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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<-'\A)
= Ln+i(Pi(A))
= Ln+i+1(Ci+1(A))
(i > 0) .
The method can also be used to express the lower L-groups as the Lgroups of multiple suspensions L^iA) = L^S'A) = Lhn+i+1(Si+iA) (i > 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
(<7ffil)(/©l) : L@X@Y A stable automorphism [/] : L
> 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))<6forally€X, (b)/(y,*) = 0 f o r d ( i , y ) > 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)
The 6-neighbourhoods are such that
16
LOWER K- AND L-THEORY
(i) Y = Mo(YJX)Ctf1(Y,X)C...C
\JMb(Y,X) = X , 6=0
(ii) the inclusion 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
such that for some integers CQ , c\,..., cn > 0 (1 - dcT - Tdc){Cr) C Cr(AfCr(Y,X))
(0 < r < n) .
Starting with bn = cn there exists a sequence 6n, 6 n _i,..., 60 of integers
3. END INVARIANTS
19
br > 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)-conirac<«6Ze finite chain complexes in ). The end invariants ofC,D,E are related by
r=0
with h : Er >Dr 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))-
PROPOSITION
• REMARK 3.10 The 1-dimensional Gz+{0}(A)-contractible chain complex C in Cz+(A) associated in 3.3 to an object (M,p) in Po(A) is Cz+,{o}(A)-contractible if and only if p = 0. For A = B-^(Z) and (M,p) = (Z, 1) this is the cellular chain complex C = C(R + ) of the tame end R+ regarded as a Z+-graded CW complex, which is not bounded homotopy equivalent to a point.
EXAMPLE 3.11 As in 3.3 let (X,Y) = (Z+, {0}). For any object (M,p)
27
3. END INVARIANTS
in Po(A) the 1-dimensional chain complex C defined in Cx(A) by 1-p 0 0 ...\ p 1-p 0 ... dc = 0 p 1-p ...
= M[z] =
is Cx,y(A)-contractible, since dc fits into the direct sum system in Po(Cx(A)) dc
i >
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)
= Afb(U,V,X) . a
In general, the inclusions
unv
—>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<0}
uuv
CI,
= x , unv = {(o,o)} c i ,
for which AT2(U,V,X) = M2(U,X)
= Af2(V,X)
= X .
Mayer-Viet or is presentations will be used in 4.8 to prove that A"i(Cx,y(A)) = Urn A"i(Cjv-6(x\y,x),M(y,^\v,x)(A)) , 6
showing that -K"*(CX,Y(A)) depends on the neighbourhoods in X of Y and the complement X\Y rather than just the quotient X/Y (with some metric). Given subobjects M+ ,M~ C M in Gx(A) let X+ = supp(M + ) = {x e X | M+(z) / 0 } C I , X~ = supp(M-) =
{xeX\M~(x)^0}CX
so that M+ = M(X + ) , M" = M(X~) . Let M + fl M~, M+ + M~ C M be the subobjects defined by
M" = M(i + nr) , DEFINITION 4.3 The Mayer- Vietoris sequence for an object M in Gx(A) and subobjects M + , M ~ C M such that M = M"1" + M~ is the exact sequence
defined using the the inclusions i* : M+ fl M~
> M± , j
±
: M±
> M ,
with i = (
EXAMPLE
)
© M"
•M .
4.4 The Mayer-Vietoris presentation of an object M in Cx(A)
4. EXCISION AND TRANSVERSALITY IN JC-THEORY
33
with respect to an expression of X as a union
x = i+ur is an exact sequence in Cx(A) M :0
> 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~<6>
> E —> 0 ,
with E±(b)r
= Ef(br)
= Er(Afhr(X±,X))
(0
The Mayer-Vietoris presentations E(6) constructed in 4.6 have the property that any Mayer-Vietoris presentation E of E can be embedded as a subobject E C E(6) for some 6. PROPOSITION 4.7 Let X = X + U X " , and let E be a finite chain complex in Cx(A) with a Mayer-Vietoris presentation t~
.
f\
.
Zy1 1 p i Tp—
y
Jp 1 /T\ Jp—
v Tp
0
with bound b > 0. (i) E is Gx,x+nx- (F\)-contractible if and only if E~*~ is GAfb(X+,X),Afb(X+,x-,X)(A)-contractible and E" is (ii) E is Cx,x+nx-(A)-contracfa'&Ze if and only if E^ is
x)M(X+,x-,x)W-contractible - ,x),Afb(x+,x-
and E
~
is
,x)W'Contractible.
\ E PROOF (i) By 3.1 the chain complex < E+ is
[E(GXiX+nX-(f\) \ G>Arb(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 <
CJV 6 (X+,X)(A)
if and only if the chain complex I D
I CAr,(x-,X)(A)
I£
36
LOWER K- AND L-THEORY
is < (CAr6(x+,X)(A),Cjv/; d(x+,x-,X)(A))-finitely dominated for some
I
d > 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 <5 M 3M 3Q 3Q)
•
M + © M~ © Q Af .
44
LOWER K- AND L-THEORY
The automorphisms '1
0
0
ip
0 0
1 i% 0 0 1 0 , 0 0 0 1 : Af+ © Af" © P + © P "
Af + © AT © P+ © P " ,
Q+ © Q" — • M+ © M " © Q + © Q~ have torsion = 0
and are such that there is defined an isomorphism of exact sequences fl M~ © P + © P " -
© Af" © P + © P " - ^ - ^ Af —• 0
fl AT" © Q + © Q~
© M~ © Q + © Q- ^ ^ > Af —> 0
with = ( lM_ \
:
M+ H M"
= ( i ^ JM) : M+ © M"
> M + © Af > Af , Af+ © Af" © Q + © Q
Applying Lemma 1.4
r(f) = rdh)so that the composite lim
4. EXCISION AND TRANSVERSALITY IN /C-THEORY
45
is the identity. This gives the exact sequence of Carlsson [16] Urn #i(Cv- 6 (x+,x-,x)(A)) —> 6
lim ^ ( C ^ x + ^ A ) ) ©lim b
b
— - ffi(C*(A)) — lim b
> lim K0(Pjsrkix+tx)W)
9 Hm K0(Pjsrb(x-,x)W)
•
The inclusions CX±(A) — f C M ( X ± ) X ) (A) (6 > 0) are homotopy equivalences in the proper eventually Lipschitz category, giving rise to identifications b
Inn 6
and hence to the exact sequence of the statement. D
The connecting map in 4.15 can be expressed as the composite d : Xi(C x (A)) — = lim i^i( lim #6 > lim A"O(PA/- 6 (X+,X\X+,X)(A))
= lim # 0 (P.V f c (x+,x-,x)(A)) »
6
6
with the connecting maps 9& given by 4.8. REMARK 4.16 Working on the level of permutative categories as in Pedersen and Weibel [53],[54] and Carlsson [16] it is possible to extend the results of §4 to all the algebraic if-groups of the idempotent completions, with natural isomorphisms tf.(Py(A)—>Px(A))
s* X*(P £ lim 6
with Px,y(A) the idempotent completion of Cx,y(A), and a Mayer-
46
LOWER K- AND L-THEORY
Vietoris exact sequence ... —> ljm i lim tfn 6
(n e Z)
§5. Isomorphism torsion In general, it is not possible to define the torsion r ( / ) £ ^i(A) of an isomorphism / : L >M in A, only for an automorphism / : M >M. This is necessary in order to define the torsion r(C) £ K\(f\) for a contractible finite chain complex C in A. The isomorphism torsion theory of Ranicki [65] is now applied to the bounded categories, for use in §§6,7. The isomorphism torsion group K{so(f\) was defined in [65] for any additive category A to be the abelian group with one generator r(f) for each isomorphism / : L >M in A, subject to the relations (i) r(gf : L
>M
(ii) r(f@f':L@
>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 [<^M,N] : M ordered pair (Af, N) of stably isomorphic objects in A, such that (i) [MM = [1M] : M—+M
,
(ii) [M,p] = [4>N,p][M,N] : M ^ N
—> P ,
(iii) [^MeM'.Neiv] = [M,N] © [M'M : M®M'
> N © N' . • A stable canonical structure [<^] on A splits the natural map
by
T ( / : M—»AT) —> 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 {£,£M for each object M of A,
>N
5. ISOMORPHISM TORSION
(iii) an isomorphism PM,N • E(M © N) objects M, N in A, such that
49
»SM © SiV for each pair of
M © TV © E(M © AT) —> E(M © N) . D
LEMMA 5.4 A flasque structure {S, <J, p) on A determines a stable canonical structure /<£/ on A, witA <£M,N = {VN®1)~1{VM®1)
: M© EM ©SiV—> N ©EM©EiV , SO
and hence a splitting map A^ (A) >Ki(Fk) allowing the definition of torsion r{C) G K\(f\) for any contractible finite chain complex C on A. • In particular, if A admits a flasque structure every object M in A is stably isomorphic to 0 (via CTM)? SO that KQ(F\) = 0. DEFINITION 5.5 A flasque structure {E, cr, p] on A is natural if the assignation M »SM on objects of A can be extended to morphisms (/ : M
>N)
> ( S / : EM
defining an additive functor E : A GM '- M © EM
>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<0andu = u - l 0 otherwise . Define a flasque structure {E,cr, p) on CR(A) by
{
oo
EM = aM : M © EM
> 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
diso : A^o(Cx,y(A)) —> 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°(Cx
such that diaor([fSR})
= [Q] - [P] € ^o(Py(A)) .
(iii) The objects of Cx,y(A) are the objects of Cx(A), so for any isomorphism / : M >N 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 {£,<7, p] of 6.3. Applying Lemma 5.4, it follows that CQ(X)(A) is equipped with a stable canonical structure, and that A'O(CQ(X)(A)) = 0 for non-empty X. The z-based torsion is defined for any contractible finite chain complex C in C o( x)(A) by rx(C) = rx(d + r : Codd >Ceven) G ifi(C o( x)(A)) , with F : 0 c± 1 : C >C any chain contraction. The filtered structure on A is used to identify C Xl xx 2 (A) - C Xl (Cx,(A)) , for any metric spaces X\ ,X2 with the max metric on the product X\ x X2 The cone and suspension of X are defined as usual by CX = X x [0,1)/X x {1} , EX = C+X Ux C'X . For X CSk identify O(CX) = O(X) x R+ , O(£X) = O(X) x R . PROPOSITION 6.4 (i) For non-empty X C Sk and any x G X the x-based flasque structure on Co(cx)(A) is natural, so that
K*(Co(cx)W) = K.(PO(cx)W) = 0 .
56
LOWER K- AND L-THEORY
(ii) The bounded F x R + -graded category Cy X R+(A) has a natural flasque structure, for any metric space Y. PROOF (i) The x-based flasque structure on Co({*})(A) = C R +(A) is natural, with S / : E M •EJV defined for any morphism / : M >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,K<>W^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
=
<
»/
v
U'i(A) ^
if
with the sums taken over all the maps « : 5°
» X + U F , i | : 5°
> X+D X
with w(±l) = x^1 € X±. For any such »/ the composites
Bo(v)
K0(f\) > ^i(C o ( x+nx-)(A)) —-> Jf,(Co( + are 0, since X and X~ are connected. Thus the reduced K-theory Mayer-Vietoris exact sequence is a quotient of the absolute if-theory Mayer-Vietoris exact sequence of 6.2 (ii). (ii) For any object M in Co 6 (x+)(^) 1^ -W' D e ^ n e object defined in Co(x+)(A) by ^ ; l M ( 0 ) e M ( O 6 ( I + ) \ O ( I + ) ) ifx' = 0 . For any base point x~ € X~ the evident regrading map 6M'-M >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>+ © <^~)~1i : E+ D E~ € im (
/
(
(
)
)
A
(
>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)[[<2:,2:~1]]
M : 0
v L"[z,z-X) -L L'\z,z-X\ - L M —» 0
8. THE LAURENT POLYNOMIAL EXTENSION CATEGORY A[Z,Z~ X ]
75
with / = / + — zf- linear. The Mayer-Vietoris presentation isfiniteif M is an exact sequence in Co(A)[z, z~l], in which case it is equivalent to an exact sequence in f\[z, z""1], and the torsion of M is defined by T(M) =
T((fh):L"[z,z-1]®M—*L'[z,z-1])eKr(F\[z,z-1)) •Z/[z,z~1] splitting g.
for any morphism h : M
• For a ring A and A = B-^(A) the finite Mayer-Vietoris presentations of objects in A[z, z"""1] = B^(J4)[Z, z""1] are 'Mayer-Vietoris presentations' in the sense of Waldhausen [82]. The restriction of an object M — i\L in AfzjZ"-"1] is an object vM in Gi(A). The induced object in'M in Gi(A)[[2,2:~1]] is equipped with commuting automorphisms ^ ( M ) = C(iiilM) , C2(M) = in'C(M) : in'M
> 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
DEFINITION 8.12 A Mayer-Vietoris presentation of a finite chain complex E in A^,,?" 1 ] is an exact sequence of finite chain complexes in
with £?', J57" finite chain complexes in Gi(A) and / linear. E is finite if it is defined in Co(A)[z, z " 1 ] , in which case it is equivalent to an exact sequence of chain complexes in A[z, z" 1 ] and the torsion of E is defined
8. T H E LAURENT POLYNOMIAL EXTENSION CATEGORY f\[zfz~x]
77
by - r((g 0) : C(/+ - zf. : E"
T(E)
r=0 D 1
Given an n-dimensional chain complex E in A^,^" ] with Er = i\Fr (0 < r < n) let N(E) be the lattice of (2n + 2)-tuples of elements of Z+ iV = (iV 0 + ,iV 1 + ,...,iV+;iV ( r,iVr,...,iV-) such that
j=-N+
k=-N+_1
N(E) has the maximum element oo = (oo,... ,oo;oo, ...,oo) € N(i3) . An element N G N(E) is yini^e if iV^ ^ oo (0 < r < n) . / Let N (E) C N(E) be the sublattice of finite elements. PROPOSITION 8.13 (i) For every n-dimensional chain complex E in f\[z,z-1) with Er = i\Fr (0 < r < n) and every N € N(E) there is defined a Mayer-Vietoris presentation of E in Gi(A)[[2, 2"1]]
E(N) : 0 —+ E"{N)[z,z-1]
f+(N)-zf-(N) — ——, 9{N)
F — >00 . • hi
(ii) Nf(E) is non-empty, and E(iV) is a finite Mayer-Vietoris presentation of E for any N G Nf(E). (iii) / / JV, N' € N(E) are such that N < N' then inclusion defines a morphism of Mayer-Vietoris presentations E(iV) »E(iV') resolving 1 : E >E. PROOF Set
E(N)r = Er(Nr) (0 < r < n) the Mayer-Vietoris presentation of Er associated to Nr = (N+,N~) G N. • The universal Mayer-Vietoris presentation of a finite chain complex E in Fk[z, z~l\ is the Mayer-Vietoris presentation of E in Gi(A)[[z, 2"1]]
, l^cc^r E(oo) : 0
> 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) = (<(*), 1)|* € * } , which has an infinite cyclic cover T(C) = X x R with generating covering translation Define a subfundamental domain to be a subcomplex Y C. X such that
U ciY = x • j=-oo
A fundamental domain is a subfundamental domain such that YC)(jY
= 0 for j ^ - 1 , 0 , 1 .
