dr

0 dc\

— Cr — C

DEFINITION 2.8 (i) The support of an object M in Cx(A) is the subspace supp(M) = {x e X | M{x) / 0 } C I . (ii) The neighbourhood category of Cy(A) in Cx(A) is the full subcategory

Ny(A) = |J C^6(y,x)(A) C Cx(A) 6>0

with objects M which are in Cjvr6(y,x)(A) for some b > 0, i.e. such that D

The inclusion F

: Cy(A)

> Ny(A)

is an equivalence of additive categories. In order to define an inverse equivalence F " 1 = G choose for each b > 0 a strong deformation retraction gb = inclusion"1 : Afb(Y,X)

>Y

in the proper eventually Lipschitz category, and set G

: N Y

( A ) —» Cy(A) ; M —+ (gb)tM

18

LOWER K- AND L-THEORY

for any 6 > 0 such that M is defined in C>v6(y>x)(A). The idempotent completions are also equivalent, allowing the identifications ffo(Po(Ny(A))) = Ko{PmY<x){!\))

= -MPy(A)) (6 > 0) .

The objects in Cx,y(A) isomorphic to 0 are the objects in Ny(A), and Cx,y(A) is the localization of Cx(A) inverting the subcategory Ny(A). The equivalence type of the filtered additive category Cx,y(A) depends only on the homotopy type of the complement X\Y in the proper eventually Lipschitz category.

§3. End invariants The algebraic theories of finiteness obstruction and torsion of §1 are applied to the graded and bounded categories of §2, to obtain an abstract account of the end invariant of Siebenmann [73] for closing a tame end of an open manifold. Let A C B be an embedding of additive categories (meaning a functor which is injective on the object sets). A morphism / : L >M in A is a B-isomorphism if there exists an inverse f~l : M >L in B. A chain complex C in A is B-contractible if there exists a chain contraction r : 0 ~ 1 : C >C in B. Let Ao C Ai C A2 be embeddings of additive categories. A chain complex C in Ai is (A2, F\Q)-finitely dominated if there exists a domination (D,f:

C^D,g

: D—*C,

h:gf~l:

C^C)

of C in A2 by a finite chain complex D in Ao. In the following result the embeddings Ao C Ai C A2 are given by Ny(A) C Cx(A) C Gx(A) for a pair of metric spaces (X,Y C X) and a filtered additive category A. 3.1 A finite chain complex C in Cx(A) is Gx,y(A)contractible if and only if it is (Gx(A),Ny(A))-yimte/y dominated. PROOF Let C be n-dimensional dc dc C : ... > 0 > Cn —* Cn-i —> . . . —> C\ —> Co . A contraction [F] : 0 ^ 1 : C >C of C in Gx,y(A) is represented by a collection of morphisms in Gx(A) PROPOSITION

r = {T:Cr

>Cr+i\0 cr such that dc(Cr(Mbr(Y,X))) C Cr-iMr_t(Y,X))

(1 < r < n) .

Let 6 = max{&0,&i,-• • ,&n} and let Z) be the finite chain complex in C/v/i(Y,x)(A) defined by

dD = dC\ : 2?r = CrWbr(Y,X)) —* Dr-l = Cr-lM^frX))

.

Define a (Gx(A), Cjy/*6(Y}x)(A))-domination of C (f : C—>D,g

: D—>C,h:

gf~\:

C^C)

by

/ = (1-dcT-

Tdc)\ : Cr —> I>r ,

^ = inclusion : D r

h = F : Cr

• Cr ,

• C r +i .

Conversely, suppose given a (Gx(A), Cjvrfc(y,x)(^))"clomination of C (/ : C—>D,ff : £ > ^ C , f t : s / ~ 1 : C—*C) for some b > 0. Passing to the germs away from Y there is defined a chain contraction of C in Gx,y(A) [T] = [&] : bf/] = 0 - 1 : C —> C . • By 2.7 every finite chain complex C in Cx,y(A) is of the type C = [D], for some finite chain complex D in Cx(A). In view of 3.1 the following conditions are equivalent: (i) C is Gx,y(A)-contractible, (ii) D is Gx,y(A)-contractible, (iii) D is (Gx(A),Ny(A))-finitely dominated. DEFINITION 3.2 The end invariant of a finite Gx,y (A)-contractible chain complex C in Cx,y(A) is the projective class [C]+ = [D] € A' 0 (P^ (y> x)(A)) = K0(Py(f\)) of any (Gx(A), Ny(A))-finitely dominated chain complex D in Cx(A) such that [D] = C. The reduced end invariant of C is the image of the end invariant in the reduced class group [C]+ G X 0 (Py(A)) = coker(AVCy(A))—*A'0(Py(A))) . • EXAMPLE 3.3 Let (X,Y) = (Z + , {0}). For any object M in A let M[z] be the object in Cx(A) defined by M[z](k)

= zkM (Jfc > 0) ,

20

LOWER K- AND L-THEORY

with zkM a copy of M graded by k G X. For any object (M,p) in Po(A) the 1-dimensional chain complex C defined in Cx(A) by /I 0 0 ..." -p 1 0 ... dc = o - p i . . .

I

\ : = M[z] =

Co = M[z] =

is Gx,y(A)-contractible, since dc fits into the direct sum system in Po(Gx(A)) M[z]

with

r=

*=o

• = (p

j =

J

p p ...) :

: M — M[z] = k=o

The direct sum system includes a (Gx(A),Cy(A))-domination (M, i, j , F) of C with ij = p : M •Af, so that C has end invariant [C]+ = [M,p]€ A'0(Py(A)) = Ko(Po(A)) . D

Given a subobject JV C M of an object M in Cx(A) define the quotient object M/N to be the object in Cx(A) with if iV(x) = M(x As usual, there is defined an exact sequence 0 —> JV — • M — • M/iV —> 0 .

3. END INVARIANTS

21

PROPOSITION 3.4 The end invariants of a pair (C,D C C) of n-dimensional Gx,Y(R)-contractible chain complexes in Cx,y(A) with C/D defined in Cjvr6(y,x),y(^) for some b > 0 are related by

[C]-[D] = [C/D] € im(#0(Cy(A)) r=0

The reduced invariants are such thai

[C] - [D] = 0 € £ 0 (Py(A)) . PROOF Immediate from the sum formula 1.1 for projective classes, identifying D

Thus the reduced end invariant [C] G KQ(PY(R)) depends only on the part of C away from Afh(Y, X) C X, for any 6 > 0. EXAMPLE

3.5 Let (X,Y)

= (R+,{0}), so that

ATb(Y,X) = [0,6] cX = R + = [0,oo) (b > 0) . Let W be a connected open n-dimensional manifold with compact boundary dW', with one tame end e + .

W

e+

Use a handle structure on W and a proper map p : (W, cW) >(X, F ) to give (W, 9W) the structure of a bounded (X, F)-graded n-dimensional pair. Let The chain complex C(W) of the universal cover W of W is an ndimensional (Gx(A),Ny(A))-finitely dominated chain complex in Cx(A). The reduced end invariant of C(W) is the finiteness obstruction of Wall [83] for the finitely dominated CW complex W

[C(W)]+ = [W] e Xo(Po(A)) = ^o(Z[ir!(W)]) , with [W] = 0 if and only if W is homotopy equivalent to a finite CW complex. The fundamental group of e~*~ is 7n(e + ) = lim 6

22

LOWER K- AND L-THEORY

with Wh = p-\X\Afh(Y,X)) = p-^ocOcW (6>0) a cofinal family of neighbourhoods of e + . The end invariant of Siebenmann [73] is such that [e+] = 0 if (and for n > 6 only if) there exist a compact cobordism (V; d-V, d+V) with d-V = dW and a homeomorphism which is the identity on dW', or equivalently such that Working as in [73] it is possible to express W as a union of adjoining cobordisms, W = Vo U V\ U ... U Vn_4 U Wn-3 with Vo, Vi,..., Vn-4 compact and Wn-^ open, such that 3_Vb = dW , Vr = d-Vr x / ( = trace of surgeries on Sr x Dn~r~1 C d-Vr ) (0 < r < n - 4) , and such that each n-4

[fr = [J Vi U Wn_3 (0 < r < n - 3) i=r

is an 'r-neighbourhood' of the end e+ with the inclusion dUr = d-Vr > VT r-connected.

The (n - 3)-neighbourhood (Un-3,dUn-3) that JIi(Wn-3,dWn-3)

= 0

= (Wn-S,dWn-3)

is such

(i^n-2),

and P = i7 n _ 2 (Wn-3,9W n _3) is a f.g. projective Z[7Ti(e+)]-module such that [6+] = [P]GAVZ[7T1(e+)]).

3. END INVARIANTS

23

[P] = 0 if and (for n > 6) only if there exists an expression Wn-3 = Vn-3UWn-2 for some compact cobordism (V n _3;d-V n -z,d+ Vn-$) with

d-Vn-3 = dwn-z

, d+vn.3

=

dwn.2,

vn-3 = a_vn_3 x j(Jui> n - 2 x D2 such that Wn-2 is an (n — 2)-neighbourhood of e + , in which case the inclusion dWn-2 >Wn-2 is a homotopy equivalence and by the invertibility of /i-cobordisms. The image of the end obstruction [e+] under the map induced by the inclusions Wb >W is the finiteness obstruction [W], In §4 below the protective class [W] G i^o(Z[7ri(W)]) is identified with the torsion T(C(W))

€ ^(Cx

of C(W) regarded as a Gx,y(A)-contractible chain complex by 3.1. • DEFINITION 3.6 The end invariant of a Gx,y(A)-isomorphism [/] : L >M in Gx,y(A) is the end invariant [/]+ = [C]+ e A'o(Pv(A)) of the 1-dimensional contractible chain complex C in Gx,y(A) defined by d = [/] : d

= L

> Co = M . D

The end invariant of isomorphisms has the following algebraic properties: PROPOSITION 3.7 (i) For any isomorphisms [/] : L M1 in Gx,y(A) [f®f

:L®L'

>M, [/'] : V

>M®M']+

= [f : L—>M]+ + [/' : L'—*M']+ G i^o(Py(A)) . (ii) For any isomorphisms [f] : L [gf : L^N}+

= [f : L^M]+

>M, [g] : M + [g : M-^N}+

(iii) For any Gx(A)-isomorphism f : L

>N in Gx,y(A) € A'0(Py(A)) .

>M in Cx(A)

[/]+ = 0 e A 0 (Py(A)) .

>

24

LOWER K- AND L-THEORY

(iv) For any morphism [e] : M

I A

>L in Gx,y(A)

:L®M

PROOF (i) Projective class is additive for direct sum of chain complexes. (ii) Let C, D be the 1-dimensional Gx,y(A)-contractible chain complexes defined in Cx(A) by dc = f : Cx = L —> Co = M , dD = gf

: Dx = L

> Do = N ,

so that [/]+ = [C]+ , [gf}+ = The 2-dimensional Gx,y(A)-contractible chain complex E defined in Cx(A) by

(gf

g) : Ei = L®M

> Eo

=N

has end invariant [E]+ = [g]+ € ffo(Py(A)) . It follows from the short exact sequence 0

> D - ^ E - ^ SC

>0

with i the inclusion and j the projection that [E]+ = [D}+ + [SC}+ = [gf}+ - [/]+ €

and so [gf}+ = [f]+ + (iii) If / : L

[g]+eKo(P

>M is a Gx(A)-isomorphism in Cx(A) then the 1-

dimensional chain complex C defined in Cx(A) by dc = f : Ci = L

> Co = M

is Gx(A)-contractible, and [/]+ = [C] = 0€K 0 (Py(A)) by the chain homotopy invariance of projective class.

3. END INVARIANTS

25

(iv) The automorphisms

/I

[g] =

0

\0

0

/I [A] =

l\

0 1 0

0

\0

: L®M®L

> L®M®L,

l)

0

0\

1

0 \ : L®M®L

—>

L®M®L

[e] l)

are such that

[flfflU = WM^W1

•

L®M®L—>L®M®L.

Applying (ii) and (iii) gives

= [g}+ + [h}+ - [g}+ - [h}+ = 0 G -MPy(A)) .

P R O P O S I T I O N 3.8 Let

f

9

0 —* C —y D —> E —>0 be an exact sequence of Gx,y(A)-coniracDr splitting morphisms in Gx,y(A) for g : Dr >Er (r > 0). PROOF Consider first the special case when E = 0, so that / : C >D is an isomorphism of contractible finite chain complexes in Gx,y(A). The sum formula in this case

r=0

is proved by repeated application of the sum formula of 3.7 (ii). Returning to the general case consider the exact sequence of contractible finite chain complexes in Gx,y(A) /' g' 0 —> C —> D' —> E — > 0

26

LOWER K- AND L-THEORY

defined by i

j

-

TV J^r

—

(~* /T\ Jy1 W vP &r

v D' ' uT—\

~

/^ ^r—1

(T\ Jp w -^r—1

?

/' = Q ) : Cr—>£>'r = C r ©E r , g' = (0 1) : £>; = CT@Er —> Er , with k : £V

^Cr_i (r > 1) the unique morphisms in Cx(A) such that y& = c?£>^ — /icf^; : Er

> D,—i .

It is immediate from the definitions that ID'U = \C}+ + \E]+ € There is defined an isomorphism (/ h) : D1 >D of contractible finite chain complexes in Gx,y(A), and by the special case oo

) r [(/ h):CT® Er—>Dr}+

€ A'0(Py(A)) .

r=0

Eliminating [D']+ gives the sum formula in general. D

Thus if / : E >E' is a chain map of contractible finite chain complexes in Gx,y(A) then / is a chain equivalence, and the algebraic mapping cone C(f) is contractible, with end invariant

3.9 A finite chain complex C in Cx(A) is Cx,y(A)contractible if and only if it is (Cx(A),Ny(A))-

M[z]

M[z]

y

,

(M,p)

with r

=

'1-p 0 I 0

p 1-p 0

0 P

1-p

... ...

M[z] =

t =

(p

0

0

...)