80
LOWER K- AND L-THEORY
For any subfundamental domain Y (e.g. X) define
(v-,u,cu) = (yx[o,i];rn(-17xo,cynyxi) C f (C) = X x R , Ty(C) = Y x [0, l]/{(y, 0) = (C(y), 1) | y € Y D C ^ } C T(C) , so that Ty(^) has infinite cyclic cover with fundamental domain V
jf=-OO
The projection PY
:
has contractible point inverses, and is thus a homotopy equivalence, corresponding to the Mayer-Vietoris presentation 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 (!<»•<») I
k=-N+_l r-l
there is defined a subfundamental domain
=
|J |JC>gOu U (Jc^u.-.u |J j=-N+ h
j=-N+ h
j=~N+
9. NILPOTENT CLASS
81
corresponding to the Mayer-Vietoris presentation E(N) of E = C(X). N is finite if and only if Y(N) is compact. •
§9. Nilpotent class The construction of the nilpotent class groups Nilo(A), Nilo(A) for a ring A of Bass [7] are now extended to define the analogues Nilo(A), Nilo(A) for an additive category A. The nilpotent class groups are required for the splitting theorem of §10 for the torsion groups Ki(F\[z, 2" 1 ]), K{so{fK[z^z~1]) of the Laurent polynomial extension category f\[z,z~1]. An endomorphism v : M >M is nilpotent if for some integer N > 0 uN = 0 : M
> M .
Let Nil(A) be the additive category with objects pairs (M
= object of A , v : M
> M nilpotent) .
A morphism in Nil(A) /
= (M.i/)—>(M>')
is defined by a morphism / : M
>M' in A such that
v'f = fv : M
> M' .
Nil(A) is given the (non-split) exact structure in which a sequence 0
> (M',i/)
v(M'V)
> (Af,i/)
> 0
is exact if the underlying sequence 0 >M" >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'
PROPOSITION 9.3 The natural maps define isomorphisms
Nilo(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\
<-
: C'o = Co — > C[ = Ci © Co 0 ... 0 Co = C\ 0 M
o a
is such that The chain homotopy nilpotent self chain map v
=
v
— yd T] -f- T] a )
is chain homotopic to 1/ : C Thus
: O
• O
•C", and such that u1 = 0 : CQ-
and there is defined an exact sequence 0 —> (C 0 ,0) —^ (C',i>;) — 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
= Offil : Lfaz-1]
= C'L+ © (jL
10. /C-THEORY OF f\[z,z~x]
DEFINITION
87
10.1 The split surjection
B®N+®N-
:
K\so(f\[z,z-1))
> tfo(Po(A)) 0 Nilo(A)ffiNilo(A) ; r(f)
—
([/]+ , [ P + , I / + ] , [ P - , I / - ] )
sends the torsion of an isomorphism in A ^ z " 1 ]
with inverse
j=-3!
to the end invariant
Br(f) = [/]+ = -[/]_
and the reduced nilpotent classes N±r(f)
= [P±,i/±]€Niio(A)
of the objects (P±,t/±) in Nil(Po(A)) given by
P_ = L+nr 1 ^") = ( j=0
v+ = PL-f^CPc-'M+f
: 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\(A) induced by the surjection of rings oo
1
oo
k
j : Alz.z' ]
> A ; ]T akz
> ] T ak
splitting the inclusion i : A >A[z, z~x\. The relative merits of algebraic and geometric significance have already been discussed in Ranicki [66]. The geometrically significant direct sum decomposition of the Whitehead group of a product TT x Z Wh(ir) <
Wh(ir x Z)
if B ® iV+ 0 NB' © N'+ 0 NL includes the split injection & : K0(Z[*\) — Wh(n x Z) ; [P] —> 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;<,0) e N'(£), so that by 8.8 there is defined a finite Mayer-Vietoris presentation of E in CO(A)[0,2~1]
E(N) : 0
> E'^Nftz^-1]
which is written as
f+(N)-zf-(N)
1
K-THEORY OF f\[z,z~l]
10.
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
= 0eWh(irxl),
so that
Br(f) - (-)n(Br(f)r € N+r(f) = (-r-\N. The splitting obstruction (B ® N+ © N-)r(f) vanishes for a simple homotopy equivalence / : M >X x 5 1 , in which case there is denned a W/i2-invariant Let B2 : Wh2(>K x Z) ^^^(Tr) be the W^2-analogue of B : Wh^ x Z) n >KO(Z[TT]). The image B2T2(f) G H (l2; Wh(ir)) is the obstruction of Slosman [78] and Wall [84,12B], such that B2r2(f) = 0 if and only >X can be chosen to be a simple if the restriction / | : f~x(X x {*}) homotopy equivalence. See §20 for the generalization to the fibering obstruction.
11. LOWER JC-THEORY
101
§11. Lower K-theory The lower If-groups K-m(A) (ra > 1) of a ring A were defined by Bass [7] using the polynomial extension rings A[z], A[z~x], A[z^z~x], to fit into split exact sequences 0 —> Ki-m(A) —> Ki-m{A[z\) ® Ki-miAlz-1})
—> Kt-niAfoz-1])
—> K-m(A) —> 0 .
The lower if-groups of an additive category A were defined by Karoubi [38] to be with S m + 1 A the (ra + l)-fold "suspension" of A, and are such that K-m(F\)
= JT_TO(P0(A)) (m > 1) .
For the additive category A = B^(A) of based f.g. free A-modules K-m(Bf(A))
= K-m(A)
(m > 1) .
m
Define a metric on Z by d(J,K)
= max{|ji - k{\ \ 1 < i < m) > 0
for J = ( i i , j 2 , • • • Jm), K = (ku i 2 , . . . , km) e Z m . Write the bounded Zm-graded category of a filtered additive category A and its idempotent completion as = Cm(A) , P 0 (C m (A)) = P m (A) ( m > l ) . Since Z m + 1 = Zm x Z the bounded Zm+1-graded category can be expressed as C m+1 (A) = C!(CTO(A)) , and by 6.2 there is an identification Pedersen [49] (for A = B-^(A)) and Pedersen and Weibel [53] expressed the lower iiT-groups of a filtered additive category A as K-m(f\) = #i(C TO+1 (A)) = K0(Pm(f\)) ( m > l ) . The lower if-groups Jf_m(A) (m > 1) of any filtered additive category A will now be shown to fit into split exact sequences > tfl-m(Po(A)) > Kl-mCPoCA^^eXi^CPoCA^" 1 ])) 0 —> 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 -fold cover of the circle S1 = R/Z by itself. Given a map c : X •S 1 let
x- = {(x,[t])eXxS1\c(x)
= [qt]eS1}
be the pullback ^f-fold cover of X
x'-—c-—> s1 -> s1. The infinite cyclic cover of X classified by c X = {(x,t)eXxR\c(x) = [t]eS1} has covering projection X
> X ; (x,t)
>x
and generating covering translation C :
X^X;
(i,t)—»(*,* + 1 ) .
X
Let X ' denote the space X regarded as the infinite cyclic cover of X! classified by c'
: X
1
; (x,[t\)
with covering projection and generating covering translation
C! - C : XX1 1 = X—*?•
= X;
107
12. TRANSFER IN K-THEORY
so that there is defined a pullback square
?•—£—> R
with q : R
•R; x
•x +q .
If X is a compact manifold and (V; U, ££/) is a fundamental domain for X then q-l
is a fundamental domain for X'. More generally, the (/-fold transfer of an R-graded CW complex (K,px ' K >R) (not necessarily of the type K = X) may be defined to be the R-graded CW complex {K\PIC)
=
(K,pK/q),
such that q-\
KT^qj
+ k.qj + k + l]
(j€Z).
k=o
Define the q-fold transfer functor q-l
ql : Ci(A) —> d ( A ) ; M -
fM , q'M(j) =
The ^-fold transfer of a morphism / : M qf : qM
+ *) .
>N in Ci(A) is the morphism
>q'N defined by f(qk,qj)
f(qk,qj+l)
\
f(qk+i,qj+q-i)
: q'M(j) = > ^!iV(fc) = JV(gJfc) © N(qk + 1) © . . . © N(qk + q - 1) . The <j-fold transfer of a linear morphism / = / + — z / _ : M Ci(A) is the linear morphism
qf
= q'f+ - zq'f- : «'Af —> 9 ' ^
•iV in
108
LOWER K- AND L-THEORY
with 0 0 0
+2)
\
V o : q'M(j)
/0 0 0 0 \0
0
0
...
0
/
!
: g A/(j) = M(qj) ® M(qj + 1) ® ... ® M(qj + q - 1) -> g!iV(j + 1) = JV(gj + «) © iV(gj + q + 1) © . . . © N(qj + 2q - 1) . The ^-fold transfer of a £-equivariant morphism is £-equivariant, so that >A[z,z -1 ]. there is also defined a functor q : A[2,z -1 ] The q-fold transfer functors induce the q-fold transfer maps in the algebraic iiT-groups
DEFINITION 12.1 The S1-transfer invariant subgroup of KI(PL[Z,Z~1}) is = r for every 3 > 2} . = {r G For any group TT there are analogously defined g-fold transfer maps in the Whitehead group of ?r x Z q' : Wh(7T X Z)
> Wft(7T X Z)
1
and an 5 -transfer invariant subgroup Wh(ir x Z)INV C W/i(?r x Z). If X is a finitely dominated CW complex then X x 5 1 is equipped with a homotopy equivalence
$
:
x x S1
> K
109
12. TRANSFER IN K-THEORY
to a finite CW complex K (Mather [46]) with infinite cyclic cover K ~ X. The map 4>~l
1
proj.
c : K —> X x S 1 > S1 has the property that for every cover q : S1 >SX the pullback cover K* is simple homotopy equivalent to K. The map representing — 1 G 7T1(51) = 5 1 - 1 : S1 —> S1 ; t
>l-t
covers itself
Thus the self-homotopy equivalence f = cf>(l x - l ) ^ " 1
> X x S1
: K
i
X x 51
K
is such that e Wh(w x 1)INV
C Wft(T x Z) (jr =
In fact, the image of the split injection of Ferry [27]
B'geo : £ 0 (ZM) —» Wft(^ x Z) ; [X] —* r(/) is precisely the 5a-transfer invariant subgroup Wh(ir x Z)INV. See Ranicki [66] for the identification of B'geo with the geometrically significant split injection B' : Ko(l[n})^Wh(7rxl);
[P] —» r ( - z : P ^ z " 1 ] — ^ P ^ z " 1 ] ) .
PROPOSITION 12.2 (i) TAe e^ec^ o/g ! on the geometrically significant direct sum decomposition of Ki(f\[z,z~1]) is given by the commutative diagram (A) 0 ^o(Po(A)) 0 Nilo(A) © Nilo(A)
„
.
.
0 Nilo(A) © Nilo(A)
i (&Bf(&N1 (\)N'
^
iiC
" ) i f ( A ^ 0""1])
110
LOWER K- AND L-THEORY
with ql : Nilo(A) —>
NT1O(A)
; [M,i/] —> [M,uq] .
(ii) T&e image of the geometrically significant split injection • r ( l — p — zp : M[z,z~l]
[M,p]
>M[z,z~x])
is the subgroup of the S1 -transfer invariant elements im(B') = ^ ( A ^ z " 1 ] ) ^ ^ . (iii) The effect of q- on ATi(Ci(A)) is PROOF (i) For a linear automorphism in A[z,
/ = f+-zf- : M^z- 1 ]— the q-iold transfer is the linear automorphism / /+ 0 0 ... -zf-\ -//+ 0 ... 0 0 - / _ /+ . . . 0
\
0
0
/+ )
0
4-1
j=o
—> (M © M e . . . e M)[«, Define an automorphism in A g = jtf = / + - / _ : M —» M . The linear automorphism in A[2,2 -1 ] is such that
f+f. = / * _ / ; , / ; - / i = i •. M —> M . Elementary row and column operations show that
r(qf) = rar+J'-^/D'rMlz.z- 1 ]—.M^z" 1 ]) € ^(A It follows that r(qf)
= r(q\gf')) = r(q'f') + T(q'-(i,g)) =
T((f'+y-z(fL)':M[z,z-1}-+M[z,z-1)) + qi,T(g : M
>M) G ^ ( A t z . z " 1 ]
12. TRANSFER IN K-THEORY
111
In particular, if / + / - = / - / + then r(q'f)
= r((/+) g — z(f-)q
: M[2,2"""1]
•M[z,z~1]) G K\(J\[z, z~l\) .