:M[z]

M ,

= k=0

ik=o

The direct sum system includes a (Cx(A), Cy(A))-domination (M, i , j , >M, so that C has end invariant F) of C with ij = p : M [C7]+ = [M,p]GK 0 (Py(A)) = A^(P0(A)) . n PROPOSITION 3.12 Let f : L >M, / ' : M >L be morphisms in Cx(A) such that that the germs [f], [f] are inverse isomorphisms in GX,Y(A)

[/'/] = 1 : L—+L , [ff] = 1 : M — • M , i.e. suc/i tta< t/iere ex^< integers 6,c > 0 KM^A ( / ' / - 1)(L) C L{Afb(Y,X)) , (ff - 1)(M) C T/ie end invariant of f is the projective class

[/]+ =

[D,p}eK0(PY(f\))

28

LOWER K- AND L-THEORY

0/ t/&e chain homotopy projection (D,p) defined in Cjvrd(y,x)(^) for

an

V

such 6,c > 0 wit& / ( L ( M ( ^ , ^ ) ) ) C M(^ c (y;X)), m t H > max(6,c) and dD = f\ : Dt = L(Afb(Y,X)) -^ Do = M{Me{Y,X)) , f / / ' - 1 : D o — • Do I / ' / - 1 : D1 — • £>0 • PROOF Let C be the 1-dimensional Cx,y(A)-contractible chain complex in Cx(A) defined by =

P

dc

f

=

The chain maps i : C

f//'-l

: Cl

= L

> Co = M ,

^Z>, j : D >C defined in Cx(A) by = M —> Do = : Co =

1/7-1 : d = . _ f inclusion : Do = M(Afc(Y,X)) ~ \ inclusion : Dx = L(Mh(Y,X))

3

> CQ = M > d = L

are such that k : ji~l

: C

> C , ij = p : D

> D

with the chain homotopy k given by k = f

: Co = M

> Ci = L .

Thus (C, z, j) is a splitting in Cx(A) of (JD,p), and the protective class of (D,p) coincides with the end invariant of /

[D,p] = [C] = [/]+ € i^o(Py(A)) .

§4. Excision and transversality in lf-theory The main result of §4 is a Mayer-Vietoris exact sequence (4.16) in the algebraic Jf-groups of the bounded categories associated to a union X = X^~ U X~ of metric spaces

K[(Cx+nX-(f\))

—> (

with the X'-groups certain generalizations of the algebraic If-groups ^ i ( C x + n x - (A)) and i^o(Px+nX-(A)). This exact sequence is obtained using algebraic excision and transversality properties of chain complexes in Cx(A). These are similar to the corresponding properties in geometric bordism and the algebraic iiT-theory localization exact sequence, which are now recalled. The Mayer-Vietoris exact sequence of bordism groups for a union

4. EXCISION AND TRANSVERSALITY IN /^-THEORY

29

X — X~*~ U X~ of reasonable spaces (e.g. polyhedra) with X~*~ fl X~ closed and bicollared in X

... —> Qn(x+ n x~) —> an(x+) e ftn(x-)

n r ) —> ... is a direct consequence of the transversality of manifolds. Any map / :M >X from an n-dimensional manifold M can be made transverse regular at X~*~ 0 X~ C X, with

N = r^nrjcM a framed codimension 1 submanifold, giving the connecting map d : nn(x) — » ! ) n - , ( i + n r ) ; (M,/) —» (AT,/|) . The Mayer-Vietoris exact sequence can also be derived from the excision isomorphisms of relative bordism groups

nn(x,x+) s*

an(x-,x+nx-)

with the connecting maps given by

d •. an(x)—>an(x,x+) s

nn(x-,x+nx-)

The localization exact sequence of algebraic If-theory (Bass [7]) for a ring morphism A >S~XA inverting a multiplicative subset 5 C A —> K0(A) —> KoiS^A) — X identifies the relative torsion if-groups K\(A >S~ A) with the class group A"i(A,S) = K0(H(A,S)) of the exact category H(A,5) of 5torsion A-modules of homological dimension 1. The following result identifies the relative torsion group #i(Py(A) >Px(A)) for a pair of metric spaces (X,Y C X) and a filtered additive category A with the torsion group i^i(Cx,y(A)) of the bounded germ category Cx,y(A). THEOREM 4.1 For any pair of metric spaces (X, Y C X) and any filtered additive category A there is a natural excision isomorphism

The connecting map in the exact sequence

is given by :M

30

LOWER K- AND L-THEORY

sending the torsion T ( [ / ] ) of an automorphism [f] : M >M in Cx,y(A) to the end invariant [/]+. PROOF The relative 7i'-group Ki(i) of the additive functor i = inclusion : Py(A)

> Px(A) ; M

> M

fits into an exact sequence

d

—> AyP identifying

Ki(Px(f\))

= Ki(Cx(f\))

, /M(P y (A)) = A'1(CY(A)) .

An element in Ki(i) is an equivalence class of triples (P, Q, / ) consisting of objects P, Q in Py(A) and a stable isomorphism [/] : P >Q in Px(A), with 0

: Kl(i)

_ _ A'0(PY(A)) ; (P,QJ)

— ^ [P] - [Q] .

By definition, (P, Q,/) = 0 € -K"i(0 if and only if there exists a stable >Q in Py(A) such that isomorphism [g] : P

rdgr'lf]

• P—+P) = 0 € A'i(Px(A)) .

Inverse isomorphisms

—

K.ii) ; r([e] : M—^M)

— . (P,Q,/)

will now be defined, as follows. Given an element (P, Q,/) G A"i(z) represent the stable isomorphism [/] : P >Q in Px(A) by the isomorphism ) JRP

p®R*Q®R

:

JRR )

in Px(A), with R defined in CX(A) C Px(A). Write the inverse of / as = g = ( \gRQ

QRR

The composite morphism in Cx(A) fRpgpR : R

> P

> i?

factors through an object in Cy(A) (e.g. M if P = (M,p)), so that it has germ [fRpgpR]

= 0 : R

Similarly, the composite gRQJQR : R

>Q

= 0 : R

>R. >R has germ >R.

4. EXCISION AND TRANSVERSALITY IN A'-THEORY

Thus the germs [/##],[###] : R

31

>R are inverse isomorphisms in

Cx,y(A), with [fRR][gnR] = [IRR9RR]

= [1-fRpgpR]

= 1 : R—> R ,

[###][/##] = [gRRfRit] = [1-gRQfQR.] = l : -R—>-R , and the morphism is well-defined. Given an automorphism [e] : M >M in Cx,y(A) let C be the 1dimensional chain complex C defined in Cx(A) by dc = e : d = M > Co = M , using any representative e of [e]. By 3.9 C is C^6(yx)(A)-finitely dominated in Cx(A) for some b > 0, so that there exists a 1-dimensional chain complex D in Py(A) with a chain equivalence g : D >C in Px(A). The algebraic mapping cone C(g) is contractible, and for any chain contraction T : 0 ~ 1 : C(g) >C(g) there is defined an isomorphism in Px(A) / = dc{g) + r : C(g)odd = ^o © Ci The morphism

> C(g)even

= D1®CQ .

is well-defined, and such that both the composites — • A'!(Cx,y(A)) — • Kx(i) are identity maps. D

DEFINITION 4.2 For subsets 17, V C X of a metric space X and 6, c > 0 write the intersection of the b-neighbourhood of U and the c-neighbourhood of V in X as

Afb,c(u,v,x) = Mh(u,x)r\Mc{v,x) = {x G X | d(x, u) MhiC(u,v,x)

32

LOWER K- AND L-THEORY

are not homotopy equivalences in the proper eventually Lipschitz category. This was first observed by Carlsson [16], using an example of the type

X = {(*, |*|/(1-|*D) I - K * < 1 } C R 2 , U = {(x, V = {(*, 1^1/(1 -\x\))\-Kx M(X+ fl X~) - ^ M(X + ) 0 M(X~) -L* M

>0

using the subobjects M(X~*"),M(-X"~) C M such that

M(i+)nM(r) = M(i + nr) ,

DEFINITION 4.5 A Mayer-Vietoris presentation with bound 6 > 0 of a chain complex E in Cx(A) with respect to a decomposition X = X + U X~~ is the Mayer-Vietoris exact sequence in Cx(A)

E : o — ^ + n r - ^ E + ® r ^ £ —> o determined by subcomplexes E^ C £? such that (i) £ + + £ - = E , (ii) £ ± is denned in Qv6(x±,x)(A), so that E^ H E~ is denned in (iii) ^ ( X i j C ^

(r>0). D

The following result is an algebraic analogue of codimension 1 transversality of manifolds: PROPOSITION 4.6 Every finite chain complex E in Cx(A) admits a Mayer-Vietoris presentation E, for any X — X + U X~. PROOF An object L in Cx(A) can be regarded as a O-dimensional chain complex. For any integer 6 > 0 define a Mayer-Vietoris presentation of L L(b) : 0

• L+(6) n r ( i ) —> L+(6) 0L-(6> — • L

> 0

by the construction of 4.4, with //*=(&) = L^b(X±,X)) Given a morphism / : L

, L+(b)DL-(b) =

L(Mb(X+,X-,X)).

>M in Cx(A) and any integer 6 > 0 there

exists an integer c > 0 such that /(L±(6))CM±(c), so that / can be resolved by a morphism of Mayer-Vietoris presentations

34

LOWER K- AND L-THEORY

• L + (6)nZT(ft)

L(fe> : 0

/+©/> M+ (c) n M~ (c)

M(c) : 0

•M-

•0.

Let E be n-dimensional. There exists a sequence 6 = (6 0 ,61,..., 6 n ) of integers > 0 such that d(Er(ArK(X±,X)))

C Er.1(Afbr_l(X±,X))

(1 < r < n) .

For every such sequence b there is defined a Mayer-Vietoris presentation of E E(b) : 0 —> £+(6) n E~{b)

> E+(b) © JE7~

> E —> 0 ,

with E±(b)r

= Ef(br)

= Er(Afhr(X±,X))

(0Arb(x+,x),Afb(x+,x-,x)(R)-contractibh

I&

if and only

if it is

(&) -finitely dominated for some

4. EXCISION AND TRANSVERSALITY IN ZC-THEORY

35

(c

< d > 0. Since £ + fl E

is defined in

CJV/- 6 ( X +,X-,X)(A)

the chain com-

e

plex E is G x ,x+nx-(A)-contractible if and o n ly if E* ^Ar6(X+,x),Ar6(x+,x-,x)(A)-contractible and E~ is

ls

(ii) As for (i), but using 3.9 instead of 3.1.

PROPOSITION 4.8 For X = X+ U l " there is a natural identification b

with an exact sequence lim i —» lim 6

Given a morphism / : L >M in Cx(A) with bound 6 let C, D, E be the 1-dimensional chain complexes denned by PROOF

dc

= f : d

-

L

v Co = M ,

dD = fx+

: Dx = L(X+) —» Do = M(Afb(X+,X))

dE -

: Ei -

with fx+, fx-

fx -

L(X~)

,

> Eo = M{Mb{X~,X))

,

the restrictions of / . The morphism of exact sequences

o —> L(x+ n fx+ © fx0 -> M(Nb(X+ ,X~,X))

-> M(Nb(X+ ,X)) ® M(Nb(X~,X))

f -> M

is a Mayer-Vietoris presentation of C. By 4.7 (ii) the morphism / : L—+M fx+ : L(X+)—+M(Afb(X+,X))

( C X , X+ (A) is a I ~

( Cx(A) isomorphism in
0. Now M(Nh(X+,X-,X))

is defined in C J v 6 ( x + j X - ) X ) (A), so

e that / is a CXjX+(A)-isomorphism if/ x + is a CAr6(x+,X)X6(^+,x- ,X)(A)isomorphism and fx- is a Cjv/;(x-,x)X6(^ + ^" > ^)(^H s o m o r P^ s m ' *n particular, if / : M >M is a Cx,x+(A)-automorphism then

fx-®0

: M(Afb(X-,X)) =

is a C0v6(x-,x),Ar6(X+,x-,x)(A)-automorphism. The morphism > lim A' 1 (C^ ( X - ) ^i(Cx,x+(A)) 6

r([/] : M—>M) — T([fx-

©0] :

is an isomorphism inverse to the morphism lim ^ i ( C M ( x - > x ) X f c ( x + , x - j X ) ( A ) )

,

6

induced by the inclusions Crtb(x-,x)M>(x+,x-,x)W The exact sequences of 4.1

—> C X)X+ (A) (6 > 0) .

-,x),ATb(X+,x-,

—* /fo(PAT.(x-,x)(A)) combined with the natural identifications = A'O(PX-(A)) give the exact sequence of the statement, on passing to the direct limit as b —> oo. • REMARK 4.9 Working on the level of permutative categories as in Pedersen and Weibel [53],[54] and Anderson, Connolly, Ferry and Pedersen [2] it is possible to extend 4.8 to all the algebraic if-groups of the idempotent completions, with natural excision isomorphisms

4. EXCISION AND TRANSVERSALITY IN K-THEORY

37

leading to the Mayer-Vietoris exact sequence of Carlsson [16]

—» ^

Kn(Px+(f\))®Kn(Px-(l\))

Km ffn

(n € Z) . This sequence will be obtained in 4.16 below for n = 1 by direct chain complex methods. D

DEFINITION 4.10 Let X = X+ u X~.

(i) A chain complex band E is a finite chain complex in Cx(A) which is Gx,x+nx-(A)-contractible, or equivalently (by 3.1) (Gx(A),Cjv/-6(x+,x-,X)(^))-fin^ely dominated for some b > 0. (ii) The positive end invariant of a chain complex band i2 is the end invariant [E\+ = [£(*+)] €Hmiiro(PM<x+,x->x)(A)) 6

of the Gjv/-6(x+,x),A/>6(^+,^~,^)(^)'con^rac^^e chain complex E(X+) in (iii) The negative end invariant of a chain complex band E is the end invariant [E\- = [E(X-)] € lim ffo(Pv»(X+,x-,X)(A)) of the Gjv/"6(x-,X),A/6(X+,x-,X)(A)"con^rac^ble chain complex £7(X~) in

EXAMPLE 4.11 Let

so that Afb(X+,X) = [-b,oo),Mb(X-,X) +

Afb(X ,X-,X)

= (-00,6],

= [-6,6],

with lim Jfo(PM(x+,x-,x)(A)) = iSTo(Po(A)) . 6

Siebenmann [74] defined a band to be a finite CW complex W together with a finitely dominated infinite cyclic cover W. (Such spaces arise in

38

LOWER K- AND L-THEORY

the obstruction theory of Faxrell [21] and Siebenmann [76] for fibering manifolds over the circle - see §20). A classifying map c : W •S 1 for W lifts to a proper Z-equivariant map p — c : W >X = R, and

W = W+UW~ = p has two tame ends e+, e~ such that with W + D W " = p'1 (X+ n X~) compact.

w+nw

w

The cellular chain complex of the universal cover W of W is a chain complex band E = C(W) in CX(B / (Z[TTI(TF)])), such that the reduced positive and negative end invariants of E are the end invariants of e + , e~ [E}±

=

EXAMPLE 4.12 A contractible finite chain complex E in Cx+uX-(A) is a chain complex band. D

PROPOSITION 4.13 For any Mayer-Vietoris presentation dimensional chain complex band E in C x + u X - ( ^ ) E : 0

> E+ DE~ +

> E+ © E~

> E

E of an n~ > 0

+

the subcomplexes j £ , i ? ~ , i £ fl E~ C JE7 are a/^o c k i n complex bands, with

- [E]- = [E-/E(X-)\ , + [E]--[E\ = [E(X+f)X-)} b

b

The reduced end invariants are such that

[E}+ + [E\- = [E] € Hm /T'o(PM(x+>x-,x)(A)) . 6

4. EXCISION AND TRANSVERSALITY IN K-THEORY

39

PROOF The subcomplexes J5+,J5~ C E are bands by 4.7 (i), and the intersection E* C\ E~ is a band since it is defined in The various identities involving the end invariants 6

are given by the projective class sum formula 1.1, using the Noether isomorphism

DEFINITION 4.14 The end invariants [f]± of an isomorphism / : L >M

in Cx+uX-(A) are the end invariants [E]± of the contractible 1-dimensional chain complex E defined in Cx+uX-(A) by dE

= f

: Ex = L

> Eo = M ,

that is [/]± = [E}± € lim

The < end invariant defined in 4.14 is the image of the end A. [ negative ° invariant in the sense of 3.6 of the isomorphism f [f(X+,X+)} : L(X+)^M(X+)

I [f(X~,X-)] : L(X-)—*M(X-) \

c

tfb(x-,

The sum formula of 4.13 gives G im(A^

0

(

x n X

()) 6

For any isomorphism / : L >M in Cx+uX-(^) ^ n e construction of 3.12 gives chain homotopy projections (D^^p^) in C>v6(x+,X-,X)(A) such that [/]± = [2? ± ,p ± ] € lim A'0( 6

Explicit representatives in Pjvrc(x+,x-,x)(^) f° r [/]+ a n ( i [/]- a r e n o w obtained, for some c > 0. For any integers c > b > 0 such that

f(L(X+)) C M(Ar6(X+,X)) , r 1 (M(7V t (X+,X))) C

40

LOWER K- AND L-THEORY

define an object in the idempotent completion

p+(/) = (i(Ar c (x + ,x)\x + ), P X such that there is defined a direct sum system in Ptfc(x+,

Px+Z" 1 with / + : L(X+) >M(A/i(^ + ,^)) the restriction of / . Similarly, for any integers c > b > 0 such that

f(L(X~)) C M(Mb(X-,X)) , r\MM{X-,X))) define an object in the idempotent completion

C

PA/"C(X+,X-,

such that there is defined a direct sum system in P.vc(x-,X)(A)

1

r

Px-/" 1 PMb(x-,x)f >M(A/6(^",^)) the restriction of / . There is also with / " : £(-X"") defined a direct sum system in

Px+nX-/" 1

PROPOSITION 4.15 The end invariants of an isomorphism f : L >M in Cx+uX-(A) are given by

6

PROOF AS in 4.10 let E be the 1-dimensional contractible chain complex in Cx+uX-(A) defined by dE

= f : Ex = L

> Eo = M .

For any integer b > 0 such that /(I(I + ))CM(M(I + ,I)) , f(L(X-))CM(Afb(X-,X)) define a Mayer-Vietoris presentation E(6) of E

41

4. EXCISION AND TRANSVERSALITY IN K-THEORY

>L(x+r\X~)

o

+ L{X+)®L{X~)

f+r\f~

•L-+0

f+®f-

o-+M(Nb(x+r\ x~))

f

M(Nb(X+)) 0 M(Nb(X-))

•M - • 0

such that

A direct application of 4.13 gives = [/]_ =

]_ =

[Pb+(f))-[M(Mb(X+,X)\X+)\, [E-(b)]-[E-(b)/E(X-)}

The end invariants of an automorphism / : M are such that

>M in

[/]+ + [/]_ = 0 € Hm A' 0 (P M (x + ,x-,x)(A)) . 6

THEOREM 4.16 The torsion and protective class groups associated to X = X + U X~ are related by a Mayer-Viet oris exact sequence lim ifi(Cv»(;c+,jt-1x)(A)) — 6 ( ( ) )

connecting map d defined by either of the end invariants 8 : tfi(Cx(A)) — Hm/ b

PROOF The relative /C-group A'i(Ai) of the additive functor M —> (M, M)

42

LOWER K- AND L-THEORY

fits into an exact sequence ) —> ^i(C^(x+,x)(A))© —+ ATo(PjVi(x+,x-,x)(A))

An element in ifi (Aft) is an equivalence class of quadruples (P, Q, e~*~, e~ ) consisting of objects P, Q in PA/" 5 (X+,X-,X)(^) a n d stable isomorphisms [c±] : P >Q in P M ( x ±,x)(A). By definition, (P,Q,e + ,e") = 0 G lfi(A&) if and only if there exists a stable isomorphism [i] : P >Q in such that

-1® : P - ^ P ) = 0 G Given objects P ^ , Q^ in PAr6(X+,x-,X)(^)? stable isomorphisms [^f1*1] P >Q± in P.v6(X±,X)(A) and a stable isomorphism [h] : P + © P •$"*" © Q~ in PA/*6(X+,X-,X)(^) there is defined an element ±

with image [P+] - [Q+] = [Q-] ~ [P~] €

Define inverse isomorphisms lim AVA 6 ) —» if,(Cx(A)) ; (P,Q,e+, e -) 6

>P) - - ^ - ( c + r 1 : Q Hm K!( 6

with P ^ , Q1*1, ^^, /i defined using the terminology of 4.15 , p - = p4-(/),

)) = Q± © (x-,x) M(X + H A"") ® P + © P" The composite Inn ^ ( A t ) —f ^i(C x (A)) —» lim ft

6

43

4. EXCISION AND TRANSVERSALITY IN /C-THEORY

is the identity, since for any (P,Q,e+,e ) G ifi(A&) the stable automorphism a = (e-)-le+ : P >P has positive end invariant W+ = [P}-[Q]€K0(Px+nX-(l\)). In order to verify that the composite #i(Cx(A)) —+ lim K!(A 6 ) —+ Xi(C x (A)) 6

is the identity consider the evaluation on the torsion T ( / ) of an automorphism / : M >M in Cx(A). Let M+ = M(X+) , M" = M(X~) C M , i ^ = inclusion : M + fl M~ j^

= inclusion : M^

•M ,

= f~lPMb(X±,X)f

JP = JMip

IQ = inclusion : Q jq

^ M± ,

• P±

> M^ ,

— inclusion : Q±

— jMiq

> M ,

> M ,

so that / is resolved by an isomorphism of exact sequences 0 —• M + fl M~ ® P+ © P~ —

•M

h U

,-

0— with

Q

up =

uQ

0 0

0

-i Q

%

° 0 1

~ Q 1 0

M+ n M~ © Q+ 0 N) = r(f : L

>M) + r(g : M

>N) ,

1

L'

>M © M )

= r(f : L

>M) + r(f : L1

>M') .

The isomorphism torsion of a contractible finite chain complex E in A is defined by T(E)

=

with

= E!®E3®E5®...

—> Eeven

= Eo 0 E2 0 Et © ...

the isomorphism in A defined for any chain contraction T : 0-1 : E

> E .

Given objects L, M in A define the sign

e(L,M) - r((5 j ) : L @M^M

® L) € K{ao{I\) .

5. ISOMORPHISM TORSION

47

Also, given finite chain complexes C, D in A define ) =

r((C@D)even—>Ceven®Deven) - r((C 0 D)odd

>Codd 0 Dodd)

The isomorphism torsion of a chain equivalence / : C >D of round finite chain complexes in A was defined in Ranicki [65] to be r(f) = r(C(f)) - 0{D,SC) € ^ " ( A ) , with C(f) the algebraic mapping cone of / . PROPOSITION 5.1 (i) For any chain equivalences f : C

>D, g : D >E

of round finite chain complexes in A r(gf : C^E)

= r ( / : C^D)

+ T(9 : D^E)

€ K['°{J\) .

(ii) Let C be a contractiblefinitechain complex in A, and let f 9 0 > C" - ^ C - ^ C > 0 be an exact sequence in A with C,C" round finite. Then f : C" • C " is a chain equivalence with torsion

r=0

for any splitting morphisms h : Cr >C'r of g : C'r >Cr (r > 0). If also C', C" are contractible then r(f) = T(C')-r(C")eKT(f\). P R O O F See [65, 4.2].

• The reduced isomorphism torsion group of an additive category A is defined by K[so(f\)

= coker(c:tf 0 (A)®tfo(A)

^ao(A)) ,

with e the sign pairing. The torsion of a chain equivalence / : C >D of finite chain complexes in A is the reduced torsion of the algebraic mapping cone

r(f) = r(C(f)) € Ki'°(f\) . The reduced automorphism torsion group of an additive category A is defined by £i(A) = coker(e: A'o with e the sign pairing.

48

LOWER K- AND L-THEORY

DEFINITION 5.2 A stable canonical structure [] on an additive category •JV, one for each A is a collection of stable isomorphisms [N,p][M,N] : M ^ N

—> P ,

(iii) [^MeM'.Neiv] = [M,N] © [M'M : M®M'

> N © N' . • A stable canonical structure [ r ( / ) = r ( [ ^ , M ] / : M ^ M ) . The automorphism torsion of a contractible finite chain complex E in A is then defined to be the image of the isomorphism torsion T(E)

= r(d+T: Eodd—>Eeven)

G A^A) .

A stable isomorphism [/] : M >iV has an isomorphism torsion r([/]) € K{so(f\) and hence also an automorphism torsion r([/]) G i^i(A). Similarly for reduced torsion. For a ring A the automorphism torsion groups of the additive category of based f.g. free A-modules are the usual torsion groups = KX{A) , If A is such that the rank of based f.g. free A-module is well-defined then Bf(A) has the canonical (un)stable structure [] with M,N ' M

the isomorphism sending the base of M to the base of N. A stable canonical structure (5.2) on an additive category A allows the definition of torsion r ( / ) G JK'i(A) for an isomorphism / : L >M in A, and hence the definition of torsion r(C) G i^i(A) for a contractible finite chain complex C in A. A 'flasque structure' on an additive category A determines a stable canonical structure; the bounded graded categories over open cones have flasque structures, allowing torsion to be defined for bounded homotopy equivalences over open cones. DEFINITION 5.3 A flasque structure {£,SiV) ,

>A such that the isomorphisms

>EM define a natural equivalence of functors a :

1A

©E

> E : A

> A. •

If A admits a natural flasque structure then if*(A) = 0, and in particular tfi(A) = 0. EXAMPLE 5.6 An additive category A with countable direct sums has a natural flasque structure {E,cr, p] by the Eilenberg swindle, with oo

EM = J^M, I

GM

: MffiSM

^ EM ; (xo,(xi,x 2 ,...))

> (xo,xux2,...)

. D

PROPOSITION 5.7 (i) C R ( A ) has a flasque structure. (ii) C R + ( A ) has a natural flasque structure. PROOF (i) For any object M in CR(A) let TM be the object defined by M(t - 1) if t > 1 Af(* + 1) if t < —1 0 otherwise ,

(

50

LOWER K- AND L-THEORY

and let T : M

>TM be the isomorphism bounded by 1 defined by a if u > 0 and v = u + 1 a ifu EM ;

> EM0EAT ; (a,6) >(a,6). pM,iV : S(M0iV) (ii) The flasque structure on CR(A) defined in (i) restricts to a natural flasque structure on CR+(A). D

LEMMA 5.8 If F : A >f\' is a functor of additive categories such that every object M1 of A' is the image M' = F(M) of an object M in A then a flasque structure {E,cr, p} on A determines a flasque structure { E V , p ' } on A', with E'M' = F(EM) , a'M, = F(aM)

: M' 0 E'M' = F(M © EM) —> E'M' = F(EM) ,

PM;N'

= F(PM,N)

: E ' ( M ' 0 iV') = F(S(M 0 N)) > E'M' 0 E'iV' = F(EM 0 UN) .

In particular, 5.8 applies to the functor F : Cx(A) duced by the inclusion.

• ^Cx,y(A) in-

PROPOSITION 5.9 Let (X,Y CX) be a pair of metric spaces, and let A be any filtered additive category. (i) The end invariant defines a morphism diao : Ar°(C x ,Y(A)) —> AyPy(A)) ; r(E) — • [E)+ such that the connecting map d in the exact sequence of J^.l tfi(Cy(A)) — - A-i(Cx(A)) — — z*3 the

composite

3 : ^(Cx

5. ISOMORPHISM TORSION

51

and im(3) C im(d"°) . (ii) The sequence

w exact. (in) If K0(Cx(f\))

= {0} then

im(a) = im(a"°) . (iv) IfCx(R) admits a flasque structure {£,cr,/»} and Cx,y(A) feas K^CX.YW)

—•> tfo(Py(A)) .

TAe automorphism torsion T(E) £ ATi(Cx,y(A)) o/ any contractible finite chain complex E is such that dr(E) = [E]+. PROOF (i) The end invariant of an isomorphism / : M >N in Cx,y(A) is the element [/]+ £ A"0(Py(A)) defined in 3.6. It is necessary to prove that for any contractible finite chain complex E in Cx,y(A) the element 3r{E) = [£]+ e A'0(Py(A)) is the end invariant [d + T]+ of the isomorphism d + F : Eodd >Eeven in Cx,y(A) used to define r(-B), for any chain contraction F : 0 ~ 1 : E >E. Consider first the 1-dimensional case, with dE

= f

: Ei

= L

> Eo

= M

an isomorphism in Cx,y(A), so that dr{[f]) G A"o(Py(A)) is the end invariant [/]+ of/. The n-dimensional case is reduced to the 1-dimensional case by the "folding" process of Whitehead [88], as follows. Let then E be n-dimensional d > 0 > En > En-i > ... > Eo . E : ... As E is contractible there exists a morphism T : En-\ >En in Cx,y(A) which splits d : En >En-\ with Yd = 1 : En

>En.

Let C be the contractible (n — l)-dimensional chain complex in Cx,y(A) defined by

C : En-\

> En-2 ® En

.0)

i

and let D be the Cx,y(A)-contractible n-dimensional chain complex in

52

LOWER K- AND L-THEORY

Cx(A) defined by

( d)

V

I

£jn

(d °)

> £jn-l

W &n

>

(d 0) T7I

771

/ ^r\

771

771

There are defined exact sequences of contractible chain complexes in 0 —> C —> D — > C —> 0 , 0

> JE7'

> D

y E

> 0,

with C, E' the elementary n-dimensional chain complexes C

: En

E' : 0

> En * En

yO

• ...

y En

»0

»0, y ...

> 0.