This is the case for each of the components of the geometrically significant decomposition of K\(f\[z, z~1]) : (I) if / _ = 0 then / r\ T\J)
==
• _ / £ \ J'~( s\ z—(( t+\q\ i\T\j+) > q Tyj) = 2 | T ^ / J J =
„—( / \ qTyj)
^(A^,^"1])) ,
e im(i\ : Kx(f\)
(II) if /-|- = 1 — p, / _ = p for a projection p2 = p : M !
>Af then
g
B'([M,p]) = r ( / ) , g B'([M,p]) = r ( ( l - p ) - z p « ) =
r(f)
(III) if /+ = i/, / _ = 1 for a nilpotent endomorphism i/ : M >M then JV4.[Af,i/] = T(1-Z~1V) = r ( / ) - B'([M]) , ^!iV|[M,i/] = r ( i / g - ^ ) - B ' [ M ] = r ( l - z - V ) G im(JV; : NTIO(A)
= iV;(9![M,i/])
^(Alz,^"1])) ,
(IV) if /+ = 1, / _ = v for a nilpotent endomorphism v : M >M then JVi[M,i/] = r ( / ) , ^!iV!.[M,i/] = r ( l - ^ 9 ) = iVl(^ ! [M,i/])
(ii) Immediate from (i). (iii) Immediate from (ii), since there is defined a commutative diagram tfo(Po(A))-
5#
go(Po(A))
B
'
)K1(A[^,z~1
an isomorphism.
• Given m > 1 let Tm = S1 x S 1 x . . . x S 1 be the m-fold torus. For each m-tuple Q = (<ji, ^2, • • •, Qm) of integers qj > 1 let be the canonical finite covering of Tm by itself. For any such Q define the Q-fold transfer functors Q' : C m (A) — . CTO(A) , Q'- : A[Z m ] — • A[Z m ]
112
LOWER K- AND L-THEORY
by sending the object M to the object Qy'M with components ,j2,...,im) i - l q2-l
qm-l
*! =0* 2 =0
km=0
and similarly for morphisms. By analogy with 12.1 and 12.2: DEFINITION 12.3 The Tm-transfer invariant subgroup of Jfi(A[Zm]) is Jir1(A[Zm])/iVI' = {re lfi(A[Zm]) | Q\T) = r for every Q} . n PROPOSITION 12.4 For any m > 1 the geometrically significant split injection B'1B't...B'm
: ^ _ m ( P 0 ( A ) ) —» if,(A[Zm])
mcpji ^Ae Zoiaer K-group J^i_m(Po(A)) (^= JFCi_m(A) /or m > 2) to the subgroup of the Tm-transfer invariant elements = K1(l\[lm])INV
im(B[B'2...B'm)
.
PROOF This follows from the case m = 1 dealt with in 12.1 and the factorization of the Q-fold transfer for Q = (qi, q2,..., qm) as the ra-fold composition of the single transfers Q] - ( « i ) W . . . ( f l m ) ! : A[Zm] — • A[Zm] with (qj)1' acting on the jth coordinate (1 < j < m).
n §13. Quadratic L-theory This section is a brief recollection from Ranicki [68, §3] of the quadratic L-theory of an additive category with involution A. An involution on an additive category A is a contravariant functor * : A
> A; M
> M*
together with a natural equivalence e :
l
>**:A
>A;M
> (e(M) : M
such that for every object M in A e(M)* = e(M*)~l
: M*
Use e(M) to identify M** = M
v M*.
>M**)
13. QUADRATIC L-THEORY
113
for every object M in A. Given objects M, JV" in A define = HornA (M*,iV) .
M®f\N
The transposition isomorphism is defined by T{M,N) : M®*N
> JV ® A M ; /
> f* ,
and is such that T(N,M)T(M,N) = 1 : M®f\N—>M®FKN
.
Given chain complexes C, D in A let C ®f\ D be the abelian group chain complex defined by (C (g)A D)r = : (C ®A # ) r > (C ®A 2?)r-l 5 / An n-cycle <j> € (C ® A -D)n is a chain map ^
Cn"*
:
> 2? ,
n
with C ~* the n-dual chain complex defined by <*C— = ( - ) ^ C : (Cn~*)r
= Cn~r = (Cn
The transposition isomorphism T : C®f\D
> D®f\C
is defined by T = (Let W be the standard free Z[Z2]-module resolution of Z l-T
l+T
\-T
W : ... —> Z[Z2] —* Z[Z2] —* Z[Z2] —^ Z[Z2] . Given a chain complex C in A define the Z[Z2]-module chain complex W%C = W ®z[z2] (C ®A C) , with TG Z2 acting by the transposition involution T : C ® A C C. The boundary of the n-chain <> / =
>C ®f\
{tl>.€(C®*C)n-.\s>O}e(W%C)n
is the (n - l)-chain d(V>) € (W%C)n_i with € (C ®fl C) n _._i (« > 0) . The quadratic Q-groups of C are defined by Q B (C) = Hn(W%C) . An n-dimensional quadratic (Poincare) complex (C, ^) in A is an ndimensional chain complex C in A together with an element xj) € Qn(C) (such that the chain map (1 + T)\J)Q : Cn~* >C is a chain equivalence).
114
LOWER K- AND L-THEORY
See Ranicki [68] for further details of the definition of quadratic (Poincare) pairs the construction of the n-dimensional quadratic Lgroup Ln(F\) (n > 0) as the cobordism group of n-dimensional quadratic Poincare complexes (C,ij> G Qn(C)) in A, and for the proof of 4-periodicity Ln(A) = Ln+4(A) . L2i(R) (resp. L2«+i(A)) is the Witt group of nonsingular (—)'-quadratic forms (resp. formations) in A. The definition of Ln(f\) is extended to the range n < — 1 by the 4-periodicity, that is Ln(f\) = Ln+4*(A) for any k such that n + 4k > 0. The surgery obstruction groups L*(A) of Wall [84] are the quadratic L-groups of the additive category with involution A = Bf(A) of based f.g. free A-modules for a ring with involution A. A chain map / : C >D in A induces a Z-module chain map f% ' W%C —^ W%D ; <> / = {tl>.\s > 0} —> M
= {fi>*f*\s > 0} .
Given an n-dimensional quadratic complex (C,tp) and a subcomplex B C C there is defined a quotient n-dimensional quadratic complex
with / : C >CjB the projection. In dealing with subcomplexes it is always required that the sequences 0
> Br
> Cr
> (C/B)r
>0
be split exact, so that there is defined a connecting chain map g : C/B >SB such that gf = 0. PROPOSITION 13.1 There is a natural one-one correspondence between the homotopy equivalence classes of n-dimensional quadratic Poincare pairs in A and the homotopy equivalence classes of n-dimensional quadratic complexes in A. PROOF This is just the categorical version of the one-one correspondence of Ranicki [61,3.4] for the additive category with duality involution P(A) — { f.g. projective A-modules} , and proceeds as follows. The boundary of an n-dimensional quadratic complex (C,ip) in A is the (n — 1)-dimensional quadratic Poincare complex in A
d(C,i>) =
(dCM)
13. QUADRATIC L-THEORY
115
with dc
(-)-(l
dCr = Cr+1 © Cn~r
\J\J
— Ks
> dCr-1 = Cr ® Cn~r+1 ,
f
\x) ^/f-J-1
t/V«/f
— v^/y_|_j KJJ KS
— r—s-l
,
G
Given an n-dimensional quadratic complex (C, t/>) in A define the thickening n-dimensional quadratic Poincare pair in A 6d(C,tl>) = (pc = projection : dC > Cn~* , (0,30)) . Conversely, given an n-dimensional quadratic Poincare pair in A X
=
(f:C—>D
, (6il>,tl>))
apply the algebraic Thorn construction to obtain an n-dimensional quadratic complex
x/dx = (DM)/C = (C(f)MM with = ({-r%af* C(f)n~r-a+1
(-r-r->T<,a+1)
= Dn~r-S+1 © Cn~r~S
•
* C(f)r = Dr © Cr-l
(a > 0) , which is homotopy equivalent to 8d(X/dX). • The boundary construction of 13.1 generalizes to quadratic pairs. An n-dimensional quadratic pair ( / : C >D,(8tl>,ip)) determines an (n — 1)-dimensional quadratic Poincare pair (dfidC—>dD,(d8*l>,dxl>)) with dC = S-'Cdl + 2 > 0 : C"- 1 -*—>C) , 3D = 5" 1 C((1 + T)6tl>0 :
116
LOWER K- AND L-THEORY
Here, (1 + T)8if>o is a n abbreviation for t h e chain m a p C ( / ) n ~ * defined b y
((i + W o / ( i + i > o ) : c(f)n~r
>D
= zr-recn-r-1—>Dr.
The quadratic complex (C, VO is Poincare if and only if dC is contractible. Similarly, the quadratic pair (/ : C •£),(&/>> VO) is Poincare if and only if dD is contractible, in which case dC is also contractible. The n-dimensional quadratic pairs in A V+ = ( / + : C + — • £ > + V- = ( / - : C r — » D are adjoining if C + = C " , V>+ = - V 7 " • See Ranicki [61, §3] for the definition of the union of adjoining n-dimensional quadratic pairs y+ = ( / + : C — + D + with common boundary The union is an n-dimensional quadratic complex in A with
The duality chain maps fit into a map of homotopy exact sequences of chain complexes
o
>D n ~*
> (D+r-* © (D-)n-*
>o
(1 + T)^o
T)6tl>0
0
> c n ~*
>D
• C(/+) © C(D
> SC
•0
so that there is defined an exact sequence of algebraic mapping cones 0 —> dD —> dD+ © dD~ —> SdC —> 0 . If dD+ and dD~ are contractible then so are dC and dD, i.e. the union of quadratic Poincare pairs is a quadratic Poincare complex. DEFINITION 13.2 A splitting ( F + , F ~ , h) of an n-dimensional quadratic complex ( £ , 0) in A consists of adjoining n-dimensional quadratic pairs
13. QUADRATIC L-THEORY
117
V + , V~, together with a homotopy equivalence
h : V+UV~
> (E,0) .
The splitting is Poincare if the quadratic pairs V + , V~ are Poincare, in which case the complex (E,6) is Poincare. • See Ranicki [63,7.5], [69, §23] for the applications of Poincare splittings to the algebraic theory of codimension 1 surgery, and Yamasaki [89] for an application to controlled surgery. DEFINITION 13.3 Given an n-dimensional quadratic complex (E,0) and a subcomplex D C E in A let (V + , F"~, h) be the splitting of ( £ , 8) given by
V+ = (f+ V-
/+ : d(E/D)r f~
= (E/D)r+i © (E/D)n~r —> (E/D)r+1 —> Dr ,
= projection :
d(E/D)r ti
(f-:d(E/D)—*(E/D)n-*,(0,dP%0)),
= 6d((E,0)/D) =
.
o
I
= (E/D)r+i I
=
\EJIU)
© (E/D)n~r (i3 O\£JI U)r—\
—> (E/D)n~r KJ) JJr
• iyr
, y £jr
.
If (-E, 0) is a Poincare complex then ( y + , V~, ft) is a Poincare splitting. • In fact, every splitting of a quadratic complex is homotopy equivalent to one constructed as in 13.3. An involution on A determines an involution on the projective class and torsion groups of A
; [M] — > [M*], i(A) ; T ( / : L—+L) > r{f* : L* *L") . In dealing with torsion it is required that the additive category A be equipped with both an involution and a compatible stable canonical structure (see Ranicki [68, §7] for details). A quadratic complex (C,V0 in A is round if the underlying chain complex C is round, i.e. such that [C] = 0 G A'o(A). The torsion of an n-dimensional quadratic Poincare complex (C, rj>) in A is defined by for any and satisfies
118
LOWER K- AND L-THEORY
There is a corresponding notion of torsion for an (n + 1)-dimensional quadratic Poincare pair V — ( / : C >D,(6iJ),ij>)) +1
= r((l
T(V)
satisfying If (V + ,V~,/i) is a Poincare splitting of an n-dimensional quadratic Poincare complex (E,d) in A the torsions satisfy T(E,6) +
-
T{V \JV~)
T(V+UV~) +
= r(h) + (-)nT(hy
= T(V ) + r ( y - )
-
r(F
+
,
nr).
The intermediate quadratic L-groups Lf((\) are defined for a *-invar( S C A'o(A) iant subgroup < S C K\{f\)
to be the cobordism groups of quadratic
I 5 C ^(A)
([C] e s Poincare complexes (C, x/>) in A with < [C] = 0, r{C^) G 5 . The in-
U(C,V)eS termediate L-groups associated to *-invariant subgroups S C S' are related by the usual Rothenberg exact sequence . . . —> Lsn(f\) —> Lsn'(A) — . Hn(l2;S'/S)
— Lf.^A) — > . . . ,
with the Tate Z2-cohomology groups defined by H»(Z2;S'/S) = {x € S'/S\x* =(-)nx}/{y + (-)ny*\y 6 S'/S} . The quadratic X-groups of A axe the special cases
The round quadratic L-groups of A are the special cases
For a ring with involution A and A = B-^(A) these are the round quadratic L-groups Ll(A) of Hambleton, Ranicki and Taylor [32]. Given an additive category with involution A and a set X it is not possible in general to define an involution on the X-graded category Gx(A). Define the locally finite X-graded category Fx(A) to be the subcategory of Gx(A) with the same objects, in which a morphism yex is required to be such that for each y G X the set {x G X \ /(y, x) ^ 0 : L(x) >M(y)} is finite. The involution on A extends to an involution
14. EXCISION AND TRANSVERSALITY IN L-THEORY
119
on F X (A) by
* : F X (A) —> F X (A) ; x£X x£X
x£X
The dual of a morphism / : L >M in Fx(A) with components /(y, x) : >M(y) is the morphism / * : M* >L* with components L(x)
r(x,y) = f(y,x)* : M*(y) = M{yy^L\x)
= L(x)* (x,y e X) .
An involution * : A >A on a filtered additive category A is required to preserve the filtration degree £, i.e. for every morphism / : L >M in A
6(f* : M*
>L*) = S(f : L
>M) .
For any metric space X the bounded X-graded category Cx(A) is a subcategory of Fx(A) which is invariant under the involution, so that it is also a filtered additive category with involution. Similarly, if (X, Y C X) is a pair of metric spaces the bounded (X, Y)-graded category Cx,y(A) of 2.6 is a filtered additive category with involution. REMARK 13.4 Given a metric space X and an n-dimensional normal map (/, b) : M >N from an n-dimensional X-bounded manifold M to an n-dimensional X-bounded geometric Poincare complex N with constant bounded fundamental group 7Ti(N) — TT Ferry and Pedersen [28] associate an X-bounded surgery obstruction
*!(/,&) ei o (c x (ZM)). By the main result of [28] for n > 5 this is the only obstruction to making (/, b) X-bounded normal bordant to an X-bounded homotopy equivalence, provided that X is a reasonable metric space (such as the open cone 0(K) on a compact polyhedron K C Sk). For compact X this is just the main result of Wall [84], with Ln(Cx(Z[7r])) = Ln(Z[?r]).
• §14. Excision and transversality in L-theory The localization exact sequence of algebraic L-theory (Ranicki [63,§3]) for a morphism of rings with involution A >S~1A inverting a multiplicative subset S C A ... —
Li(A)
— > Ln{S~lA)
— > Ln(A,S)
—> L ^ A ) —
•••
relates the free L-groups L n (5~ 1 A), the intermediate projective L-groups LTn(A) with / = kei(K0(A)
tKoiS^
120
LOWER K- AND L-THEORY
and the L-groups L n (A, S) of cobordism classes of (n — l)-dimensional quadratic Poincare complexes (C, tp) over A such that C is an S~XAcontractible f.g. projective A-module chain complex, corresponding to an excision isomorphism A similar exact sequence
£
— • L»(Cx(A))
will now be obtained for any pair of metric spaces (X, V C X) and filtered additive category with involution A, with J = ker(AyPy(A))—»£ 0 (Px(A))) , by analogy with the A"-theory exact sequence of 4.1 Jfi(Cy(A)) — A^(Cx(A)) —-» A'1(Cx,y(A))
—> /fo(Py(A)) —» A0(Px(A)) . The exact sequence is obtained from quadratic L-theory excision isomorphisms X^(Py(A)—>C X (A)) a I n (C X ) y(A)) analogous to the A'-theory excision isomorphism of 4.1 This requires the following L-theory analogue of the result of 2.7 that every finite chain complex in the bounded germ category Cx,y(A) is of the type C = [D] for a finite chain complex D in Cx(A). LEMMA 14.1 (i) For every n-dimensional quadratic complex (C,ip) in Cx,y(A) there exists an n-dimensional quadratic complex (D,d) in Cx(A) such that
(CV) = [DA(ii) L n (Cx,y(A)) is naturally isomorphic to the cobordism group of ndimensional quadratic Cx,y(A)-Poincare complexes in Cx(A). PROOF (i) As in the proof of 2.7 for any representatives in Cx(A) dc : Cr > Cr-i (1 < r < n) there exists a sequence b = (bo,bi,... ,bn) of integers br > 0 such that (a) dc{CT{MK(Y,X))) C C r _, (b) (dc)2(Cr) C CT-2{M (c) the n-cycle
Yl
C r)
14. EXCISION AND TRANSVERSALITY IN L-THEORY
121
is represented by an n-chain
V>'€(W % C)' n =
£ p+q+r=n
with boundary
yW) € For any such sequence 6 the n-dimensional quadratic complex (D, 0) in defined by
°
V
) '
D
-
c
—>D r = Cr = Cr(ATbr(Y, is such that [D,0] = (C,^). (ii) Immediate from (i) and its analogue for quadratic pairs. Let J = im(a"° : Ar°(C x ,y(A))—^A^ 0 (Py = ker(£ 0 (Py(A))—+£o(P*(A))) C and let i/(Py(A) ^Cx(A)) be the relative L-quadratic groups in the exact sequence
THEOREM 14.2 For any A, (X, y C X) tfeere are natural excision isomorphisms =* L B (C x ,y(A)) , LJn(PY(f\)—-Cx(A)) Ae connecting map in the exact sequence
given by d : £ n (C x ,y(A))
122
LOWER K- AND L-THEORY
with (C, ip) any n-dimensional quadratic Cx,y(A)-Pomcare complex in Cx(A). PROOF The relative L-group L^(Py(A) >Cx(A)) is the cobordism group of n-dimensional quadratic Poincare pairs (/ : C >D,(6iJ>,tl>)) in Px(A) such that (C, VO is defined in Py(A) and D is defined in Cx(A). The algebraic Thorn construction (13.1) on such a pair gives an n-dimensional quadratic Poincare complex (C(/),£V7V0 m Px,y(A) with reduced projective class [(/)] = [D] = 0 € £o(Px,y(A)) . The complex is homotopy Cx,y(A)-finite, and represents an element
The boundary of an n-dimensional quadratic Cx,y(A)-Poincare complex (C, xl>) in Cx(A) is an (n — l)-dimensional quadratic Poincare complex 9 ( C » in C X (A) with dC = S ^ C ^ l + T)^ o : Cn~* >C) a Cx,y(A)-contractible chain complex. By 3.9 there exists a bound 6 > 0 such that dC is (Cx(A),Cjv/-6(y,x)(A))-finitely dominated, with reduced projective class [dc] =
-BUOT{C^)
€ J = im(di9° : iJr'"'(Cx I y(A))—»^ 0 (Py(A))) . The n-dimensional quadratic Poincare pair (dC • Cn~*,(0,9t/>)) obtained by thickening (13.1) represents an element in the relative L-group
X;J(Py(A)—>Cx(A)). The algebraic Thorn construction and thickening define inverse isomorphisms [
—
*
I n (C x ,y(A)) ;
Ln(CXlY(f\)) a As in §4 write the complement of Y in X as
Z = X\Y . The .L-theory analogues of the excision isomorphism of 4.8 = lim Jfi(
14. EXCISION AND TRANSVERSALITY IN L-THEORY
123
will now be obtained, along with the exact sequence Inn Jfi(C M( y,z,x)(A)) —> ffi( 6
—* lim 6
COROLLARY 14.3 TAere w a natural
identification
= lim L» itA an exact sequence
—> lim 6
J 6 - ker(£o(PA/V(y,z>JO(A))—>#o(Pz(A))) c PROOF Given an n-dimensional quadratic Poincare complex (C,ij>) in there exists a sub complex D C. C such that D is defined in for some b > 0 and the quotient C/D is defined in )- T n e morphism i n (Cx,y(A)) —> lim 6
is an isomorphism inverse to the morphism lim ^ ( C M ( z > x ) i M ( y | Z > X ) ( A ) ) —> X n (C x 6
induced by the inclusions CM(zlx),M(y,s,x)(A) —> Cx,y(A) (6 > 0) . The exact sequences of 14.2
combined with the natural identifications give the exact sequence of the statement, on passing to the limit as b —• o o .
124
LOWER K- AND L-THEORY
THEOREM 14.4 For a metric space X expressed as a union X = X + U X~ of subspaces X + , X ~ C X the quadratic L-groups of the bounded X-graded category Cx(A) of a filtered additive category with involution A fit into a Mayer-Vietoris exact sequence 6
L n (C x (A)) —+ Km 6
connecting map is defined by
0 : L n (C x (A)) —> H m I * (£,) — •
d((E,6)/E+)
with E^ C J5 any subcomplex appearing in a Mayer-Vietoris presentation of E
E : o — ^ + n r - ^ E + ® E - ^ E —> o . PROOF For each 6 > 0 let L^b(Ab) be the cobordism group of adjoining pairs ( V + , y ~ ) of n-dimensional quadratic Poincare pairs in Cx(A)
such that C is C^6(x+,x-,x)(A)-finitely dominated in Cx(A) with projective class and such that D^ is homotopy Cx± (A)-finite. The common boundary of y + and V~ is a Cj\fb(x+,x- ,x)(^)"fin^ely dominated (n—1)-dimensional quadratic Poincare complex in Cx(A) By construction, L^b(Ab) is the relative L-group in the exact sequence — » LJn"(Ab)
with
a:
14. EXCISION AND TRANSVERSALITY IN L-THEORY
125
The abelian group morphisms defined by union U : Hm£*(A6) —+ Ln(Cx(A)) ; ( V + , r ) ^ F + u F 6
will now be shown to be isomorphisms by exhibiting explicit inverses. Define an abelian group morphism V : L n (Cx(A)) —-> limLJn»(Ab) ; (E,0) —> (V+,V-) 6
with the adjoining n-dimensional quadratic Poincare pairs
given by the construction in 13.3 of a splitting (V + , V~) for (E, 0) using the subcomplex D = £*+ C B, so that
(C,V0 = d((E,0)/E-)
, D+ = E+ , D~ = (E/E+)n~* .
It will be proved that V = U~x. The chain complexes D± are defined in CJV/-6(X±,X)(A), and hence homotopy Cx±(A)-finite. In order to verify that the chain complex C = d(E/E+) is Qvfc(x+,x-,X)(A)-finitely dominated in Cx(A) it is convenient to embed the given Mayer-Vietoris presentation of E as a subobject E C E(6) in one of the Mayer-Vietoris presentations constructed in 4.6 E(6> : 0 —> E+(b) H E-(b)
> E+(b) 0 E'(b) —> E —> 0 ,
which is possible for some sufficiently large bound b > 0. The (n — 1)quadratic Poincare complex d((E, 0)/E+ (b)) is obtained up to homotopy equivalence from d((E, 0)/E~*~) by algebraic surgery (Ranicki [61,§4]) on the n-dimensional quadratic pair with (E+(b)/E+)n-* defined in C J V 6 ( X + ) X - ) X ) ( A ) . Thus d(E/E+) is Qv6(X+,x-,X)(A)-finitely dominated if and only if d(E/E+(b)) is CAr6(X+,x-,X)(A)-finitely dominated, and there is no loss of generality in assuming that E = E(6). The same argument also shows that it suffices to verify that d(E/E+(b)) is C^6(x+,x-,X)(A)-finitely dominated for any bound b > 0. It will now be shown that for sufficiently large b the Mayer-Vietoris presentation E(6) is compatible with the quadratic structure 0 G Qn(E), and proceed with the verification for such 6. Define the quadratic Q-groups of a Mayer-Vietoris presentation of a chain complex E in Cx(A) E : 0
> £ + fl E~
> E+ 0 E~
by
Qn(E) =
Hn(W%(E+nE-)—
• E —• 0
126
LOWER K- AND L-THEORY
to fit into an exact sequence . . . —» Qn(E+ D E~) —» Q n (E+) 0 Qn(E-) —> QB(E) -^Qn.,(£+n£-)^ .... The long exact sequence of Z[Z2]-module chain complexes
0 _ > (£+ n £?-) ® c V { x + x . x ) ( A ) ( £ + n£T) —» ~) —+ E ® C j r W £ — (E/E+ ®Cx(A) W ~ ) © ( £ / £ " ®C*(A) ^/-E + ) —"» 0 (with the free Z[Z2]-action on (E/E+ ® £/JET) 0 (E/E~ ® E/E+)) induces an exact sequence of homology Z-modules . . . —» Hn+1(E/E+ ® JS/E-) —-* QB(E) —» —» Hn(E/E+ with Qn(E) >Qn(E) the union map. If the bound 6 in the construction of the Mayer-Vietoris presentation E = E(6) is chosen so large that the composite chain map (1 + T)00 (E/E+)n~* —> En~* —» E —> E/E~ is zero then 0 eker(Qn(E)^Hn(E/E+
®E/E~)) = im(Qn(E)—>Qn(^)) .
Choosing a lift of 0 G Q n ( ^ ) to an element ( 0 + , 0 ~ , 0 + fl 0") € Qn(E) there is defined a splitting (J7 + , U~ ,g) of (E,6) with
$ : £+0(£+n£-)r_i®£-
> Er ; (x + ,y,x")
• z + + ar" .
(The splitting (J7+,{7~,^) is not Poincare in general). The chain complexes 9i£ + , dE~ fit into an exact sequence of chain complexes in Cx(A)
o —> d(#+ n £ - ) —> 5E + e a^~ —> as1 —> o with BE' = S-iCiCiE+DE-—>E+©E-)n-*—>E) +
.