Now T(C)

= T(D) = T(E) € ifi(C X l y(A)) ,

and by the sum formula 3.8 [C]+ = [D}+ = [EU € /fo(Py(A)) . As C is (n — l)-dimensional this gives the inductive step in the proof that the connecting map d of 4.1 sends the torsion T(E) to the positive end invariant [d + T : Eodd >Eeven}+. ^ (ii) The objects P,Q in Py(A) are such that [P] = [Q] G ^o(Px(A)) if and only if there exists an isomorphism in Px(A)

{QP

f QR f )

JSP

JSR )

:

P®R-^Q®S

for some objects R,S in Cx(A), in which case [/SR] ' R isomorphism in Cx,y(A) with isomorphism torsion r([fsR]) G

>S is an

hr°(CxN in Cx,y(A) the element dis°T(f) G i^o(Py(A)) has image d"°r{f)

= [N] - [M] G im(^o(C x (A))—,A' 0 (Py(A))) = {0} .

(iv) By 3.7 (iii) the end invariants of the isomorphisms <JM • M 0 EM >TtM in Cx(A) have end invariants [aM\+

= 0 € AVP

6. OPEN CONES

The torsion of an isomorphism / : L

>M in

53

CX,Y(A)

is defined by

so dr([f]) € JiTo(Py(A)) is the end invariant of the stable automorphism kAfK/lki]" 1 : 0 >0 in Cx,y(A). By the sum formula 3.7 (ii) d(r[f}) = [kM][/]K]- x ] + = [aM]+ + [f}+-[aL}+ = [/]+ € KO(PY(I\)) , verifying the 1-dimensional case of dr(E) = [E]+. The n-dimensional case for n > 2 is reduced to the case n = 1 by folding as in (i). D

§6. Open cones The open cone of a subspace X C Sk is the metric space O(X)

=

{txeRk^\teR^,xeX}.

In particular, 0(Sk) = R*+1 . For the empty set 0 it is understood that 0(0) = {0} , Co(0)(A) = A . k For any Y C X C S and any b > 0 let Ob(Y,X) = M ( 0 ( F ) , 0 ( X ) ) = {z G O(X) | d(x, y) < b for some y G Also, for any Y,Z CX C Sk and any b > 0 let = {iG O(X) | d(x, y) 0 the inclusion

w a homotopy equivalence in the proper eventually Lipschitz category. PROOF Following a suggestion of Steve Ferry, replace X + , X ~ by Lipschitz homotopy equivalent cubical subcomplexes of S*, for which there is an identity

The results of §l-§5 will now be specialized to the 0(X)-bounded categories CQ(X)(A).

54

LOWER K- AND L-THEORY

PROPOSITION 6.2 (i) Let Y C X C S*. T/ie torsion and class groups of the bounded categories over the open cones are related by an excision isomorphism

tfi(PO(Y)(A)—>PoW(A))

S i

an exact sequence

(ii) For a compact polyhedron X = X + I I I " C Sk there is defined a Mayer- Vietoris exact sequence

0 A'o(P 0 ( x-)(A)) . PROOF (i) A special case of 4.1. (ii) A special case of 4.16, identifying + +

o ( i u i " ) = o(x )uo(X"), o(x+nx~) =

and using 6.1 to identify

= A0(PO(x+,x)no(x-,x)(A)) . D

Pedersen and Weibel [54] identified the algebraic if-theory of Po(X)(A) for a compact polyhedron X C Sk with the reduced generalized homology groups of X with coefficients in a non-connective delooping of the algebraic A'-theory of the idempotent completion Po(A) The excision isomorphisms and the Mayer-Vietoris exact sequence of 6.2 are particular consequences of this identification. However, the elementary techniques used here in the cases * < 1 give an explicit formula for the connecting map relating torsion and projective class, and are better suited for the generalization to L-theory obtained in §14 below. Next, it is shown that C Q ( X ) ( A ) admits a flasque structure for nonempty X , so that the isomorphism torsion theory of §5 applies. Given a non-empty subspace X C S*, and a base point x € X define for any object M in Co(x)(A) an object TM in C Q ( X ) ( A ) by (M((t-l)y) (TM)(ty) = I M(0) K0

if * > 1 if t = 1 and y = x otherwise .

6. OPEN CONES

Let T : M

55

>TM be the isomorphism in Co(x)(A) defined by

T(uz,ty) : M(ty)

> TM(uz) ; a a 0

if u = £ + 1 and z = y if t = 0 and w = 1 and 2 = x otherwise .

DEFINITION 6.3 The x-based flasque structure {E,cr,/9} on C O (x)(A) is defined by

EM = Yl T'M > oM

•• M @ E M —> E M ; (ao,(«i,a2,...))

> (Tao,Tai,Ta2,...)

,

^M>N : S(M ® iV) ^ EM ® SiV ; (a,6) > (a,6) . The x-based torsion of an isomorphism / : M >N in CQ(X)(A) is the torsion of / with respect to the #-based flasque structure r.(/) =

T^NMWM]'1

: 0 — > M - ^ J V ^ 0 ) € K!(C o ( x ) (A)) . • in 5.6

In particular, the natural flasque structure defined on CR+(A) is the 1-based flasque structure {£,N by

E/ : SM —> XN ; (aua2,...)

—+ (f(ai),f(a2),...)

.

k

It follows that for any non-empty X C. S and any x € X the x-based flasque structure on C O (cx)(A) = C R +(C O (x)(A))

is natural. (ii) Immediate from (i) and the identifications Cy X R+(A) = C R +(Cy(A)) , R+ = O({pt.}) .

PROPOSITION

6.5 The functor

CxuYxR+(A) — C x ,y(A) ; M —> M(X) = induces an isomorphism Ki(CXoYxR+(f\))

S 2fi(C X

and there is defined an exact sequence

-» /Co(Px(A)) the connecting map

d : AV r(f:M—*M)—+\f(X,X)]+ sending the torsion r(/) of an automorphism f : M >M in CxuYxR+(A) to the end invariant [f(X,X)]+ of the automorphism [f(X,X)] : M(X) >M(X) in C x ,y(A) defined by the restriction of f to the subobject M(X) C M. PROOF Theorem 4.1 gives an exact sequence

By 6.4 (ii) CyXR+(A) has a natural flasque structure, so that the idempotent completion PyXR+(A) also has a natural flasque structure and #i(Cy x R + (A)) = tfo(PyxR+(A)) = 0 .

6. OPEN CONES

57

Thus there is a natural identification -frl(CxuYxR+(A))

=

^l( C XuYxR+,YxR+(A)) .

Every object M in CxuYxR+(A) is stably isomorphic to the object M(X) in Cx(A). The germs away from Y of morphisms in Cx(A) are the germs away from Y x R+ of morphisms in CxuYxR+(A). It follows that the functor CXUYXR+,YXR+(A)

> CX,Y(A) ; M

> M(X)

induces isomorphisms in algebraic if-theory, allowing the natural identification D

In fact, 6.5 is a special case of a general result: the functor CxuYxR+(A)

> Cx,Y(A)

induces isomorphisms in all the algebraic if-groups tfn(PxuYxR+(A)) 3 Kn(Cx,Y(f\)) with a consequent long exact sequence —> Kn(PX(t\))

(n e Z) ,

(Ferry and Pedersen [28], Hambleton and Pedersen [31]). DEFINITION 6.6 (i) Given a map u> : 5° 5 0 (w) : K0(f\)

y KO(PO(I\))

>X let

>

= ifi(C o(s .)(A)) - ^ (ii) The Whitehead group of Co(x)(A) is defined by r coker(E Bo(w) : E A'o(A)—-•/f 1 (C o(x) (A))) i (A) with the sum taken over all the maps UJ : S°

if X / 0 i

>X.

The class at 0 of a finite chain complex C in Co(x)(A) is

r=0

58

LOWER K- AND L-THEORY

PROPOSITION 6.7 (i) If u : S° >X extends to a map D1 Bo(w) = 0 : K0(t\) —> (ii) If X is non-empty and connected If X has n components

Jfi,X2,... , X

with

n>2

then

n-l

n-l

Wh(Co(x)(f\)) =

n

>X then

cokerC£Bo("i)--Y,KW^K 1=1

1=1

for any maps u>j : S° >X such that u>j(+l) G -X"i, o;t(—1) G -Xj+i. (iii) For any a:,^' E l /et {E,a,/o}; {E',^',/)'} 6e the x- and x'-based flasque structures on Co(x)(&)- The difference of the x- and x1 -based torsions of a contractible finite chain complex C in CQ(X)(^) *5 TX{C)-TX,(C) = Bo(u>)[C(O)] e / CO

UJ

: 6

V

I 1

>X ; +1

Ml

1

> a: , - 1

> x .

(iv) TAe Whitehead torsion of a contractible finite chain complex C in Co ( x)(A) T(C)

= TX[C) e wh(cO(X)W)

is independent of the choice of base point x G X used to define the torsion rx(C) G i£i(Co(x)(A)). PROOF (i) If u> extends to a map D1 >X there is a factorization —» A' 1 (C O(X) (A)) and ffi(Co(Di)(A)) = 0 (by 6.4 (i)). (ii) Immediate from (i). (iii) The automorphism defined for any object M in Co(x)(A) by has torsion r(aM) - Bo(a;)[M(0)] G J (iv) Immediate from (iii). • REMARK 6.8 In view of 6.7 (ii) and Anderson [1] the Whitehead group Wh(Co(x)(R)) ls a special case of the Whitehead groups defined by Anderson and Munkholm [4, p. 146]. In particular, for X = 5° = Wh(CR(f\)) - coker(50 : A0(A) >A'1(CR(A))) . •

6. OPEN CONES

59

For any object M in Co(x+r\X-)(R) [M] = [M(0)]eim(tf0(Co<x+nx-)(A))

f {0}

if x+ n x - ? 0

\ im(/f o (A)—KMPO(A)))

if X+ n X~ = 0 .

Given a map w : 5°

>X+l)X~

; +1

> x+ , - 1

> aT ,

such that x* € X* define the abelian group morphism B0(w) : A'o(C0(x+nx-)(A)) — • / [M]—>r([^]-1[^]:

with {E^ja"^,/?^} the x^-based flasque structure on CQ(X±)(^) by 6.3. The morphism Bo(u>) of 6.7 (i) is the composite B0(u)

: K0(f\)

given

> A'0 BQ{UJ)

By 6.7 (iii) the x^-based and a:"-based torsions of a contractible finite chain complex E in Co(X+uX-)(R) differ by TX+(E)-TX-(E)

= B0(u>)[E(0)]

The composite

» is independent of a;, being the natural map [M] —» [M] = [M(0)] . PROPOSITION 6.9 TAe connecting map

— Ao(P0(x+nx-)(A)) sends the x-based torsion rx(E) £ -&a(Co(x+uX-)(^)) ° / a contractible

60

LOWER K- AND L-THEORY

finite chain complex E in CO(x+uX-)(A) t° the protective class Orx(E) = [£]+ r=0

=

j \

PROOF By construction, d is the composite d : J

—> A'0(Po(x+),o(x+nx-)(A)) • For x G X+ the image of r ^ E ) in ifi(CO(x+),o(X+nX-)(A)) is the zbased torsion TX(E(O(X^))) of the contractible chain complex £(O(X+)) in C o( x+),o(x+nx-)(A), and the identity drx(E) = [E]+ e ^ 0 (Po(X+nx-)(A)) is immediate from 5.8. For x £ X"^ choose a base point x1 G X + , and define u; : 5° >X by w(+l) = x', o>(-l) = x, so that TX(E) = r and

If X + fl X~ is non-empty then ifo(Co(x+nX-)(A)) = 0, since there exists a flasque structure on Co(x+nX-)(A) (by 6.3). Thus if E is round at 0 or if X + fl X~ is non-empty then dr(E) = [E]+ e Jfo(Po(X+nX-)(A)) for all base points x G X + U X~, in agreement with 6.7 (iii). PROPOSITION 6.10 The image under the connecting

map d of the x-based

torsion rx(f) G i f i ( C o ( x + u X - ) ( A ) ) of a chain equivalence of band complexes in C Q ( X + U X - ) ( A ) is given by

f :C

>D

PROOF The x-based torsion of / is the a;-based torsion of the algebraic mapping cone C(f) r,(/) = Tt(C(f)) € KtiCoix+ux-W) • (The sign term fi in the definition of r ( / ) in Ranicki [65, p.223] is 0, since X + U l " is non-empty and A"o(Co(x+uX-)(A)) = 0.) The result

6. OPEN CONES

61

follows from 6.9 and the sum formula of 3.12 for the end invariants of the band complexes in the exact sequence 0

> D

> C{f)

> SC

> 0.

For any X = X+ U X~ C 5* and 6 > 0 write Ob(X+) = Afb(O(X+),O(X))

= 0(x+,x)uAfi(0(x+

nx-),o(x-)),

0b(X-) = Afb(O(X-),O(X))

PROPOSITION 6.11 (i) The Whitehead torsion and reduced protective class groups associated to X = X+ U X~ C Sk are related by a MayerVietoris exact sequence

(ii) If E is a contractible finite chain complex in Co6(x±)(A) for some 6 > 0 then

T(E) e MWh(CO(x±)m—+Wh(Coix+ux-)(m

.