Now C(.E fl E~ >E+ © E") is chain equivalent to E, so that dE' is chain equivalent to dE, which is contractible. Thus dE+ and dE~ are complementary homotopy direct summands of d(E+ H -B"), and are both Cjvr6(x+,x-,X)(^)-fin^ely dominated in Cx(A). The quadratic complex (E, 0)/E~*~ is obtained from the quadratic Poincare pair U~~ by the algebraic Thom construction collapsing the boundary, so there is defined a chain equivalence d(E/E+) ~ C(S-\E+nE-)—>BE~) ,
14. EXCISION AND TRANSVERSALITY IN L-THEORY
127
and d(E/E+) is also C^(x+,x-,x)(A)-finitely dominated in Cx(A). The reduced end invariant of the isomorphism torsion of (1?, 0) T(E,$)
= r((l + T)do:En-*
>E) = -T{8E)
is given by dr(Et6) = [dE~\ = [d(E/E+)\
[
= [C]
iao
€ h = \m{d : Kr(Cx(f\))^ Since [M] € Jb for every object M in CJV 6 (X+,X-,X)(A) (assuming and X~ are non-empty) it follows that [V+ n V-] = [C] €JbC Ko(P*>(x+,x-,x)W) , as required for V(E, 6) = (V"1", F " ) to represent an element in Lj^Afc). For any n-dimensional quadratic Poincare complex (J5, ^) in Cx(A) let (V+, V") be the pair constructed above to represent V(£, 6) 6 L^6(A6). The union V + U V~ is homotopy equivalent to (E,6). Homotopy equivalent quadratic Poincare complexes are cobordant, so the composite UV : L n (Cx(A)) > lim L^(A 6 ) > i»(Cx(A)) ; 6
(E,9) —>V+UV~ is the identity. For any adjoining pair of n-dimensional quadratic Poincare pairs in Cx(A) V+ = ( / + : C — > £ > +
V- = ( / - : representing an element (y 4 ",^") € L*(Ab) let F + U T = (E,0) be the union n-dimensional quadratic Poincare complex in Cx(A). Let (V' + , V~,/i') be the splitting of (E,0) obtained by the construction of 13.3 using the sub complex D = D+ CE
=
The pair (V / + ,V'~) is homotopy equivalent to (V + ,F~). Homotopy equivalent pairs are cobordant, so the composite VU : lim LJn>(Ab) —+ Ln(Cx(f\)) —» lim 6
6
is the identity. Thus U, V are inverse isomorphisms, and the L-theory Mayer-Vietoris sequence is exact.
•
128
LOWER K- AND L-THEORY
REMARK 14.5 The algebraic Poincare transversality used in the proof of 14.4 can be developed further, giving a purely algebraic proof of Proposition 7.4.1 of Ranicki [63], and hence of the Mayer-Vietoris exact sequence of Cappell [15] for the L-groups of an amalgamated product of rings with involution A = A + *# A~ ... —» LJn(B) © UNil n+1 —> Ln(A+) © Ln(A~) —> Ln(A+ *B A~) —» LJn_x{B) © UNiln — • . . . with J = kei(K0(B) >K0(A+) © K0(A~)), and the analogue for an HNN polynomial extension A = A+ *B [Z,Z-1] ... —> LJn(B) 0 UNil n+1 —> Ln(A+) —>
•K0(A+)). See §16 below for the L-theory of with J = ker(K0(B) the Laurent polynomial extension A = B[z^z~1]. See Connolly and Kozniewski [20] for an expression of the UNil-groups as the L-groups of an additive category with involution. • COROLLARY 14.6 For a compact polyhedron Y C Sk expressed as a union Y = Y^ U Y~ of compact polyhedra F + , F ~ C Y the quadratic L-groups of the bounded O(Y)-graded category Co(Y)(A) of a filtered additive category with involution A fit into a Mayer-Vietoris exact sequence ... —> I ^ ( P o ( y + n y - ) ( A ) ) — • £ B (Co(y+)(A))ffiI n (C O (y-)(A)) —> £ B (C o ( y ) (A)) — with J =
i
^
C A' 0 (Po(y+ny-)(A)) .
PROOF Apply 14.4 to the expression of the open cone X = O(Y) as a union X = X + U X~ of the open cones of subpolyhedra F + , F ~ C F , with X± = O(Y±). By 7.1 the inclusions
i
+
nr
= o(r+ny-)—>Mh{x+,x-,x) (5>o)
are homotopy equivalences in the proper eventually Lipschitz category. The induced isomorphisms in the projective class groups
£
(b > 0)
14. EXCISION AND TRANSVERSALITY IN L-THEORY
129
send J to J&, so that there are also induced isomorphisms in the L-groups LJt(Px+nx-W) —> i* 6 (PM(x + ,x- ) x)(A)) (b > 0) , and the direct limits in the exact sequence of 14.4 are given by = LJ,(Px+nX-(f\))
.
b
• In order to formulate the intermediate versions of 14.4 it is convenient to assume that the bounded categories Ciy(A) for W = -X", -X"+, X~ have flasque structures (as is the case for the open cones W = O(F), O(y+), O(Y~) of 14.6) so that the intermediate L-groups Lj are defined for any *-invariant subgroup T C Ki(Cw(R))> COROLLARY 14.7 For X = X + U X~ and any ^-invariant subgroup T C KxiCxi^)) such that kev(d : IdiCxW)—+lmi
Ko(Ptfklx+9x-,X)W)) ^
T
b
there is defined a Mayer-Vietoris exact sequence of intermediate L-groups
6
PROOF Exactly as for 14.4, the special case T = ifi(Cx(A)) with dT = lim J&. D
Given *-invariant subgroups 5+ C ^,(C X + (A)) , 5 " C ^ ( C X define the *-invariant subgroups
= ker(lim
C lim Ari(C A r t(x +, x - >X )(A)) , T = i
130
LOWER K- AND L-THEORY
such that there is defined an exact sequence 0
> R/Ro
> S+®S~
> T
> 0.
COROLLARY 14.8 The intermediate quadratic L-groups fit into a MayerVietoris exact sequence
—> lim i £ 6
PROOF The proof of 14.4 shows that for any n-dimensional quadratic Poincare complex (E,0) in Cx(A) such that the torsion T(E,8) €
has reduced end invariant BT(E,0)
=
OGlim 6
there exists a Poincare splitting (V~*~,V~,h) with F + , V~ defined in for some b > 0, and V+ fl F ~ consequently defined in (A). (Specifically, replace d(E/E+) by any chain equivalent finite chain complex in Cjvrfc(x+,x-,X)(A)), for some b > 0.) The torsions riV*) € K1(Cx±(f\)) , T(V+nV~) 6 are related by T{E,6)
-
T(V+UV~)
T(V+UV~)
= r(h) + (-)nT(h)* ,
=
If (E, 6) is such that T(E, 6) € T then 5 T ( E , 0) = 0 and there exists such a Poincare splitting (V+, V~, h) with r(y±)G5±CA'1(Cx±(A)), r(F+ D V-) € i? C AVC x + n X -(A)) . Working as in the proof of 14.4 define the cobordism groups L* ' (A&) of adjoining cobordisms (V + , V~) with r ( y ± ) G S1*1 tofitinto the exact
15. L-THEORY OF Ci(A)
131
sequence
—• i f (C x + (A))©lf (CX-(A)) — Lsn+'s'(Ab) pass to the limits as 6 —* oo to obtain an exact sequence ... —» lhn L«
i f (C x + (A))©lf (C*-(A)) —> l i m i f >s' 6
6
and define inverse isomorphisms
[/ : lunif- S "(A 6 ) —» 2£(C 6
V : ^ ( C x ( A ) ) — . Urn Lsn+>s~ (Ah) ; (£, 6
§15. i-theory of d(A) The identification ^ ( C ^ A ) ) = A"0(P0(A)) of §7 will now be generalized to L-theory, and it will be proved that X n (d(A)) = Xn_i(Po(A)) for any filtered additive category with involution A. The isomorphisms £ n -i(Po(A)) >Z/n(Ci(A)) will be defined by the following construction, which is a direct generalization of the product (Ranicki [61], [62]) of a quadratic complex and the symmetric Poincare complex of 5 1 . DEFINITION 15.1 (i) Given a finite chain complex (C,p) in Po(A) define a finite chain complex in Afz,^"1] (C,p)®R = C a - p z i C I * , * - 1 ] — ^ [ z , * - 1 ] ) . (ii) Given an (n — 1)-dimensional quadratic (Poincare) complex (C,p, if?) in Po(A) define an n-dimensional quadratic (Poincare) complex in
132
LOWER K- AND L-THEORY
>((C,p)®R)r = C r k s - ^ e C V - i l * , * - 1 ]
(s>0). D
In order to apply the L-theory Mayer-Vietoris exact sequences of §14 express the bounded R-graded category as CR(A) = C o( x+uX-)(A) , with
i + u r = 5° - {+1,-1}, x± = {±i}, o(x±) = R± , O ( i + n r ) = O(x+)no(x-) = {0}, o ( i + u r ) = O(X+)DO(X-) = R as in §6, using the equivalence Ci(A) « CR(A) to identify A Mayer-Vietoris presentation of a finite chain complex E in Ci(A) is the exact sequence E
:
0
> E+ fl E~ - ^ E + 0 £ T - ^ ^
• 0
+
determined by subcomplexes i£ , .B" C £ such that for some b > 0
£r+ C J £r(i) , E-C "£ Er(j) (r > 0) . j=—b
j = —oo
THEOREM 15.2 TAe quadratic L-groups of the bounded Z-graded category Ci(A) are such that up to natural isomorphism
MCi(A)) = £ n -i(Po(A)) (n € Z) . TAe map 5 : Ln{d(l\))
= I B (C R (A)) —» I n _,(P 0 (A)) ;
(E,0) -+ d((E,6)/E+) defined using any Mayer-Vietoris presentation E of E is an isomorphism with inverse B-1 = - ® R : LB_!(P0(A)) — I n ( PROOF The natural flasque structures on the categories CR± (A) are compatible with the involutions, so that £.(C R± (A)) = 0 .
CQ(X±)(^)
=
15. L-THEORY OF Ci(A)
133
The exact sequence of 14.4 includes 0 —» Ln(Cx(A)) —-• X n -!(P 0 (A)) —-> 0 , so that B — d is an isomorphism. For any (n — l)-dimensional quadratic Poincare complex (C,p, t/>) in Po(A) let (E,0) = (C,p,tf)®R and define a Mayer-Vietoris presentation E for E by E+ = C(l-zp:C[z] >C[z]),E- = C(l-zp:C[z^] >zC[Z-1)). The Co(A)-finitely dominated (n — l)-dimensional quadratic Poincare complex defined in Ci(A) by d((E,0)/E*) is homotopy equivalent to (C,p,^)®R, so that B(E,9) = (C,p,0) €L n -i(Po(A)) and D
The isomorphism B : A'i(Ci(A)) B* = -*B : /fi( since the inverse
is such that *5[M,p] =
^A"0(P0(A)) of §7 is such that
r(l-p*+z-ip*:M*[z,z-1]
It follows that =
J ff*"
1
(Z 2 ;ii:o(Po(A))).
COROLLARY 15.3 (i) The intermediate quadratic L-groups o/Ci(A) are such that up to natural isomorphism /or any ^-invariant subgroup S C ATi(Ci(A)) (resp. Wh(Ci(f\))),
with
dS C ATo(Po(A)) (Vesp. iifo(Po(A))^
(ii) For any ^-invariant subgroups S C T C A"i(Ci(A)) ^Ae exact 5e—»
Hn(l2;T/S)
134
LOWER K- AND L-THEORY
is naturally isomorphic to the exact sequence
Similarly forSQTC Wh(Ci(f\)). PROOF (i) As for 15.2 (the special case 5 = lfi(Ci(A))), but using 14.6 and 14.7 instead of 14.4. (ii) Immediate from (i). D
In particular, 15.3 gives
with LI the round L-groups. REMARK 15.4 The total surgery obstruction groups S*(X) of a space X are defined in Ranicki [60], [69] to fit into an exact sequence ... ^Hn(X;L.)
—•Ln(Z[*])—»SB(X)—>#„_!(*;L.)—*
...
with 7T = fti(X) and TT*(L.) = £*(Z). The total surgery obstruction s(X) G S n (X) of a finite n-dimensional geometric Poincare complex X with n > 5 is such that s(X) = 0 if and only if X is homotopy equivalent to a compact n-dimensional topological manifold. The total projective surgery obstruction groups Sj(X) are defined to fit into an exact sequence ... —> Hn(X;L.)
—> 2£(Z[*]) —
with a Rothenberg-type sequence
The following conditions on a finitely dominated n-dimensional geometric Poincare complex X with n > 5 are equivalent: (i) the total projective surgery obstruction sp(X) £ S£(X) (Pedersen and Ranicki [52]) is such that sp(X) = 0, (ii) X x S 1 to be homotopy equivalent to a compact (n + 1)-dimensional topological manifold, (iii) there exists an open (n + 1)-dimensional manifold W with an Rbounded homotopy equivalence h : W ~ X x R. Thus if X is a finite n-dimensional geometric Poincare complex with
15. L-THEORY OF Ci(A)
sp(X)
135
= 0 the total surgery obstruction is such that
s(X) G ker(Sn(X)—->S>n(X))
= im(Hn+\l2;K0(1[K)))—>Sn(X))
.
n l 1
The corresponding element of if " " (Z2;i^o(Z[7r])) is determined by a choice of R-bounded homotopy equivalence h : W ~ X x R, and may be regarded as either the class of the Siebenmann end obstruction [W]+ G #O(Z[TT]) or the bounded Whitehead torsion r(h) G W ^ C I ^ T T ] ) ) , as follows. In the first instance, note that for any finite n-dimensional geometric ».Ko(Z[7r]) Poincare complex X the isomorphism B : W/I(CI(Z[TT])) sends the torsion T(X X R) to Br(X
x R) = [X x R]+ + (-)n([X
x R]+)* = 0 G K0(l[*])
so that T(X x R) = 0 G W7i(Ci(Z[7r])) and X x R is a simple (n +1)-dimensional R-bounded geometric Poincare complex. Given an R-bounded homotopy equivalence h : W ~ X x R make it transverse regular at X x {0} C I x R, so that there is defined a normal map
(/,&) = h\ : M = h-\Xx{0})
>X
from a compact n-dimensional manifold M. By definition, h splits if there exists a normal bordism with (V; W , W ) a bounded /i-cobordism and ti : W •X x R an Rbounded homotopy equivalence such that the restriction
(/',*>') = h'\ : M' =
h'-\Xx{0})—>X
is a homotopy equivalence. By analogy with the splitting theorems of Farrell and Hsiang [24] and Wall [84, §12B] the obstruction to splitting h for n > 5 can be identified with the Tate Z2-cohomology class of the bounded torsion using the isomorphic exact sequences
(ZW)) —> L B+2 (C,(ZW))
136
LOWER K- AND L-THEORY
The primary obstruction to splitting h is the simple bounded surgery obstruction of Ferry and Pedersen [28], which is determined by the torsion r(h) € and is just the finite surgery obstruction of (/, 6)
<(h) = am(f,b) € L^idilln]))
=
L»(Z[T]) .
Working as in Pedersen and Ranicki [52] it is possible to identify X up to homotopy equivalence with the homotopy inverse limit of a system of neighbourhoods of the end W+ = ft-1(R+) of W X ~
^
and to extend (/, 6) to a normal map offinitelydominated n-dimensional geometric Poincare bordisms (e,a) : (W+;M,X) — X x ([0,1]; {0},{1}) >X. The finite bounded surgery obstruction of h with e| = id. : X and the projective surgery obstruction of (/, b) are both zero a,{h) = (/,6) = 0€L B + i(C,(ZM)) - L'n(Z[*}) . The finite surgery obstruction of (/, b) is determined by the bounded torsion r(h) € VTMCi(ZW)) , which is the Siebenmann end obstruction of = [w}+ = [w+] e Both W and X x R are simple R-bounded geometric Poincare complexes BT{K)
T(W)
- T ( X x R) = 0 € WA(C!(Z[jr])) ,
and = r{h) + (-)n+1r(h)* so that The finite surgery obstruction of (/, b) is = ker(L n (ZM)— If this vanishes then (/, b) is normal bordant to a homotopy equivalence; let
15. L-THEORY OF Ci(A)
137
be such a normal bordism, restricting to a homotopy equivalence M1 ~ X. The open n-dimensional manifold w' = W-\JMLUM>-LUMW+ is equipped with a split bounded homotopy equivalence h! : W related to ft by a normal bordism (g,c) : (V; such that the finite bounded rel d surgery obstruction
»X,
[
For n > 5 it is possible to make a choice of (L, M, M1) with (F; W, W) a bounded /i-cobordism if and only if <j.{g,c) € i m ( I ^ + 2 ( C 1 ( Z M ) ) ^ X n + 2 ( C 1 ( Z H ) ) )
= im(L n+1 (ZW))—-LJ +1 (ZW)) , giving the secondary obstruction. (This is just the appropriate version of the trick used by Browder [10] for dealing with an embedding problem by first solving it up to cobordism). In the bounded version of the terminology of Wall [84, §12B] the bounded splitting obstruction is r(h) = [W]+€LShn{*)
=^+2(Z
REMARK 15.5 By the realization theorem of Ferry and Pedersen [28] for a finitely presented group ?r and n > 6 every element x € Ln(Ci(Z[7r])) is the bounded surgery obstruction x = cr*(e,a) of an n-dimensional bounded normal map > (X,dX) x R (e,a) : (W,dW) with (W, 9W^) an R-bounded n-dimensional manifold, (-X", dX) a compact (n — l)-dimensional manifold with boundary such that 7Ti(dX) = TTI(X) = 7T and h = e\ : dW >dX x R an R-bounded homotopy equivalence. Making (e, a) transverse regular at (X, dX) x {0} C (X, dX) x R there is obtained an (n — 1)-dimensional normal map of pairs (/,&) = e\ : (M,dM) = e~\(X,dX) x {0}) > (X,dX) . As in 15.4 it is possible to extend df : dM >dX to a normal map of finitely dominated (n — 1)-dimensional geometric Poincare bordisms (g,c) : (dW+;dM,dX)
— dX x ([0,1]; {0}, {1})
138
LOWER K- AND L-THEORY
with g\ = id. : OX >dX, and dW+ = e~\dX x R+) C dW. The if-theory isomorphism B : W&(C!(ZW)) —> Xo(ZW) sends the bounded torsion r(h) € W^(CI(Z[TT])) to the end obstruction Br{h) = [dW+] e K0(l[*]) . The L-theory isomorphism
B : MC,(ZM)) —» ^-i(ZW) sends the bounded surgery obstruction x = a*(e,a) 6 Ln(Ci(Z[7r])) to the projective surgery obstruction of the normal map of finitely dominated (n — l)-dimensional geometric Poincare pairs (f',b') = (f,b)U(g,c) : (MUdW+,dX)^(X,dX), with the identity on the boundary, that is
Bvl(e,a) =
§16. L-theory of A^z"1] The splitting of K\(f\[z, z~1]) obtained in §10 will now be generalized to L-theory, and it will be proved that up to natural isomorphism
LniAfoz-1]) = I«(A)©XB_,(Po(A)) for any additive category with involution A. The duality involutions act on the split exact sequence of §10 ?!
B®N+®N-
0 —>
^o(Po(A)) 0 NT1O(A) © NUo(A) —> 0 by so there is induced a split exact sequence of the Tate Z2-cohomology groups
0 —» ir(Z 2 ;
^ ^
Similarly for the Whitehead group W ^ A ^ z " 1 ] ) , with KQ replaced by Ko- In particular, for any group % there is defined a split exact sequence 0 —> Hn(l2;Wh(*))
—* Hn(l2;Wh(n x Z))
16. L-THEORY OF A[*,*- x ]
139
The geometric model for the L-theory splitting is the Kiinneth theorem for manifold bordism and its analogues for geometric Poincare bordism. Here, X is any reasonable space (such as a finite CW complex) and iln(X) is the bordism group of pairs (M, / ) with M a compact n-dimensional manifold and / :M >X a reference map. The manifolds can be in any category with transversality, such as smooth, PL or topological, and can be taken to be oriented if required. The splitting of the manifold bordism groups is given by a direct sum system
with i : Qn(X) — Sln(X x S1) ; (M,/) —> (M, 9 /) (g : X >X x 5 1 ; x >(x, *) for a base point • € S1) , B : Qn(X x S1) > fin-i(X) ; (N,g) > (ff" 1 ^ x {*}),ff|) (assuming g : N >X x S1 transverse at X x {*} C X x S1) , j : Qn(X x S 1 ) ^ iln(X) ; (N,g) —» (JV,W) • X) , (p = projection : X x S 1 1 C : fin_i(X) > fin(X x 5 ) ; (M,/) —> (M x S\f x 1) . Let Q^(X) (resp. Q^(X)) be the bordism group of pairs (M,/) with M a finite (resp. finitely dominated) n-dimensional geometric Poincare complex and / : M >X a reference map. For n > 5 the finite and finitely dominated geometric Poincare bordism groups are related by the exact sequence of Pedersen and Ranicki [52]
The map fiP(X) — , J»(Z 2 ;^ 0 (ZW)) ; (M,/) —» [M] sends a finitely dominated Poincare bordism class to the Tate Z2~cohomology of its projective class, and the map
£"(Z2;£o(ZM)) —• n^X)
; x —» (N,g) n
sends the Tate Z2-cohomology of a self (—) -dual projective class
x = (-)"*• e ^o(ZM) to the bordism class (N,g) of any finite (n — 1)-dimensional Poincare complex N with a map g : JV ^X for which there exists a finitely
140
LOWER K- AND L-THEORY
dominated Poincare nuU-bordism (6N, 6g) with projective class [SN] = x. Geometric Poincare complexes do not have transversality: if N is a finite n-dimensional geometric Poincare complex and g : N >X x S1 is a map it is not in general possible to make g Poincare transverse at X x {*} C X x S 1 , with g~x(X x {*}) C N a finite codimension 1 Poincare subcomplex. For n > 6 the obstruction to the existence of such a subcomplex up to bordism is the image of the Tate Z2-cohomology class of the torsion r(N) G Hn(l2] Wh{ic x Z)) under the projection B : Hn{l3;Wh(*
x Z)) —> Hn~l {12; KQ{T.[K})) .
The obstruction vanishes for (JV,ff) x S 1 = (N x S\ x 1) € fi*+1(X x S 1 x S 1 ) , and there exists a finite codimension 1 Poincare subcomplex Q = (gx
1)-\X
x {*} x S 1 ) C N x 5 1
such that Q ~ P x S1 with P = Q the infinite cyclic cover of Q classified by Q
— , N x S1 -^^+
S1 .
Now P is a finitely dominated (n — 1)-dimensional geometric Poincare complex with a codimension 1 Poincare embedding P C iV, and the projective class is such that
[p] = iMiv)G ^ ( z ^ A y z M ) ) . The construction defines a map
B : tlkn(X x S1) —» ftJUW ; (JV.y) — . (P,ft) (ft = g\) which fits into a split exact sequence o —* nhn(x) - ^ n*(x x with
51) i
^_!(x) — o
t : fi*(X) — ^ Q*(X x S1) ; ( M , / ) —
(M,9/)
as in the manifold case. In accordance with the distinction established in Ranicki [66] there are two natural choices for splitting this sequence, given by the 'algebraically significant' direct sum system i B h h 1 a n(x) ] > n n(x x s ) j B and the 'geometrically significant' direct sum system B h 1 n n(x x s )
B'
16. L-THEORY OF A!*,*" 1 ]
141
with j : Slhn(X x S1) —» Qhn(X) ; W J ) —> (tf.pg) , 5 ' : fi*_x(X) —>fi*(Xx 51) ; (M,f) —» (M x 5 \ / x 1) using the mapping torus trick of Mather [46] to replace M x S1 by a homotopy equivalent finite n-dimensional geometric Poincare complex in the definition of B1'. The difference between the algebraically and geometrically significant split injections B, B' is given by
B'-B
: il^iX)
Hn-\l2;Ko(l[n])) = £
^
^ fi*(x x s1). The bordism groups il^(X) of pairs (M, / : M >X) with M a simple n-dimensional geometric Poincare complex fit into an exact sequence and a split exact sequence
0 —> fi«<X) - ^ - fisn(X x S1) i
ft*_,(X)
—» 0 .
Again, B has both an algebraically significant splitting B and a geometrically significant splitting B', with B'-B
: Q^iX)
-^
Hn-\l2;Wh(n))
= Hn+1(Z2;Wh(n))
The chain complex proof in Milgram and Ranicki [47] of the L-theory splitting theorem for the Laurent polynomial extension A ^ z " 1 ] of a ring with involution A Lkn{A[z,z-x\) =
LlWQL'^iA)
depended on an algebraic transversality result: every free quadratic Poincare complex over Afz,^"1] has a 'fundamental domain' projective quadratic Poincare pair over A, by an algebraic mimicry of the geometric transversality by which every infinite cyclic cover of a compact manifold has a compact fundamental domain. The same method works for L n (A[ 2 ,z- x ]) = £ n (A)9£ n _ 1 (P 0 (A)) for any additive category with involution A, using the locally finite Zgraded category with involution Fi(A) = F2(A) . DEFINITION 16.1 (i) An n-dimensional quadratic cobordism in A X =
((fg):C®Cl^
142
LOWER K- AND L-THEORY
is fundamental if (ii) The union of a fundamental n-dimensional quadratic cobordism X in A is the n-dimensional quadratic complex in A^,;?" 1 ] U(X) = (E,6) with
Er =
V(-) B - r -v.r (-r- r - s TV' S+ i; • £;»-••-• =
Dn-r-a[z,z-1]®Cn-r-s-1[z,z-1]
—v Er = u ^ r ' i e c - i ^ r 1 ] . (iii) A fundamental cobordism X in Fi(A) is finitely balanced if C and £> are (Gi(A),Co(A))-finitely dominated and [C] = [D] € ^o(Po(A)) , in which case
[U{X)\ = [D^z-^-lC^z-1]}
= OeAVPolA^r1]))
and the union U(X) is homotopy Afz^'^-finite. D
The union of a fundamental cobordism X can be viewed as the 'infinite cyclic cover' U(X) = ( j=-oo
obtained by glueing together a countable number of copies of X.
If X is a fundamental quadratic Poincare cobordism in A (resp. Fi(A)) then the union U(X) is a quadratic Poincare complex in A[2, z~l\ (resp.