(iii) If E is a contractible finite chain complex in CO(x+uX-)(^) that §T(E)

suc

^

= 0 G #o(Po<x+nx-)(A))

then for any Mayer-Vietoris presentation of E

E : o — ^ + n r ^ £ + © r i £ —> o there exist finite chain complexes F~*~, F~ in Co(x+nx-)(&) and chain equivalences <jy^ : F^ >E^ in Cob(x±)(^) for some b > 0, with T(E) = r(^ + ) + r(^-) + r ( ( ^ + © ^ - ) - 1 i : E + n £ -

>F+ © F")

e im(Wfc(Co(x+)(A)) © W/i(Co(x-)(A))—^W/i(CO(x+ux-)(A))) . PROOF (i) By 6.7 there is no loss of generality in assuming that X + and

62

LOWER K- AND L-THEORY

X~ are connected, so that

\ coker(B0(uO :

f coker(£B0(>?) : E

=


M' is an isomorphism in Co6(x+)(A) with x~-based torsion TX-(BM)

=

rt-(M(Ot(I+)nO(I-))—>M'(06(I+)n0(I-)))

€ im(tfi(Co»(;c+)no(X-)(A)) so that the Whitehead torsion is

^i(C o( x+uX-)(A))) = {0} ,

T{9M) = 0 € Wh(Coix+uX-)W) • Thus for any isomorphism / : M >N in CQ 6 (X+)(A) there is defined an isomorphism in CO(x+)(^) f = 0Nf(0M)-1 : M1 > M > N > N1

7. /^-THEORY OF Ci(A)

63

such that

rx-(f) = rz-(f) + rz-(0N)and the Whitehead torsion of / is r(f) = r(f) E im (Wh(Co(x Similarly for chain complexes. (iii) The reduced end invariant [E]± is the Co(x+nX-)(A)-finiteness obstruction of E^1 [E}± = [E±] € ^o(Po(x+nx-)(A)) . By the sum formula and (ii) the Whitehead torsion of E is T(E)

= T(i:E+f\E-

>E+®E~)

= r{<j>+) + r{(f>-) + r((<j>+ © F+ © F~)

(

§7. iiT-theory of d(A) We now consider the Tf-theory of the bounded Z-graded category Ci(A) = C Z (A) , using resolutions in the Z-graded category Gi(A) = Gz(A) . The bounded R-graded category CR(A) equivalent to Ci(A) is used to obtain a chain complex interpretation of the isomorphism #i(CR(A)) S /TO(PO(A))

originally obtained by Pedersen [49] (for A = B-^(A)) and by Pedersen and Weibel [53] (for any A). In order to apply the if-theory exact sequences of §4 to CR(A) use the expression of X = R as a union X = X* U X~ with x + = R + = [o,oo), x~ = R - = (-00,0], i

+

n r

= {o},

or the equivalent subsets of Z. The 6-neighbourhoods for 6 > 0 are the intervals X) = [-6,oo),M(X-,X) = (-00,6], X - , X ) = [-6,6] C R . The various graded categories of a filtered additive category A associated to Z, Z+ = {n G Z I n > 0} and Z~ = {n £ Z | n < 0} are denoted by B Z (A) = Ba(A) , B Z + (A) = B+(A), B Z -(A) = B_(A), B z+){ o}(A) = B+,o(A) , B Z -, {O} (A) = B_,0(A) .

64

LOWER K- AND L-THEORY

with B = C or G. The inclusion Z

>R is a homotopy equivalence in

the proper eventually Lipschitz category, with homotopy inverse R and similarly for Z*

> Z; x

> [x] ,

•R*. Thus Ci(A) is equivalent to CR(A) and

The flasque structure defined on CR(A) in 5.7 (i) restricts to a fiasque structure on Ci(A), and this will be used to define torsions in Ki(Ci(f\)) for isomorphisms in Ci(A). The flasque structure on Ci(A) restricts to natural flasque structures on C±(A), so that A'»(C±(A)) = A%(CR±(A)) = 0 . The induced flasque structure on C±{0}(A) is not natural, in general. The inclusions C±,o(A)

> C R ±, { 0 } (A)

are also equivalences of additive categories, so that jr.(C±,o(A)) =

ff.(CR±>{0}(A))

.

Given an object M in A let M[z] be the object of C + (A) denned by M[z](k) = zhM

(Jfc > 0) ,

k

with z M a copy of M. Given a collection of morphisms in A such that {j E Z | fj ^ 0} is finite let oo

j=o

be the morphism in Cf (A) defined by =

zkM.

The polynomial extension category f\[z] of A is the subcategory of C+(A) with objects M[z] and morphisms Ylz* fj- Similarly for Af^"1]. j

The end invariant [E]+ € A'o(Po(A)) of a G+jo(A)-contractible finite chain complex E in C+?o(A) is defined as in §3. PROPOSITION 7.1 The end invariant defines an isomorphism 8 : ^(C+.oCA)) —> Ko(Po(A)) ; T(E) — * [E}+ , with inverse B : ffo(Po(A)) —-» K1(C+t0(f\))

;

[M,p] —-> T(1 - p + zp : M[z]—*M[z]) .

7. /C-THEORY OF Ci(A)

65

PROOF d is the connecting map in the exact sequence of 4.1 for (X, Y) = —> — with P+(A) = Po(C+(A)). It follows from the natural flasque structure on C+(A) given by 5.7 (ii) that #i(C+(A)) = /fo(P+(A)) = 0 , so d is an isomorphism. The identification dr(E) = [E]+ is given by 5.9. In order to verify that B is the inverse of d it is necessary to prove that for any object (M,p) in Po(A) the automorphism in C+^A) 1 - p + zp : M[z) > M[z] has positive end invariant [l-p + zp]+ - [M,p] € ^o(Po(A)) . This has already been done in 3.11. • x Given an object M in A let M[z, z~ ] be the object of Ci(A) defined by Mlz.z- 1 ]^) = zkM (fc€Z), k with z M a copy of M. Given a collection of morphisms in A {/ i GHom A (L,M)|JGZ} such that {j G Z | /j; ^ 0} is finite let

j=-oo

be the morphism in Ci(A) defined by

f(k,j) = /*_,- : LI*,*-1^") = z'i—^MI*,*" 1 ]^) = *kAf. The Laurent polynomial extension category A^,^" 1 ] of A is the subcategory of Ci(A) with objects M[z,z~1] and morphisms Ylz^ fjj

The end invariant of a G_5o(A)-contractible finite chain complex E in C_j0(A) is denoted by [£]_ G Ko(Po(f\)). Let No (A) be the full subcategory of Ci(A) with objects M of finite support (2.8 (i)), corresponding to the neighbourhood N{0}(A) of C{0}(A) in CR(A) (which is the full subcategory with objects of compact support). Similarly, let N±,0(A) be the neighbourhood of C{0}(A) in C±(A), the full subcategory with objects of finite support. The inclusions A = C{0}(A) —> N0(A) , A —> N±,0(A)

66

LOWER K- AND L-THEORY

are equivalences of additive categories. Recall from \ that a finite chain complex CinC±(A)is< n ' )nx ^ 3.7 ^ ^±,ov^v ' /n\\ -finitely dominated. contractible if and only if it is < ) ~ )n\\i i(C±(A),N±, 0 (A)) A chain complex band E in Ci(A) is a finite chain complex which is (Gi(A),N0(A))-finitely dominated, so that E is a chain complex band in CR(A) in the sense of 4.10 and the end invariants [E]± £ Ko(Po(f\)) are defined for X = Z = Z~*"UZ~, with sum [E}+ + [E\- = [E] + [E(0)} e ffo(Po(A)) . The end invariants of an automorphism / : L >L in Ci(A) are defined as in 4.14 to be the end invariants of the 1-dimensional (contractible) chain complex band E given by d,E — f

: Ei

= L

• EQ = L ,

that is [/]± = [E}± € Ko(Po(A)) with [/]+ + [/]_ = 0 € ffo(Po(A)). PROPOSITION 7.2 (i) The end invariants of automorphisms in Ci(A) define an isomorphism

; r(f : L^L)

— [/]+ = -[/]_

inverse : JT o (Po(A))

(ii) Tfee isomorphism d of (i) sends the torsion r ( / ) 0/ a chain equivalence f : E >E' of bands in Ci(A) to the difference of the positive end invariants dr(f) = [E'}+-[E}+€Ko(Po(f\)). PROOF (i) d is the connecting map in the Mayer-Vietoris exact sequence of 4.15 for I = I + u r given by Z = Z + U Z". The categories Cx±(A) = Cz±(A) have natural flasque structures (by 5.7 (ii)), so that tf.(Cz±(A)) - tf.(Pz±(A)) = 0 . The exact sequence of 4.16 includes 0 —» ^(CxCA)) —> ifo(Po(A)) —^ 0 ,

so that d is an isomorphism. In order to verify that B is the inverse of d note that by 3.10 (again) for any object (M,p) in Po(A) the automorphism in Ci(A)

l-p + zp : Mfaz-1]

>M[s,*-1]

7. tf-THEORY OF Ci(A)

67

has positive end invariant [l-p + zp]+ = [M,p] € ^o(Po(A)) . (ii) It is immediate from 5.9 that d sends the torsion T(C) G -K"I(CI(A)) of a contractible finite chain complex C in Ci(A) to the positive end invariant dr{C) = [C]+ € ifo(Po(A)) . The algebraic mapping cone C(f) is contractible, and fits into an exact sequence of band in Ci(A) 0 > E1 > C(f) > SE > 0. Applying the sum formula of 3.8

dr(f) = dr(C(f)) + [SE]+ = [E']+ - [E}+ € DEFINITION 7.3 The Whitehead group of Ci(A) is Wfc(d(A)) = coker(S0 : K0(f\) ^ with Bo : tfo(A) —^ Xo(Po(A)) —» ^ (

PROPOSITION 7.4 (i) T/ie reduced end invariants of isomorphisms in Ci(A) define an isomorphism

8 : Wfc(Ci(A)) —* A'o(Po(A)) ; r(/ : L—>M) — [/]+ = - [ / ] _ inverse

B : £ o (Po()) [ M , p ] — v ^ l - p + zptM^z- 1 ]—•Mlz.i:- 1 ]). (ii) The Whitehead torsion r ( / ) G H/rft(Ci(A)) 0/ a chain equivalence f :E >E' of bands in Ci(A) is sent by the isomorphism d to the difference of the reduced positive end invariants dr(f) = [E']+ - [E\+ G Ko(Po(A)) . PROOF (i) Immediate from 7.2 (i). (ii) Immediate from (i) and 7.2 (ii). • REMARK 7.5 (i) Browder [9] studied the problem of homotoping a homeomorphism W >M x R to a PL homeomorphism, with W an open ndimensional PL manifold and M a closed (n — 1)-dimensional PL manifold, proving that there is no obstruction in the case n > 6, n\(W) = {1}.

68

LOWER K- AND L-THEORY

(ii) If W is an open n-dimensional manifold with an R-bounded homo>X x R (e.g. a homeomorphism) for some topy equivalence h : W finite (n — 1)-dimensional geometric Poincare complex X then W is a band with two tame ends. The cellular chain complex C(W) of the universal cover W is a band in CR(B^(Z[TT])), and [W]± = [C(W)]± € ffo(ZM) with 7T = Ki(W) =

TTI(X).

The end invariants are such that

[W}+ + [W). = [W] = [X] = 0 6 £o(ZM) • By the main result of Siebenmann [73] (cf. Pacheco and Bryant [13, §4]) for n > 6 the following conditions are equivalent:

(a) [W]+ = 0 e £ 0 ( Z M ) , (b) the R-bounded homotopy equivalence h : W

>X x R can be made

transverse regular at I x {0} C I x R with the restriction / = h\ : M = h~\X x {0})

> X

a homotopy equivalence, (c) W is homeomorphic t o M x R for a compact (n — l)-manifold M homotopy equivalent to X. See 15.4 below for the surgery-theoretic interpretation. (iii) For any filtered additive category A a band chain complex E in CR(A) is simple chain equivalent to C(l — z : Ffz^" 1 ] >F[z^z~1]) for a finite chain complex F in A if and only if [£]+ = [E]- = 0 G K0(P0(f\)), where simple means r = 0 € Wh(CR(f\)) = W7*(Ci(A)). • Let C R ( A ) Z be the subcategory of C R ( A ) with objects L which are invariant under the Z-action on R L(x) = L{x + 1) (x € R) , with the Z-equivariant morphisms. Let Afz,^"1] be the subcategory of Ci(A) with one object L ^ z " 1 ] for each object L in A, and morphisms the Z-equivariant morphisms in Ci(A) oo

£

jfi : L[2,z-1]-,M[z,z-1}

j=-OO

defined by collections {fj € Horn | j € Z} with {j \ fj ^ 0} finite. The functor A ^ z " 1 ] —> CR(A)Z ; M\z,z-X\ is an equivalence of additive categories.

— M[ Z , 2 - J ]

8. THE LAURENT POLYNOMIAL EXTENSION CATEGORY f\[z,z~l]

69

PROPOSITION 7.6 For any filtered additive category A the map

B : # o (Po(A)) —> A'!(CR(A)2) ; [M,p] — • r(zp + 1 - p : M[z, z" 1 ]—+M[z, z" 1 ]) . w a split injection. PROOF The composite ^(C

B : Ko( is an isomorphism, by 7.2 (i).

• For a ring A and the additive category A = B-^(A) of based f.g. free A-modules there is an equivalence of additive categories with A[z, z 1] the Laurent polynomial extension ring of A. The map B is the original injection due to Bass, Heller and Swan [8] of the projective class group of a ring A in the torsion group of the Laurent polynomial extension A ^ z " 1 ] > KtiAfaz-1]) ; [P] > r(z : P[z,z~l] >P[z,2~1]) . B : K0(A) The If-theory of the polynomial extension category A[z,z -1 ] will be examined in greater detail in §10 below.

§8. The Laurent polynomial extension category A^z"1] The finite Laurent polynomial extension category A[z, z~1] of a filtered additive category A was defined in §7 to be the subcategory of Ci(A) with objects the polynomial extensions of objects L in A oo

L[z,z-X] = Y,

ziL

j=-oo

and with Z-equivariant morphisms. The infinite Laurent polynomial extension category Gi(A)[[z,z~1]] is defined to be a subcategory of G£2(A) containing A[z, z""1] as a subcategory. The Z-equivariant version of the algebraic transversality of §4 is used to construct 'finite Mayer-Vietoris presentations' in Afz,^"1] for finite chain complexes in A[z, 2" 1 ], as subobjects of canonical 'infinite Mayer-Vietoris presentations' in Gi(A)[[z,2r~1]]. This algebraic transversality is an abstract version of the existence of fundamental domains for free cocompact actions of the infinite cyclic group Z on manifolds. Algebraic transversality will be used in §10 to express the torsion groups of A[2, z~l\ in terms of the algebraic if-groups of A.

70

LOWER K- AND L-THEORY

DEFINITION 8.1 (i) The finite Laurent extension of an additive group A is the additive group A ^ z " 1 ] of formal power series

oo

]T) Q>kZk with *=-oo

coefficients a* G A such that {k G Z | a* ^ 0 € A} is finite. (ii) The infinite Laurent extension of a countable product of additive oo

groups A = J | A(j) is the additive group A[[z, z'1]] of formal power j=-oo oo

series

^ *=-oo

a^z with coefficients

«* = n a*toe ^ = n

such that {k G Z | ajt(j) 7^ 0 G A(i)} is finite for each j G Z. D

x

In particular, if A is the additive group of a ring then A[z,z~ ] is also a ring, and if M is an A-module then Mf^,^"1] is an A[z,2;~1 For a f.g. free A-module M and any A-module N 00

For a countable direct sum M = Yl M(j) of f.g. free A-modules j=-oo 00

EomA(M,N) = J ] EomA(M(j),N) , jz=-00

Hom >1[ , fl -. ] (M[z,z- 1 ],JV[z,z- 1 ]) = Homii(Af,JV)[[z,z-1]] . Given an additive category A let Gi(A) = Gz(A) be the Z-graded category. EXAMPLE 8.2 For any ring A let B-^(A) = {based f.g. free A-modules} , B(A) = {countably based free A-modules} . The forgetful functor —> B(A) ; M —+ j=-OO

is an equivalence of additive categories. • For any object L in A let L[z,z~ ] be the object in Gi(A) defined by x

L[z,z-'}{j)

= z*L ( J € Z )

with z^L a copy of L. A morphism / : L[z,z~~l) is homogenous of degree k if

>L'[z,z~1] in Gi(A)

8. T H E LAURENT POLYNOMIAL EXTENSION CATEGORY f\[z,z~x]

(i) for all jj' e 1 with j ' f(j'J)

71

-j^k = 0: ^ I - ^ / l ' ,

(ii) for all i,j £ Z

/(« + *,«) = f(j+k,j) {

= fk : i+k

z L = z'L = L —> z L' for some morphism fk : L >V in A, in which case / is written as / = zkfk :

= zj+kL'

= L'

L[z,z-1]—+L'[z,z-1].