16. L-THEORY OF f\[z,z~x]
143
THEOREM 16.2 The quadratic L-groups L J|t (A[z,z" 1 ]) of a Laurent polynomial extension category A^,^"" 1 ] fit into a split exact sequence
0 —> Ln(f\) —+ M A l v " 1 ] ) - ^ In_1(P0(A)) —> 0 , witA 2? tAe composite of the map Ln(f\[z,z~1]) >Ln{C\(f\)) induced x by the inclusion I\[z,z~ \ C Ci(A) and the isomorphism of 15.2 *± Ln-tiPoW) . B : LnidW) PROOF Let Ln(V) be the cobordism group of funamental n-dimensional quadratic Poincare cobordisms in Po(A)
X = ((/0):C©C—>D,(W,tf©-tf)) which are finitely balanced. The groups L*(V) fit into split exact sequences 0 —-> Ln(A) —> Ln(V) —> £ n -i(P 0 (A)) —> 0 , with LB(A) — L B (V) ; (£>,^) —» ( 0 — i J . W . O ) ) , I n ( V ) —» X n _!(P 0 (A)) ; X — • (C,V) • It is now possible to work exactly as for the case A = B^(A) in §§14 of Milgram and Ranicki [47], using the L-theory of the locally finite additive category with involution Fi(A). Specifically, it is claimed that the union defines an isomorphism U : I n ( V ) —* Ln{IK[z,z-l\) ; X —» U(X) . For any finite n-dimensional quadratic Poincare complex (E,0) in A[z,z -1 ] and any Mayer-Vietoris presentation of E in Ci(A) > E+ fl E~ - ^ E + 0 E~ -^ E >0 E : 0 there exists a finitely balanced fundamental n-dimensional quadratic Poincare cobordism X in Fi(A) with such that (E,0) is homotopy equivalent to U(X). The inverse of U is defined by U~x : Lnibfoz-1])
—• L n (V) ; (£,0) —> ^ D
For a ring with involution A and A = B-^(A) 16.2 recovers the split exact sequence of Ranicki [57]
0 —> Lhn(A) —> LltUIz.z- 1 ]) " ^ XB_1(A) —» 0 . As for K\(l\[z,z~1]) and S)J(X x S1) there are two distinct natural
144
LOWER K- AND L-THEORY
choices of splitting Ln(f\[z,z~1}), direct sum system it
given by the algebraically significant B 1
MA) ,
MAI*,*- ]) ,
Ln-xf B
j<
and the geometrically significant direct sum system *!
B 1
MA) •
MAf*,*" ]) ,
if
Xn_!(P0(A))
&
with ji induced by j : f\[z, z~l]
>A; L[z, z" 1 ]
*L and
B' = - ® R : L n - ^ See Ranicki [66] for further details concerning the two types of splitting. REMARK 16.3 Assume now that A has a a stable canonical structure which is compatible with the involution, so that an n-dimensional quadratic Poincare complex (E,6) in At^,^"1] has a torsion T(E,6)
= T(60 : En~*
>E) € WhiP^z^-1]) .
The proof of 16.2 gives afinitelybalanced fundamental quadratic Poincare cobordism in Fi(A) X = ((fg):C®C^D,W,1>®-1>)) such that (E,0) ~ U(X) and BT(E,8)
= [C] = [D] € AVPo(A)) .
The Tate Z2-cohomology class that BT(E,9) -
T(E,0)
£ Hn(Z2;Wh(f\[z,z-1}))
is such
OeH^^KoiPoim
if and only if there exists a fundamental quadratic Poincare cobordism X in A such that (E,0) ~ U(X). D
As usual, there are also versions of the splittings MAI*,*" 1 ]) = MA)©£»-i(Po(A)) for the intermediate and round L-groups, such as
with LI = Li 'simple L-groups.
the simple L-groups and JDJ* = LI
x
the round
17. LOWER L-THEORY
145
§17. Lower L-theory The lower quadratic L-groups L* m '(A) (m > 0) are defined for any additive category with involution A, and the connections with Laurent polynomial extensions and the Rothenberg exact sequences are obtained. By convention, write >
= I n (A) ( n £ Z ) .
DEFINITION 17.1 The lower quadratic L-groups of A are defined for m > 0 by
• It is an immediate consequence of the 4-periodicity of the quadratic L-groups L*(A) = L*+4(A) that the lower quadratic L-groups are also 4-periodic L<~m>(A) = L ^ ^ A ) ( m > 0 , n e Z ) . The duality involutions act on the split exact sequence of §10 i. B 0 iV+ 0 N0 > Ki(f\) > Ki{f\[z,z~1]) > A'o(Po(A)) 0 NUo(A) 0 Nilo(A)
>0
by it* = *zi , B* = — * B , N±* = *N^ ,
so there is induced a split exact sequence of the Tate Z2-cohomology groups 0—» J"(Z2;if1(A)) —^
Hn(l2;K1(f\[z,z-1}))
Define the duality involution on the lower if-groups * : K.m(F\) —> ff_ m
to agree with (—) the projective class duality or equivalently with (—)m+1 the torsion duality ( - ) m + 1 * : ifi(CTO+1(A)) — ifi( With this convention there are defined split exact sequences 0
> ff"(Z 2 ;Ki_
#
146
LOWER K- AND L-THEORY
for all m > 0. Define the reduced lower K-groups of A by ( = K-m(l\)
for m > 1) .
THEOREM 17.2 The lower quadratic L-groups are such that for all m > — 1 and n £ Z
= X m+n (P m (A)) = im+n+l(C m + and there are defined Rothenberg exact sequences
PROOF By induction on ra, with the initial case m = 0 already obtained in §16. As usual, there are two preferred ways of splitting the exact sequence
0 —» 2£-m+1>(A) —^ ^ - " ^ ( A ^ z - 1 ] ) i
Li-T^PoCA)) —» 0 ,
an algebraically significant one compatible with the functor j : A[2, z~*] >A and a geometrically significant one involving the split injection
& = - ® R : xilT^PoCA)) _4-»+ 1 >(A[^2 ? - 1 ]). D
PROPOSITION 17.3 For any m>\ the quadratic L-groups L* — L^ of the m-fold Laurent polynomial extension A[Zm] of A are such that up to natural isomorphism t=0
PROOF Iterate the case m = 1 obtained in §16. Again, the natural isomorphism can be chosen to be either algebraically or geometrically significant. The sum of the algebraically significant summands for i; > 1 is the subgroup ker(ii : Ln(A[Zm])—>Ln(A)) C I n (A[Z m ]) with j : A[Zm] > A ; L[Zm] > L the augmentation functor. D
17. LOWER L-THEORY
147
The geometrically significant summand im(B[B'2 ...B'm = -®Rm:
I<,I- +1 >(A)-^L n (A[Z m ])) C Ln(f\[Zm})
will be identified in §18 below with the subgroup of the Tm-transfer invariant elements. The lower quadratic L-groups of a ring with involution A L*-m(A\A)
= 4"m)(A)
(m>0,n£Z)
were defined in Ranicki [57] inductively to fit into the algebraically significant splittings
L<-m\A\z,z-1]) = 4—>(A)e4iT~1>(A) ( m > - l ) . Here, A[2, z" 1 ] is the Laurent polynomial extension ring «A[z, z""1], with the involution extended by z = z'1 . For m = 0,1 these are the projective and free L-groups of A
L\A) = Ll{A) , L?\A) = Lt(A) . EXAMPLE 17.4 The lower quadratic L-groups of a ring with involution A are the lower quadratic L-groups of the additive category with involution B-f (A) of based f.g. free A-modules
Li~m\A) = L{-m)(Bf(A))
(m > -1) .
This is immediate from 17.2 and the identifications
Cm(A) = dCCn D
REMARK 17.5 Given a ring with involution A let A[x], A[x~x], A[x, x~x] be the polynomial extension rings with the involution extended by x = x. m
The lower i-groups Ll~ '(A) (m > —1) were shown in Ranicki [59,4.5] to fit into split exact sequences
0 —• L<-m>(A) —> 4" m> (^M) © 4 ~ m > ( ^ - 1 ] ) — . 4-m>(A[x,x"1]) —» 4 - l - 1 > U ) — 0 , by analogy with the lower if-theory split exact sequences 0 —* K-m(A) —+ X_ m (A[ a; ])eK_ m (A[x- 1 ]) —» ^ ^ ( ^ [ x ^ - 1 ] ) — . K-m-x{A)
—» 0 .
148
LOWER K- AND L-THEORY
DEFINITION 17.6 The intermediate lower quadratic L-groups Lf (A) are
denned for a *-invariant subgroup 5 C K-m(f\)
(m > 1) by
C
if(A) = im°+n(Pm(A)) = i£ + »+i( «+i(A)) (n € Z) with So C Ko(Pm([\)), S\ C -ftTi(Cm+i(A)) the *-invariant subgroups isomorphic to 5. D
The intermediate lower L-groups associated to *-invariant subgroups 5 C 5 ' C K-m(f\) are related by the usual Rothenberg exact sequence
... —> I f (A) —» i f (A) —» ff"(Z2;S'/S) —»
^
and
DEFINITION 17.7 The ultimate lower quadratic L-groups of an additive
category with involution A are defined by the direct limits =
lim Li"m>(A) („ € Z) , m>oo
with L{n~m\f\)
>Lfc"m~1>(A) the forgetful maps.
The ultimate lower L-groups are such that
The following identification of the functor X >L^i (Co(x)(A)) with a non-connective delooping of L-theory was obtained by Ferry and Pedersen [28] using geometric bounded surgery methods. (A lower Ltheory delooping was previously obtained by Yamasaki [89] in the context of controlled surgery). See Ranicki [69, Appendix C] for the connection with the assembly map in algebraic L-theory. PROPOSITION 17.8 For any filtered additive category with involution A the functor {compact polyhedra}
> {Z-graded abelian groups} ;
x _ » L<; is a reduced generalized homology theory with a 4-periodic non-connective coefficient spectrum L^~°°^(A) such that
= JSf.(X;L<-~>(A)). PROOF For homotopy invariance it suffices to consider the ultimate lower L-theory of Co((7x)(A), with O(CX) the open cone on an actual cone
18. TRANSFER IN L-THEORY
149
CX of a compact polyhedron X. The natural flasque structure defined on CQ(CX)(R) m 3.3 is compatible with the involution. For each ra > 0 the category C m (C O (cx)(^)) n a s a natural flasque structure which is compatible with the involution, so that ii" m > (C o ( cx)(A)) = £ m + . + 1 (C m (C o ( cx)(A))) = 0 , and passing to the limit ii" oo> (C O (cx)(A)) = Jm^ LS"i 1 (C m (C o( cx)(A))) = 0 . For exactness let X = X + U X~. For each m > 0 define the *-invariant subgroup •tf-m-i(Po<x+nx-)(A))) F_m = im(0:tf_ m (C o( x+uX-)(A)) C #_,„_!(P o( x+nx-)(A)) , which is naturally isomorphic to the *-invariant subgroup YLm = im(a:JST1(Co(x,+uX,-)(A))—»^0(Po(X'+nx-)(A))) C A'0(Po(X'+nX'-)(A)) m+1 /=t 1 ± with X' = E X, X = E™+ X the (m + l)-fold reduced suspensions. By 14.6 there is defined a Mayer-Vietoris exact sequence
Passing to the limit as m —> oo and identifying
gives the ultimate lower L-theory exact sequence
r\
§18. Transfer in L-theory In §12 the lower K-group J^1_m(P0(A)) (= A"i_m(A) for m > 2) was identified with the subgroup Ki(f\[lm])INV of the Tm-transfer inm variant elements in Ki(Fk[Z ]). The same is now done for L-theory, identifying the lower L-group Ln~m'(A) (m > 1) with the subgroup
150
LOWER K- AND L-THEORY
Lm+n({\[Lm])INV of the T m -transfer invariant elements in L m + n (A[Z m ]). Again, start with the case m = 1. As in §16 consider first the case of manifold bordism. The (/-fold •S 1 ; z >zq induces transfer maps covering map q : S1 Qn(X x S1) ; (W,f)
using the pullback diagram 41x5' 1xq
f
w
If / : W >X x S1 is transverse regular at X x {*} C X x S1 with inverse image the codimension 1 framed submanifold M = f-^X !
!
x{*})cW
1
then / : W >X x S is also transverse regular at X x {•} C X x S1 with inverse image a copy of M
M S {fTl{Xx{*})cW- , so that = £(W ! ,/ ! ) = (M,/|) = B(W,/)Gfi B -i(X).
Bq\W,f)
Moreover, if / : W
>X x {*}
>X x 5 1 then
[
q\WJ) = (W'J ) = \}(WJ) = q(W,f)eQn(XxS1). q
Thus the bordism transfer map fits into a morphism of split exact sequences 1
0
-^(XxS1)
B
.1
•ftn-l(X) 1
B
0
•0
•0
and the image of the split injection C : fin-i(X) —+ Qn(X x S 1 ) ; (M,/) — • (M x 5 1 , / x 1) coincides with the subgroup of transfer-invariant bordism classes
nn(x x 5 1 ) / N V = {(W, f) € fin(X x S1) I ?!(W, / ) = (W, / ) for all q > 1} . The identity im(C :
fin-i(X)
>Sln(X x S1)) = Sln(X x S 1 )
18. TRANSFER IN L-THEORY
151
allows the identification Similar considerations apply to geometric Poincare bordism, with a morphism of split exact sequences
o
o
> a*(x) — ^ ft*(x x s1) - - * _ > QZ-dX)
•o
^
It follows that there is an identity with B' the geometrically significant split injection, allowing the identification
nhn(x x s1?™ =
si'^ix).
For an additive category with involution A and any integer q > 1 the g-fold transfer on the bounded Z-graded category g-l
]
q : d(A) _ > d(A) ; M —> q'M , q'M(j) = £ M(^j + i) ik=0
is a functor of additive categories with involution. The #-fold transfer functor on the Laurent polynomial extension category
q- : At*,*" 1 ]—>A[z,*- 1 ];
L[z,z-*] —» JiLfrz-1]) =
(EL^z-1]
induces the q-fold transfer maps in the algebraic X-groups ql : Ln(f\[z,z-1))
-^
Ln(f\[z,z-1])
.