DEFINITION 8.3 The finite Laurent extension Afz,*"1] of an additive category A is the subcategory of Gi(A) with one object L[z, z~l] for each object L in A, with morphisms finite sums of homogenous morphisms

/ = f ; zkfk : LI*,*" 1 ]—•I'!*,*" 1 ], k=-oo

so that

• 1

1

For each object M = L[z,z~ ]inA[z,z~ ] define a homogenous degree 1 automorphism C(M) = z : M

> M

by C(M)i = identity : L > L. 1 The morphisms in A ^ z " ] are the morphisms / : M which are ^-equivariant that is

>M' in Gi(A)

/((M) = C(M')f : M > M' . The finite Laurent extension Atz,^" 1 ] of a filtered additive category A is a subcategory of the bounded Z-graded category Ci(A), namely the subcategory

AM" 1 ] = d(A) z with objects L^z"" 1 ] and the (-equivariant morphisms. EXAMPLE 8.4 For any ring A the finite Laurent extension of the additive category B-^(A) of based f.g. free A-modules is the additive category of based f.g. free A[z, 2:~1]-modules

72

LOWER K- AND L-THEORY

For any ring A the inclusion defines a morphism of rings A\

A

i : A

> A[z,z

-ii

V* > a= ^

]; a

j ajz>

(a if j =0 , aj = i Q . f , Q

j=-oo

r

^

"

Induction and restriction define functors ij : (A-modules)

> (A[z, 2 "^-modules) ,

i' : (A^z^J-modules)

• (A-modules).

1

An A-module L induces an A[z,z~ ]-module

= L\z,z~x\ =

i\L = Afaz-^toAL

j=-oo 1

The restriction I'M of an A[z, 2r~ ]-module M is the A-module defined by the additive group of M with A acting by the restriction of A[z, 2:~1]-action to A C Atz,^" 1 ]. An A^z" 1 ]-module morphism / : L[z^z~1] >L'[z,z~1] can be expressed as a polynomial

OO

OO

i

fc= —oo

=

OO

— ° °fc=—oo

/

with coefficients / j G Hom^(L,L ) (j G Z) such that for each x G L the set {j G Z |/j(z) 7^ 0} is finite. For f.g. L this condition is equivalent to the set {j G Z | fj =^ 0} being finite, but this need not be the case for infinitely generated L. In particular, if L = ii\K for an A-module K then the A[z, z"1]-module morphism p =

V^ jz-'pj : iiL = i\i'i\K

> i\K ; a ® (6 ® x)

>ab®x

3=-co

(a,beA[z,z-1] , xeK) has each coefficient p^ : L

>K non-zero, namely oo

Pj

: L = iuK = J^ k=—oo

oo

zkR

*

K

5 E

0 x

**

' x> •

/:=—oo

For eachfcG Z there is only a finite number of ^ G Z with pj(zkK) (only j = A:), so that

^ {0}

j=-co

with the infinite Laurent extension defined as in 8.1 (ii) with respect to

8. T H E LAURENT POLYNOMIAL EXTENSION CATEGORY f\[z,z~l]

73

the countable product structure oo

EomA(L,K) = U

KomA(zkK,K).

k=-oo

DEFINITION 8.5 For any additive category A the induction and restriction functors are given by oo

ti : A

>f\[z,z-1}', L

> uL = L\z,z~x\ = ] T zjL , j=-oo

r : Af*,*-1] — G ^ A ) ; M = Llz.z-1}

> I'M , (i'M)(j)

= zjL .

The object v(L[z, z~x]) will ususally be written L[z^z~x]. A morphism / :L >L' in A induces a homogenous degree 0 morphism i\f : i\L > i\V in FKlz.z'1) with (ii/)o = / : L >L'. A morphism / : L\z,z~x\ >L'[z,z~l] in A[z,z -1 ] restricts to a morphism in Gi(A) i'f : with 'j,k)

L^z-^^L'^z-1}

= fj-k : L[z,z-l](k)

= zkL-^L'[z,z

*

EXAMPLE 8.6 For a ring A and the additive category A = Bf(A) of based f.g. free A-modules the induction and restriction functors of 8.5 are the induction and restriction associated to the inclusion i : A >A[2,2""1] t'i : A = Bf(A)

>A[2,s"1] = B / (A[z^" 1 ]),

i! : Afz^" 1 ] = &{A[z,z-1])—^Gx(A)

« B(A) . D

2

Let G2(A) = Gz2(A) be the Z -graded category. Given an object L in Gi(A) let Lfz,*"1] be the object in G2(A) defined by

Lfoz-ifok)

= zkL(j)

0\*€Z).

For any objects L, V in Gi(A) use the countable direct product structure oo

oo

HomGl(fl)(I,I') = I ] ( E nomn(L(j),L'(j'))) to identify the infinite Laurent polynomial extension

74

LOWER K- AND L-THEORY

with the image of the injection HamGl(A)(L, !/)[[*, a"1]] — H o m e ^ M * , * - 1 OO

E *=-oo

defined by i'> *').(;>*)) = fk'-k(j'J)

:

DEFINITION 8.7 The infinite Laurent extension Gi(A)[[z,2:""1]] of Gi(A) is the subcategory of G2(A) with one object L[z,z~1] for each object L in Gi(A), and morphisms

EXAMPLE 8.8 The additive category B ^ z , * " 1 ] ) of countably based free A-modules is such that the forgetful functor is an equivalence. D oo

A morphism / =

^

z* fj : M

1

>M' in Gi(A)[[z,2:~ ]] is homoge-

j=-oo

nous of degree k if fj = 0 for j ^ k.

A morphism / : M

>M' in

1

G^A)!^,^- ]] is linear if fj = 0 for j ^ 0,1, so that

/ = fo+zfr

: M

> M' .

In the applications it is convenient to introduce a change of sign, writing linear morphisms as

f = U-zf-

:M

> M'

with /+ = / 0 , / _ = - / i . The Laurent polynomial extension category A[2, z~l] will be identified with the full subcategory of Gi(A)[[^,2:~1]] with objects L[z, 2"1] such that L(j) = 0 for j ^ 0. Also, CoCA)^,*"1] is identified with the full subcategory of Gi(A)[[2:,2:~1]] with objects Z/fz,^"1] such that {j G 0} is finite. DEFINITION 8.9 A Mayer-Vietoris presentation of an object M in A^,^" 1 ] is an exact sequence in Gi(A)[[ in'M

such that

i,i'M = Y, E j= — oo k= — o

Define a morphism in p(M) = Yl

zJ

P(M)i

• W'M —* M =

ilL

5

j=-oo

by p(M)j

= jth projection : i'M =

N

(>{M) L

>L ;

k=-oo

jr k=-oo

DEFINITION 8.10 The exact sequence . M(cx)) : 0

>• in'M

P(M)

>• in'M

>M

> 0

is the universal Mayer-Vietoris presentation of M in D

Let Z+ = {r e l\r > 0} U {oo}, and let N be the lattice of pairs (iv+,iv-)ez+ xz+, with (NUN')*

= m a x f ^ , ^ ) , (N n N')± = m i n ^ , ^ ) .

N has the maximum element oo = (oo; oo). An element N = (AT+,iV~)

76

LOWER K- AND L-THEORY

G N is finite if iV"+ ^ oo and N~ ^ oo. Let N-^ C N be the sublattice of finite elements. PROPOSITION 8.11 (i) For every object M = i\L in A^,*" 1 ] and every N = (iV + ,iV~) G N there is defined a Mayer-Vietoris presentation of

M(N)

: 0

f() g(N) > Af'^Iz.z- 1 ] ——-• M ' ^ z , * " 1 ] ——U M

•0

with M'(N)

=

= f+(N)-zf-(N) : f+(N) : M"(N) —> M'{N) ; M"(iV) -—+ M'(N) ;

9(N)j

: M'(N)

M"(N)[z,z-1]^M'(N)[z,z-1), ^

zfcxfc — k

xk

J]

z*xA ,

> h=-N+ ^2 +l

-

(ii)//AT e N is fini^e i\iE

, > i,rE

P(E) *E

>0

78

LOWER K- AND L-THEORY

associated by 8.12 to the maximal element oo G N(E). The universal property is that for any chain map / : E' >E of finite chain complexes in f\[z, z~x\ and any Mayer-Vietoris presentation E' of E' there is defined a unique morphism / : E' >E(oo) resolving / . DEFINITION 8.14 (i) For any filtered additive category A let B : tfo(Po(A)) —> KxWz.z-1])

;

[M,p] — • r(zp + 1 - z : M ^ J T 1 ] — • M [ z , * " 1 ] ) ,

Bo : tfo(A) — tfo(Po(A)) — . ^ ( A ^ , ^ 1 ] ) ; [M] —> r(z : M ^ " 1 ] —•MI*,*" 1 ]) . (ii) The Whitehead group of A[z, z" 1 ] is defined by Wh{f\[z,z-1])

= cokei(B0 : K0(f\)

^(Af*,*"1])) . •

For a ring A and the additive category A = B-^(A) of based f.g. free A-modules this is the original injection due to Bass, Heller and Swan [8] of the project ive class group of a ring A in the torsion group of the Laurent polynomial extension A^,,?" 1 ]

B : K0(A) —> KtiAfoz-1]) ; [P] — r(z : P^,*- 1 ]—P^z" 1 ]) . The Whitehead group of the polynomial extension category

is z" 1 ]) - coker(B0 : A'o(A) since ifo(A) is generated by [A]. (If A is such that the rank of f.g. free A-modules is well-defined then Ko(f\) — Z). The Laurent polynomial extension of a group ring A = Z[TT] is the group ring

Afoz-1]

= Z[ixZj,

and the actual Whitehead group of ?r x Z is Wh(nxl)

=

K1(l[TrxZ})/{T(±zjg)\geir,jel}

= Wh(Bf(l[ir})[z,z-1])/{r(g)\g e^r} . A chain complex band E in A[z, z~l] is a finite chain complex such that the restriction vE is Co(A)-finitely dominated in Gi(A), so that E is a chain complex band in Gi(A) and the end invariants [E]± G ifo(Po(A)) are defined as in 4.10. EXAMPLE 8.15 Let X be a finite n-dimensional CW complex with fundamental group 7Ti(X) = 7T x Z, so that Z[TTI(X)] = Z[7r][z,2"1]. The

79

8. T H E LAURENT POLYNOMIAL EXTENSION CATEGORY

cellular chain complex of the universal cover X is a finite n-dimensional chain complex C(X) in B^(Z[TTI(X)]). The Mayer-Vietoris presentations of C(X) constructed in 8.12 are obtained by an algebraic transversality which mimics the geometric codimension 1 transversality for maps p : X >SX inducing p* = projection : ^i(X) = TT X Z • ^lC^1) = Z . Let X = X /TT be the infinite cyclic cover of X classified by p, and let £ : X *X be a generating covering translation. If X is a compact n-dimensional manifold it is possible to choose p : X •S 1 transverse regular at a point * € X, so that V — p -1 ({*}) is a codimension 1 framed submanifold of X, and cutting X along V defines a compact cobordism (V; 17, CJJ) between U and a copy (JJ of U such that oo

X - |J j=-oo

Cj+2U Thus codimension 1 manifold transversality determines a finite MayerVietoris presentation of C(X)

0 —* C{V)[z,z-x\

C(X) — 0 ,

with {/, V the covers of 17, V induced from the universal cover X of X. More generally, for any finite CW complex X it is possible to develop a codimension 1 CW transversality theory using the mapping torus of ( T(C) = X x [0,l]/{(*,0) = ( C(U)[z,z-l\

f+ zf

, CiV^z-1}

~ ~

—» C(X) — 0 .

In particular, for Y = X V = X x [ 0 , l ] , U = X , TY(Q = T(C) , corresponding to the universal Mayer-Vietoris presentation 0 —> C(X)^,^- 1 ]

1-zC1

> C(X)^,^- 1 ] —> C(X) —> 0

with £ : X >X a 7Ti(X)-equivariant lift of £ : X »X. Subfundamental domains Y C. X can be constructed in exactly the same way as the subcomplexes E'(N) C ixE used to define E(N) in 8.12. Let J r (0 < r < n) be an indexing set for the r-cells e r C X, so that

X = Ue°u|Je1U...u|Jen, h

h

In

and choose a lift e r C X for each cell e r C X so that

X

= U j=-OO Jo

Jl

/„

For any element N - (N such that

r c x(-2) u j=-iVr

+ J

r

U

U < igr " 1 (!M' >M >0 is exact in A. The nilpotent class group Nilo(A) of an additive category A is the class group of Nil(A) Nilo(A) = #0(Nil(A)) . The inclusion i : A

> Nil(A) ; M

> (M,0)

is split by j : Nil(A)

> A ; (Af,i/)

> M .

The reduced nilpotent class group Nilo(A) is defined by Nilo(A) = coker(i : K0(f\)

>Ni

and is such that Nilo(A) =

KO(A)©NT1O(A) .

82

LOWER K- AND L-THEORY

EXAMPLE 9.1 The Nil-groups of a ring A are the Nil-groups of the additive category B^(A) of based f.g. free A-modules Nilo(A) = Nilo(B/(A)) , Nilo(A) = NTI O (B / (A)) . D

A finite chain complex C in A and a chain homotopy nilpotent chain map v : C >C have an invariant [C, u] G Nilo(A), which will now be denned. In fact, Nilo(A) will be expressed in terms of such pairs (C, i/), by analogy with the following expression of Ko(f\) in terms of finite chain complexes C in A. Let KQ (A) be the abelian group with one generator [C] for each finite chain complex C in A, subject to the relations (i) [C] = [C] if C is chain equivalent to C", (ii) [C] — [C] + [Cn] = 0 if there is defined an exact sequence in A > C" >C > C > 0. 0 For any finite chain complex C in A there is defined an exact sequence of finite chain complexes in A 0 with (7(1 : C

> C

> C(l : C

>C)

> SC

> 0

>C) contractible, so that [SC] = -[C] e ^ ^ ( A ) .

PROPOSITION 9.2 The natural maps define inverse isomorphisms ^ ^ ( A ) ; [M] — [M] , —> K0(t\) ; [C] —> [C] . The morphisms are defined by regarding objects M in A as O-dimensional chain complexes, and by sending finite chain complexes in A to their class. PROOF Let KQ (A) (n > 0) be the abelian group defined exactly as KQ (A) but using only n-dimensional finite chain complexes in A. In particular, ^ 0 ) ( A ) = K0(FK). An n-dimensional chain complex C is also (n + 1)-dimensional, so there are defined natural maps tf(»)(A) —» A^ n+1) (A) ; [C] —» [C] (n > 0) . Given an (n + l)-dimensional chain complex C let C be the n-dimensional chain complex d d f C : C n +i —> Cn — • ... — • C2 —* C\ ,

9. NlLPOTENT CLASS

83

and let C" be the (n + 1)-dimensional chain complex

u\ d

\0J (dl) C : Cn+i > Cn > . . . — • C2 > C\ © Co > Co . Now C" is chain equivalent to SC, and there is defined an exact sequence 0

> C"

>C

> SC0

> 0,

so that

[C] = [C"\ - [SCo] = [SC] - [SCo] = [Co] - [C] e K(on+1\f\) Thus the natural map is an isomorphism .Kg (A)

>Ko

.

(A) , with

inverse A^ n+1) (A) —, K{on\l\)

; [C] —

[Co] - [C] (n > 0) . D

Define NUQ (A) to be the abelian group with one generator [C, v\ for each finite chain complex C in A with a chain homotopy nilpotent self chain map v : C >C, subject to the relations (i) [C, v] = [C, v'\ if there exists a chain equivalence / : C >C such that fu - I/1/ : C

• C" ,

n

(ii) [C, 1/] — [C, i/'] + [C", v ] = 0 if there is defined an exact sequence 0 —> (C",u") —

(C',1/') — * ( O ) —» 0 .

The suspension 5(C, 1/) = (SC,v) is such that there is defined an exact sequence 0 —> (C, 1/) —> (C(l : C—>C), 1/ © 1/) —> 5(C, 1/) —> 0 with C(l) contractible, so that [ S ( C » ] = -[C,^] € The reduced Nil-group defined by N i l ^ A ) = coker(i : A ^ ^ is such that

Nii°°\f\)

= A' Nil^CA) ; [M,u] —» [M,u] , NTlo(A) — • miT\f\)

; [M,v\ —» [Af.i/] .

PROOF In view of 9.2 it is sufficient to consider the absolute Nil-groups.

84

LOWER K- AND L-THEORY

Let Nilg (A) (n > 0) be the abelian group defined exactly as Nilg° (A) but using only pairs (C, v) with C an n-dimensional chain complex in A. In particular,

Nitf°(A) = Nilo(A) . The proof that the natural maps Nil o n) (A) ^Nil^ n+1) (A) (n > 0) are isomorphisms proceeds by analogy with the proof of 9.2. Given an (n + 1)-dimensional pair (C, v) let N > 0 be so large that there exists a chan homotopy rj : uN ~ 0 : C »C, so that 77 : Co >C\

is such that vN

= drj : Co

> Co .

Let (M, /i) be the object of Nil(A) defined by /0 1 0 ...\ 0 0 1 ... Co = C o 0Co0...©Co : M = 0 \ : N-l

M

= ^

Co = Co © Co © . . . © Co .

Also, let e, / be the morphisms in A defined by e = (vN~l vN~2 . . . i/) : M = Co © C o © . . . © C o / = (ry 0 . . . 0) : M = Co © Co © . . . » Co

> Co ,

> Ci ,

1

and let ( C , v ) be the (n + 1)-dimensional pair defined by

C

: Cn+1

y Cn

> ...

(0) y C2 — A

e) d®M

Co ,

„' = v : c; = cr —> c; = c r (r ^ 1), : CJ = Ci 0 M —>• C[ = Ci@M The iVth power of v' : C[ type

v,N =

>C[ has an upper triajigulax matrix of the

/^ r \ . c, = C i 0 M ^ c /

with /zN = 0. The chain map u" : C" v" = ( ( 0 ^ O ^ )

.

: CI =

10 : c ; —-> c ; if r ^ 1

>C" denned by ^1®^ —

^

9. NILPOTENT CLASS

85

is nilpotent, with v"2 = 0, and there is denned a chain homotopy r/©0 : v'N ~v"

: C"

> C" .

Powers of chain homotopic self chain maps are chain homotopic, so that 2 1 >C is chain vi2N = (uiNy i s c h a i n homotopic to i/" = 0, and v : C" homotopy nilpotent. The morphism /0\

;) — S(C",i>") —» 0 with (C",i>") the n-dimensional pair defined by C? = C^+1 (0 < r < n) , v" = u1 . There is also defined an exact sequence It follows that

• The inclusion A >P0(A);M reduced nilpotent class groups Nilo(A)

>(M, 1) induces an isomorphism of

> NII O (PO(A)) ; [M,i/]

> [(M,l),i/]

with inverse Nilo(Po(A))

> NT1O(A) ; [(M,p),i/]

> [Af,i/] .

These isomorphisms will be used to identify Nilo(Po(A)) = NT1O(A) . Given a pair (C, v) with C a Co(A)-finitely dominated chain complex

86

LOWER K- AND L-THEORY

in Gi(A) and v : C >C chain homotopy nilpotent it follows from 9.3 that there is defined a reduced nilpotent class invariant [ O ] € Nilo(Po(A)) - Niio(A). This will be used in §10 below.

§10. A-theory of A^,*"1] The splitting theorem of Bass, Heller and Swan [8] and Bass [7] for the torsion group of the Laurent polynomial extension A[z, z~x] of a ring A

IdiAfaz-1]) = K1(A)®Ko(A)®mo(A)®mio(A) will now be generalized to the torsion group of the finite Laurent extension A ^ z " 1 ] of a filtered additive category A Kii^z-1]) = Jri(A)©ff o (Po(A))eNao(A)©ffiio(A). The proof makes use the Mayer-Viet or is presentations of §8 and the nilpotent objects of §9 to obtain a split exact sequence B © N+ © ATit 0 > Kiso(f\) —> K[ao{l\[z,z''1]) > A^o(Po(A)) 0 Ifiio(A) ® Nilo(A) —> 0 and the analogue with K{ replace by K\ i\ B®N+®N0 > A^(A) — - K^z^z-1]) > so

A^o(Po(A)) © Nilo(A) © Nilo(A) > 0, as well as a version for the Whitehead torsion and reduced class groups ti B © iV+ © N-. 0 —> Wh(f\) —> Wh{l\[z,z-1]) > Ao(Po(A)) © Nllo(A) © NT1O(A) —> 0 . Given an object L in A and j £ Z define objects in Gi(A)

k=j

CjL~ = C\L[z,z-l]T = £

zkL.

k=-oo +

The projections onto &L and &L~ define morphisms in Gi(A) = lffiO : L^z-1] = (jL+ p P+ ,

There are two distinct ways of splitting i f i ^ A ^ z " 1 ] ) , as the algebraically significant direct sum system Zt

3\ B@N+®N_ iiro(Po(A))®Nilo(A)©Nilo(A)

88

LOWER K- AND L-THEORY

and the geometrically significant direct sum system i\ -t

3\ B®N+®N,

""" tfo(Po(A))®Nuo(A)©Nilo(A)

Similarly for K\ instead of K\'°. The algebraically significant splitting j< of i\ is induced by the functor j

:

Alz.z" 1 ] — A ; M = L[z,Z-1] — j,Af = £ ,

k=-oo

DEFINITION 10.2 (i) The algebraically significant injection £ e i V + © J \ L : lfo(Po(A))©Nrio(A)©i«io(A)

>

K{90(t\[z,z-1

is the splitting of B © N+ © iV_ with components B([P]) = rCsiPfz.z- 1 ]—PI*,*" 1 ]), iV+[P,i/] -

ra-C^-lJi/rPlz.z-1]—^P^z"1]),

JV_[P,,/] =

TCI-CZ-IJI/IPIZ.Z-1]—^Plz,^1])-

(ii) The geometrically significant injection splitting of B © JV+ © A^_ B# © iV; © iVi : A'o(P0(A)) © NTIO(A) © NTIO(A)

> ^i' 0 (A[^, ^~ 1

has components B'([P}) = rC-ziPlz.z- 1 ]—fPlz.z" 1 ]), N'+[P,u] NL[P,v] =

rO-z-^^Iz.z-1]—>P[z,z-1]), TCI-ZI/^IZ.Z-1]—.Plz.z"1]).

REMARK 10.3 For a ring A and the additive category A = Bf(A) of based f.g. free j4-modules the algebraically significant direct sum decomposition of the automorphism torsion group Ki(f\[z,z~1]) =

10. ^-THEORY OF Af*,*" 1 ]

89

B®N+®N"* K0 B © iV+ © Nis the original decomposition of Chapter XII of Bass [7], with j \ K\(A[z, z-1\) >K\(A) induced by the surjection of rings oo

1

oo

k

j : Alz.z' ]

> A ; ]T akz

> ] T ak

splitting the inclusion i : A >A[z, z~x\. The relative merits of algebraic and geometric significance have already been discussed in Ranicki [66]. The geometrically significant direct sum decomposition of the Whitehead group of a product TT x Z Wh(ir)
T ( - * : Pf*,*" 1 ]—>P[*,;T 1 ]) which was identified in [66] with the split injection defined geometrically by Ferry [27], sending the finiteness obstruction [X] G ifo(Z[?r]) of a finitely dominated CW complex X with fti(X) = n to the Whitehead torsion B\[X])

= r(l x - 1 : X x 51

^X x S1) G

W/I(TT X

Z) .

It is also geometrically significant that the image of B' : i^o(Z[7r]) • Wh(7r x Z) is the subgroup of the transfer invariant elements, as will be explicitly verified in §12 below.

•

90

LOWER K- AND L-THEORY

The objects P± of 10.1 fit into direct sum systems in Po(Gi(A)) f"1 f PC—M+f

(L-,1) ,

(CM-,1) ,

P- .

The nilpotent endomorphisms v± : P± >P± of 10.1 fit into endomorphisms of exact sequences in Po(Gi(A))

c ff

> L+ —I—v CaM+

0

•v

o

c

—-—>

> P+ > P-

CM~

-1

>0 »• o

v-

and are such that = 0 : P + —> P+ , (v-)

t+tl

= 0 : P_

> P- .

PROPOSITION 10.4 For any object M — L[z,z-1] in A ^ ^ " 1 ] and any N = (N+,N~) € N ' the torsion T(M(N)) € K\so{i\[z,z-1)) of the finite Mayer-Vietoris presentation M{N) of M is such that (B © N+ © iV_)r(M(AT)) = ([ i=-N+ G Jfo(Po(A)) © Nilo(A) © Nno(A) . PROOF The map #(iV) in the exact sequence M(N) : 0

> M"{N)[z,z-1]

f(N)

> M'(iV)^,^" 1 ] • M

is split by the homogenous degree 0 map h(N) : M = L[z,z~

> 0

10. /f-THEORY OF A^,*" 1 ]

91

defined by N~

h(N)0

= inclusion : L

> M'(N) =

zkL ,

^ k=-N+

The isomorphism / = (f(N)h(N)) : M"(N)[z,z-l]®M is such that T(M(N))

• M'iN^z.z'1]

= rf/JGCfAlv" 1 ]).

The nilpotent objects (P±,^±) associated to / in 10.1 are given up to isomorphism by I/+ : P+ =

Y" z*L —* P+ ; j=-N+

V

^ ^—

j=-N+

Y

z^xj

j=-N+

P_l- and P_ fit into exact sequences in Gi(A) P+ / 0 , M"{N)+ 0 L+ —> M'(iV) + > P + —> 0 / P0 > M"(N)~ © I T — • (M'(N)- —> P_ > 0 with p+, p_ the projections. The matrices of i/+ and V- are upper triangular, so that [P+,i/+] = [P_,i/.] = OGNUo(A). Thus the components of (B © iV+ © JV_)r(M(iV)) are given by Br(M(JV)) = Br(f) = [/]+ = [P+] € Jfo(Po(A)) , = N±r(f)

= [P±,v±] = 0 G Nilo(A) .

REMARK 10.5 The algebraic K-theory splitting theorem of Bass, Heller and Swan [8] used the linearization trick of Higman [34] to represent every element of K\(A[z, z~1]) as the difference of the torsions of linear automorphisms. Atiyah [6,2.2.4] proves the Bott periodicity theorem in topological if-theory by applying the linearization trick to polynomial clutching functions of bundles. See Bass [7, IV] and Karoubi [39, III.l] for the connection between the algebraic A"-theory of polynomial extensions and Bott periodicity. D

so

x

For any additive category A every element of K\ (J\[z, z~ \) is the difference of the torsions of linear isomorphisms:

92

LOWER K- AND L-THEORY

PROPOSITION 10.6 An isomorphism in A^,^" 1 ] t j=-s

determines a commutative diagram of isomorphisms in Afz,^"1] (1

0\

f=(e'i) f" =

(M © with f

and f" linear, so that r(f)

=

T(f')-r(f")eKr(l\[z,z-1}).

The nilpotent objects in Po(A) associated to f', f, f" are such that

J

zkM

zkxk t-1

u" : Pi = ik=0

ik=0

and Br(/') = [P+] , Br(f")

= [P?] € ffo(Po(A)) ,

iV±r(/') = [P ± ,i/ ± ] , N±r(f")

=

OGNTIO(A)

.

PROOF Define a contractible 1-dimensional chain complex E in f\[z, z"1] by dE = f : Ex = L[z,z-1] —» £J0 = M ^ , ^ 1 ] . Let iV = (6,0; M"[z, z-1) ' ~ *+

93

"~> M'[z, z-1]

'I

%

> M[z, z-1)

-

>0 .

Passing from C0(A) to A note that E'(N) and E"(N) are 1-dimensional chain complexes in A with

E"(N)1 = 0 , E'iN^

= L,

f+(N) = i_|_ = inclusion : t

E"(N)0 = M" = f-(N)

t

Z M

* —> E'(N)0 = M' = J ] zkM ,

J2 k=-s+l -1

k=-s

= i_ = £ (inclusion) : t

E"(N)0 = M" =

t

^

^ ^ — ' ^ ' W o = M' =

g(N)j = i'j = jth projection :

= M' = V **M —> Eo = M ; t

d>E'(N)

=

e

' =

zk k :

z2

f

k=-s

The homogenous degree 0 morphism h : M[z, a:"1]

>M'[z, z~x] defined

by t

h0

= inclusion : M

> M

1

=

V^

Z

*M

k=-s

splits i' : M'[z,z~l]

>M[z,z~1]. The morphism denned in Afz,^"1] t

e =

V^ k=-s

zke

k

'• L[z,z~x]

> M'^z^z"1]

94

LOWER K- AND L-THEORY

* M" j=0 0

M"

I j=-3-k

if k > 0 if k < - 1

is such that it =

THEOREM 10.7 The isomorphism torsion group K[so(f\[z, z~1]) fits into the split exact sequence 0

i\

.

£0iV+0iV_

> K[30(f\) —> K^iPiiz.z-1})

>

A^o(Po(A)) 0 Nilo(A) 0 Nilo(A)

> 0,

with B the composite iso

-l

B

riso

induced by the inclusion A ^ z " 1 ] C Ci(A). Similarly, the automorphism torsion group K\(f\[z^z~1]) fits into the split exact sequence B i(A)

A'o(Po(A)) 0 Nilo(A) 0 Nilo(A)

> 0,

and the Whitehead group VF/^A^z" 1 ]) fits into the split exact sequence N+ © iV_ i (A)

^o(Po(A)) © Nilo(A) © Nilo(A) —> 0 . PROOF For any linear isomorphism in A[-j,0-1]

there is defined a commutative diagram of isomorphisms in Po(A)[z, z ~l\

95

1 0 . JOTHEORY OF

0

1-zv-J

with : M

•-(%):L~P.*rthe isomorphisms in Po(A) defined by fe'+ =

PL-/"

fe^[. = ^

-1

1

: M

> P+ , fe'_ = P M + Z "

(inclusion) : L

1

: M

> P-

>- P4. , h"_ = inclusion : L

,

> P-

The torsion of / can thus be expressed as

JV;[P+,i/+] + NL[P-,v-] € ^r(A[z,z- x ]) with (P±,i^±) ^ n e nilpotent objects associated to / in 10.1 and fe = fe'~1fe" : L >M an isomorphism in A. By 10.6 every element of K[so(f\[z, z~l\) can be expressed as a difference of the torsions of linear isomorphisms in A[z, z" 1 ], thus establishing the geometrically significant

96

LOWER K- AND L-THEORY

direct sum system

Ko(Po(f\)) © Nilo(A) © Nilo(A) . B' © N'+ © N'_ The geometrically significant surjection jl splitting it is thus given for linear isomorphism / : . L ^ z " 1 ] ^ M f z ^ " 1 ] by

Define abelian group morphisms u : ifo(Po(A)) —» ^r°(Po )) = r ( l - 2p : M ^ M ) , H : Nflo(A) • Kiso(f\) ; [Af,i/] » T(1 - 1/: M >M) . The splitting maps in the algebraically significant direct sum system Kia°(f\) 3\

B ^o(Po(A)) © Nilo(A) © Nilo(A) B © JV+ © Nare related to the geometrically significant splitting maps by B'

= B +u

, N'± = N± + / A ,

j[ = j \ + BLJ

so that the algebraically significant direct sum system has also been established. For the automorphism torsion groups note that for an automorphism

in Afz,*-1] the objects M ' , M " in the commutative diagram of 10.5

(L © M")[z,z-1)

.(L®M")\z,z-l\

f 0 0

f'=(e'i)

1

{M@M")[z,z-i]

f

"-(hi">

10. K-THEORY OF A^,*" 1 ]

97

axe defined by t

M' = ^

t zJM

'

M

" =

and there exists a homogenous degree 0 isomorphism in A ^ z " 1 ] : M'[z,z-1)

> (MffiM")!*,*" 1 ] .

Thus <j>f and f" are linear automorphisms such that

r(f) =

Titn-ritneKii^z-1)),

and every element of Ki(f\[z1z~1]) is the difference of the torsions of linear automorphisms in Afz,^" 1 ]. The verification of split exactness now proceeds as for the isomorphism torsion groups. • PROPOSITION

10.8 The projection

B®N+®N-

: Ki9O(f\[z, 2"1])

> A^0(P0(A)) © Nilo(A) 0 Nilo(A)

so

1

sends the torsion r(E) G K{ (f\[z1z~ ]) of a contractible finite chain complex E in f\[z^z~1] with Er = i\Fr to

e Ko(Po(A)) © Nilo(A) © NT1O(A) , with i/+, v- the chain homotopy nilpotent self chain maps of the C 0 (A)finitely dominated chain complexes £~N E~*~, £N E~ in Gi(A) defined for any N G Nf(E) by

j=-N+ oo

oo

j=-oo

-i — E *i"1*;' j——o

r=0

, = _/V +

98

LOWER K- AND L-THEORY

the 'positive end invariant (4-10) of E regarded as a chain complex in

PROOF Applying the torsion sum formula of 5.1 (ii) to the exact sequence of finite chain complexes in f\[z, z~l]

E(N) : 0-*E"(N)[-1}

f = f+(N) J + -K zf'

gives

(E) =

T(f)r=0

The sign term 0 does not contribute to (B © N+ © N-)T(E),

since

= ker(B © N+ © N- : ^{"(A^.z- 1 ]) —>A'o(P o (A)) © NT1O(A) © Niio(A)) .

From 10.4 -l

j=-N+

e A^o(Po(A)) © ffilo(A) © Nilo(A) .

The linear chain equivalence / fits into a commutative diagram of chain equivalences in

10. if-THEORY OF

f\[z,z~l]

99

(1-*-*»+

0

E"(N)[z,z with ft', ft" the homogenous degree 0 chain equivalences with components h'+ = z-1 (inclusion) : E'(N)

> C " 1 ^ ' ^ ) = (~N+E+ , = (N~E~ ,

h'_ = inclusion : E'(N)

• E'{N~)

h'[ = inclusion : E"{N)

> E"{N+) = (~N+E+ ,

h"_ = inclusion : E"(N) so that

> E"(N-)

= tN~E~ ,

(S©iV + ®iV_)r(/) = ([CN+E+}+,[CN*E+,»+},[(»-

E-,v.])

e A^o(Po(A)) © Nilo(A) © NT1O(A) .

REMARK 10.9 The following conditions on a contractible finite chain complex E in A[2, z" 1 ] are equivalent: (i) T{E) e im(ti : Wh(f\) ^W^A^^" 1 ])), _ _ (ii) (B © 7V+ © N-)T(E)

= OeA'o(Po(A))©Nilo(A)©Nilo(A) ,

(iii) there exists a finite Mayer-Vietoris presentation of E : U —> hi [z,z

\ —> hi \z,z

J —> hi —> u

with E' and E" contractible in A and r(E) = 0 € so that T(E)

= i\(r(E') - T(E")) e im(t! : Wh(f\)

This is the abstract version of the codimension 1 splitting theorem of

100

LOWER K- AND L-THEORV

Farrell and Hsiang [24], which in its untwisted form states that a homotopy equivalence / : M >X x Sx of compact n-dimensional manifolds is such that T ( / ) € im(i. : Wh(n) >Wh(n x Z)) = ker( B&N+&N-:

Wh(n x Z)

^

H) e NUO(ZM) )

if (and for n > 6 only if) / splits, i.e. is homotopic to a map (also denoted by / ) which is transverse regular at I x {*} C l x S 1 and such that the restriction is a homotopy equivalence of (n — l)-dimensional manifolds. The components of the splitting obstruction (B © iV+ ® N-)r(f) are given by Br(f) N+r(f)

= [M-] = \i'-C(M)/CN+C(M)+] =

<E K0(l[*]) ,

\i'C(M)/CN+C(M)+,(],

for any N = (N+,N~) £ N'(C(M)), with M = f*(X x R) the pullback infinite cyclic cover of M. By duality X 1 _ m (Po(A[z^- 1 ])) —+ K-m(f\)

> 0,

exactly as in the original case A = B^(A). As in §7 define the polynomial extension A [z] of an additive category

102

LOWER K- AND L-THEORY

A to be the additive category with one object =

L[z]

k=0

for each object L in A, and one morphism

/ = f V / * : L[z]—*L'[x] for each collection {/* G Horn A (£,£') | k > 0} of morphisms in A with {fc > 0|/* ^ 0} finite. Regard f\[z] as a subcategory of Ci(A) with objects M such that u;

r if i < o.

\o

The functor j+ : f\[z] —* A M " 1 ] ; L[z] — • i ^ . z " 1 ] defines an inclusion of f\[z] as a subcategory of A[2,2""1]. The polynomial extension Af^"1] is defined similarly, with one object

k=-OO

for each object L in A, with morphisms

*=-oo

and with an inclusion The inclusions define a commutative square of additive functors A

^±

>A[*1

i-

Given a functor > {abelian groups} f : {additive categories} define the functor > {abelian groups} ; A > LF(f\) LF : {additive categories} by = coker(0 + j _ ) :

1

1

11. LOWER K-THEORY

103

DEFINITION 11.1 The lower K-groups of an additive category A are defined by tf-m(A)

= LmKo(Po(f\))

(m > 1) . D

Following Bass [7, p.659] define a functor > {abelian groups} F : {additive categories} to be contracted if the chain complex

0

> F(A)

> F(f\[z}) ( 3+3

~ > J^A^,*" 1 ]) — • LF(f\) — • 0 has a natural chain contraction, with B the natural projection. 11.2 The functors : { additive categories) > { abelian groups] ; A

PROPOSITION m

L K\

> Lm

are contracted. For m > 1 there are natural identifications ^K^fK) = Kt-rniPoW) and for m > 2

= ffi(Cm(A)) = JJTo(Pm-l(A))

PROOF Consider first the case m = 0. Working as in the proof of 10.7 there are defined naturally split exact sequences t+ N-j+ _ 0 —> ^ ( A ) —> /fi(A[«]) • Nilo(A) —-» 0 , *_ N+j_ > Nilo(A) —> 0 0 —> Ki(K) —* K!(l\[z-1]) and hence a naturally contracted chain complex

0

so that For m > 1 replace A by C m (A) and use the identifications of filtered additive categories 1

] , C ro (A[z,z" 1 ]) -

C m (A)[z,z- x ]

104

LOWER K- AND L-THEORY

to obtain a naturally contracted chain complex

K0(Pm(l\))

—» 0

which can be written as 0—

tfo(Pm-l(A))

—-» ifolPm-HflW^ifolPm-llfllr 1 ]))

—» JfoCPm-lCA^.Z-1])) — tfo(Pm(A)) —» 0 . Thus the functor F : A >Ko(Pm-i(Fk)) is contracted with LF(f\) = K0(Pm(A)) = tf_m(A) . n EXAMPLE 11.3 Given a ring A let A[z] (resp. A^" 1 ]) be the subring oo

of Af^,^" 1 ] consisting of the polynomials ]>^ akZk (resp. k=0

0

]T] Q>kzk) k=-oo

such that cik = 0 for k < 0 (resp. A: > 0). For the additive category A = Bf(A) of based f.g. free A-modules f\[z) = B'(A[z]) , A^- 1 ] = BfMz-1]) , and the functors *± : A—.AI**] , j± : A[^] —-» A ^ z " 1 ] are induced by the inclusions of rings , j± . A I z * ] — • ^ . z " 1 ] . i± : A—+A[z±] Bass [7] defines a functor > {abelian groups} F : {rings} to be contracted if the functor LF : {rings} • {abelian groups} ; A • LF(A) defined by LF(A) = coker(0 + j _ ) : F{A[z\) ® is such that the chain complex (_•;) 0

y F(A)

• F(A[z])

—-> F(A[z,z-1]) —> LF(A) —> 0 has a natural chain contraction, with B the natural projection. The "fundamental theorem of algebraic if-theory" ([7]) is that the functors Ki-m ' {rings} • {abelian groups} ; A > K\-m(A) (m > 0)

11. LOWER K-THEORY

105

are contracted, with natural identifications LKt-miA) = K-m(A) . This is the special case of 11.2 with A = B / (A). • For m > 1 define the m-fold Laurent polynomial extension of an additive category A inductively to be the additive category

A[Zm] = AIZ-^Hw- 1 ] , A[Z] = l\[zuz?\ , and write f\[z1,zr1,z2,z^1,...,zmjz-1]. A[Zm] = m Alternatively, A[Z ] can be viewed as the subcategory of the bounded Zm-graded category Cm(A) = CZm(A) with one object M[lm] for each object M in A, graded by M[lm](jl,J2,

• • • , Jm) = Z»Z»

...ztM

,

m

with the Z -equivariant morphisms. THEOREM 11.4 The torsion group of the m-fold Laurent polynomial extension of A is such that up to natural isomorphism

(7)

*=0 ^

'

t=l

with Nil^A) = coker(A^(A) >NiU(A)) , NiU(A) = lf*(Nil(A)) PROOF Iterate one of the splittings of §10 (As usual, there are algebraically and geometrically significant splittings.) D

For a ring A and A = B^(A) there are evident identifications A[Zm] = B/(A[Zm]) , NiU(A) = NiU(A) , NTI,(A) = Nil(A) , with A[Zm] = A[z\,z^ 1,z2,Z21,... ,zm,z^} the m-fold Laurent polynomial extension of A. The KQ- and K\-groups of Z are - Z , ^j(Z) = Z2 . EXAMPLE 11.5 The lower if-groups and the lower Nil-groups of Z are K.m(l) - Nil_m(Z) = Nll_m(Z) = 0 (m > 1) , by virtue of the computation Wh(Zm) — 0 of Bass, Heller and Swan [8].

106

LOWER K- AND L-THEORY

§12. Transfer in X-theory The lower AT-group -RTi_m(Po(A)) (= ifi-m(A) for ra > 2) of an additive category A will now be identified with the subgroup Ki(f\[Zm])INV of the Tm-transfer invariant elements in Ki(f\[lm]). This is significant because the reduced lower K-group ifi_m(Z[7r]) (= ifi_m(Z[7r]) for m > 2) arises geometrically as the Tm-transfer invariant subgroup of the Whitehead group Wh(7r x Z m ), in connection with the 'wrapping up' procedure for passing from Rm-bounded open n-dimensional manifolds with fundamental group TT to closed (ra + n)-dimensional manifolds with fundamental group n x Z m . In the first instance consider the case ra = 1. For each integer q > 1 let q :

5 1 —+ S1 ; z = [t] —» z" = [qt]

be the canonical