By analogy with 12.1 and 12.2: DEFINITION 18.1 The S1 -transfer invariant subgroup of i ^ A ^ z " 1 ] ) is Lni^z-1])1™
= {x e Ln(f\[z,z-1))\q\x) = x for every q> 2} . D
PROPOSITION 18.2 (i) The effect of q' on the geometrically significant direct sum decomposition of Ln(fK[z,z~l\) is given by the commutative
152
LOWER K- AND L-THEORY
diagram £ n (A)©X n _ 1 (P 0 (A)) gffil i n (A)©£ n _ 1 (P 0 (A)) (ii) The image of the geometrically significant split injection is the subgroup of the S1-transfer invariant elements L n -i(P 0 (A)) = im(B') = Ln(f\[z,Z-l))INV . (iii) The effect of q' on L n (Ci(A)) is q] = 1 : L n (Ci(A)) —+ Ln(C!(A)) . PROOF (i) By the proof of 16.2 it is possible to regard Ln(A[z, ^ - 1 )) as the cobordism group of n-dimensional quadratic Poincare cobordisms in Po(A) such that
[C] = [D]ek The <j-fold transfer of X is the union of q copies of X
q'X = [JX = ((f'_f+):C®C
>D',(6tl>',
Q
with D' = C(g) the algebraic mapping cone of the chain map / /+ 0 0 ... 0 \ -//+ 0 ... 0 0 - / - /+ ... 0 9 = 0 \ 0
0 0
0 0
... ...
q-l
/+ -/_/ q
]
The relations B'q' = J5', i\q = i\q are now clear. (ii)+(iii) Immediate from (i). Let m > 1. For any m-tuple Q = (^1,^2? •••^m) of integers qj > 1 the Q-fold transfer functor of additive categories with involution
<
] —• A[Zm] ; L[lm) — . ( ^ £ ii=oj 2 =o
. . . ' £ I)[Zm] jm=o
19. SYMMETRIC L-THEORY
153
induces transfer maps in the L-groups Q! : L,(A[Zm]) — L,(A[Z™]) . By analogy with 12.3 and 12.4: DEFINITION 18.3 The Tm-transfer invariant subgroup of Ln(A[Zm]) is Ln(f\[Zm])INV = {x € Ln(A[Zm]) | Q\x) = x for every Q} . D
PROPOSITION 18.4 For each m > 1 tAe geometrically significant split injection
B[B'2...B'm : j t f - ^ A ) — Lm+»(A[Zm]) maps tfee lower quadratic L-group to the subgroup of the transfer invariant elements in L m + n (A[Z m ])
i t f - ^ A ) = im(B'1B'2...B'm) = Lm+n(f\[Zm])INV . PROOF This follows from the case m = 1 dealt with in 18.2 and the factorization of the Q-fold transfer for Q = (q\, #2, • • • ? Qm) as the m-fold composition of the single transfers Q' = (qi)\
n
§19. Symmetric L-theory The symmetric L-groups i*(A) of an additive category with involution A are defined in Ranicki [68] to be the cobordism groups of symmetric Poincare complexes in A. The symmetric L-groups of A = B^(A) for a ring with involution A are the symmetric L-groups L*(A) of Ranicki [61]. Except for 4-periodicity all the results on the quadratic L-groups L*(A) of §13-§18 have symmetric L-theory analogues, with entirely analogous proofs. In particular, there are defined lower symmetric L-groups -k/Lm\(A) (m > 0, n € Z). The one noteworthy feature is that for n < — 3 the lower symmetric L-groups coincide with the lower quadratic L-groups L{n~m)(A). Let e = ±1. Given a finite chain complex C in A let Z2 act on C ®f\ C by the e-transposition involution T€ : C ® A C —> C 0 A C ; <j> —> eTcj> . f e-symmetric
The < , . Q-groups of C are defined by ^ e-quadratzc (Qn(C,e) = i \Qn(C,e) =
156
LOWER K- AND L-THEORY
PROPOSITION 19.4 The lower e-symmetric L-groups are such that for d!m>0;n6Z m
Ln(f\[Zm},e)
=
,«) = I m + n (P m (A),e) =
X"i-m>(A,6) = Lm+n(f\[Zm],e)INV
,
and there are defined Rothenberg exact sequences
DEFINITION 19.5 The ultimate lower e-symmetric L-groups are defined by the direct limits £"-oo>(A,e) = ^ with L ? m v ( A , 6 )
L£_ ro) (A,€)
(n€Z),
l
>i/ _ m _ 1 \(A, c) the forgetful maps. n
PROPOSITION 19.6 For any A, e the functor {compact polyhedra} * {Z-graded abelian groups} ;
» a reduced generalized homology theory with a non-connective coefficient spectrum L/_ v(A,e) such that
§20. The algebraic fibering obstruction The chain complex approach to the K- and L-theory of the Laurent polynomial extension category Af^z" 1 ] developed in §10 and §16 will now be used to give an abstract algebraic treatment of the obstruction theory for nbering n-dimensional manifolds over 5 1 for n > 6. Following the positive results of Stallings [79] for n = 3 and Browder and
20. THE ALGEBRAIC FIBERING OBSTRUCTION
157
Levine [11] for n > 6 and TTI = Z a general fibering obstruction theory for n > 6 was developed by Farrell [21], [22] and Siebenmann [74], [76], with obstructions in the Whitehead group of an extension by an infinite cyclic group. See Kearton [38] and Weinberger [85] for examples of non-fibering manifolds in dimensions n = 4,5 with vanishing Whitehead torsion fibering obstruction. >F is defined by The mapping torus of a self-map h : F T(h)
= F x [0, l]/{(*,0) = (h(x), 1)|* € F) ,
as usual. If F is a compact (n — 1)-dimensional manifold and h : F >F is a self homeomorphism then T(h) is a compact n-dimensional manifold such that T(h) —- 1/(0 = 1) = S 1 ; (*,*)—>[*] is the projection of a fibre bundle over S1 with fibre F and monodromy h. A CW complex band is a finite CW complex X with a finitely dominated infinite cyclic cover X. Let ( : X >X be a generating covering translation. For the sake of simplicity we shall only consider CW complex bands with
C. = 1 : *i(X) — so that However, there is also a version of the theory which allows a non-trivial monodromy (* : Ki(X) •TTI(X) in the fundamental group, corresponding to the exact sequences of Farrell and Hsiang [23] and Ranicki [58] for the K- and L-groups of a-twisted Laurent polynomial rings A a ^ z " 1 ] with az = za(a) for an automorphism a : A >A ^
K^A) —> ~
K^A^z.z-1}) ~ ^eNilo^a"1)
(l-a)eoeo > K0(A) ,
— Ln(Aa[z, z-1)) — , L ^
with J = ker(l - a : ^ 0 (A) ^A'0(A)) and z = z~l. A finite structure on a topological space X is an equivalence class of pairs (finite CW complex K , homotopy equivalence / : K subject to the equivalence relation
(KJ) ~ (K',f) if rCT1/ : K^K')
=0 €
>X)
158
LOWER K- AND L-THEORY
The mapping torus T(Q of a generating covering translation £ : X >X for a CW band X has a preferred finite structure (Mather [46], Ferry [27] and Ranicki [67]), represented by (T(f(g),h) for any finite domination of X (K,f:
K-^X
, g : X-^K
, gf ~ 1 : K—>K) ,
with h : T(fCg) ~ T(gfC) ~ T(Q defined as in [67, §6]. Let p : X be the covering projection, and define homotopy equivalences (x,t)—+p(x), q+ : T ( C ) _ X ;
>X
q~ : TiC1)—>X; (x,t) —-» p(x) . The intrinsic finite structure on X is not in general compatible with either of the extrinsic finite structures X inherits via # + , q~ from the preferred finite structures on T(£), T((~x). The geometric fibering obstructions of a CW band X with respect to a choice of generating covering translation £ : X >X are the torsions = r(q+ : T(C)—>*) , measuring the difference between the intrinsic and the two extrinsic finite structures on X. The effect on the fibering obstructions of the opposite choice of generating covering translation £ is given by Further below, the fibering obstruction of a manifold X with finitely dominated infinite cyclic cover X will be identified with the geometric fibering obstructions of the underlying CW band — in the manifold case there is a Poincare duality $~(X) = ±<3>+(X)*, so the two fibering obstructions determine each other. Let (B,A C B) be a pair of additive categories. Given a self chain >C of an A-finitely dominated chain complex C in equivalence h : C B define the mixed torsion-class invariant The mixed invariant has components
[CM = (r(/0,[C],0,0) G W f A l v " 1 ] ) = W/i(A)0A^o(Po(A))eNrio(A)eNrio(A) with respect to the geometrically significant direct sum decomposition of Wh(f\[z, z~1]). The mixed invariant of a self-homotopy equivalence h : X >X of a finitely dominated CW complex X with ft* = 1 : TTI(X) = 7T
>7r is defined by
[X,h] =
20. THE ALGEBRAIC FIBERING OBSTRUCTION
159
with h : C(X) >C(X) the induced self chain equivalence of the finitely dominated cellular Z[?r] -module chain complex of the universal cover X. In Ranicki [67] the mixed invariant was identified with the torsion [X,h] = r{h x - 1 : X x S 1
>X x S 1 ) e Wh{K x Z) .
Assume now that A has a stable canonical structure. An f\-finite structure on a chain complex C in B is an equivalence class of pairs (D,f) with D a finite chain complex in A and / : D >C a chain equivalence, subject to the equivalence relation
(C,/)~(C',/') if rCT 1 f: D—+D') = OeWh(H). Given a self chain equivalence h : C >C of an A-finitely dominated chain complex C in B (as before) define a preferred A[z, z~1]-finite structure (E,k) on the algebraic mapping torus T+(h) = T(h) = Cfl-zftiCIz,*- 1 ]—>C[z,z- 1 ]) as in Ranicki [67, §6], using any A-finite domination (D,f : C >D,g : D >C,gf ~ 1) of C, with E = T(fhg : D >D). Define similarly a preferred A[2, z~^-finite structure on the opposite algebraic mapping torus T~(h) = Cil-z^h-.Ciz.z-1] For any chain homotopy inverse h~1 : C x
v : T-{hT )
>C[*,2- 1 ]). >C of h let
> T+(h)
be the chain equivalence defined by r = (~*k
e
A : T-(/i- x ) n = Cn®Cn-i
>T+(h)n = Cn®Cn-i
for the appropriate e : Cn-\ >Cn- The torsion of r with respect to the preferred A[z,^~1]-finite structures is the mixed invariant of h
r(r) =
[CKleWh^z-1]).
For any CW band X there is defined a homotopy commutative diagram
with
r : TiC1) —» T(C) ; (x,t) — . (x, 1 - *) . With the conventions of 8.15 the cellular Z[TT X Z]-module chain complex
160
LOWER K- AND L-THEORY
of the universal cover T(() of T(() is the algebraic mapping torus of C~-i : C(X)
C(f(C)) = = Cil-zC1 •• C{X)[z,z-l}^C{X){z,z-1]) . Now r induces the isomorphism of fundamental groups r. : TnCTOT1)) = » x Z — n n ( r ( C ) ) = « Z ; (g,n) —» ( f f ,-n) , and the induced Z[?r x Z]-module chain equivalence
f : r»C(f(C-x)) —» C(T(C)) is given algebraically by r : T~(C : C(X) >C(X)) with torsion the mixed invariant
> ^ ( C " 1 : C(X)
>C(X))
r(r) - [CfX),^1] = [IX-'je^CrxZ). Applying the sum formula for Whitehead torsion gives $-(X)-$+(X) = r{r) = [XiC^eWhivxl). Let now E be a chain complex band in A[z,z~a], that is a finite chain complex such that rE is Co(A)-finitely dominated in Gi(A). The positive and negative end invariants [E]± are such that The universal Mayer-Vietoris presentation E(oo) : 0 > iii'E determines chain equivalences in q+ = (p o)
>
«" = (p 0) : T-(C) = Working as in Ranicki [67, §6] use a Co(A)-finite domination of vE in Gi(A) (D,f: i'E^D,g : D^iE, gf ~ 1) to define a preferred A[z,z~1]-finite structure (C, h) on T + (C - 1 ) with and similarly for T~(£). The chain equivalence r : T-(C) — + 1 is defined by
20. T H E ALGEBRAIC FIBERING OBSTRUCTION
161
with torsion the mixed invariant r(r : T " ( C ) — ^ ( C 1 ) ) = [ i ^ C * ] € Wh{l\[z,z"1]) . The algebraic fibering obstructions of a chain complex band E in A ^ z " 1 ] are defined by
and are such that
The geometric fibering obstructions of a CW band X with TTI(-X') = 7T x Z are just the algebraic fibering obstructions of the cellular Z[TT x Z]module chain complex C(X) of the universal cover X ^(X)
= ^ ± (C(X)) € Wh(7r x Z) .
The algebraic mapping torus of a self chain equivalence h : F a finite chain complex F in A T(h) =
^
>F of
1
has T(h) ~ J^, ^ ~ ft"1, so that the algebraic fibering obstructions are given by
e im(i! : Wfe(A)
A chain complex band E in A[2,z -1 ] is simple chain equivalent to T(h) for a simple self chain equivalence h : F *F of a finite chain complex F in A if and only if has image
G +
for any N = (N ,N~)
x0(Po(A)) e NUo(A) e NTIO(A) ,
€ N^(E). Similarly, $>~(E) has image
G A'o(Po(A)) 0 Nil<,(A) © NUo(A) . The algebraic fibering obstructions are absolute invariants of chain
162
LOWER K- AND L-THEORY
complexes which determine the Whitehead torsion, by analogy with Reidemeister torsion. (In the geometric context this was already observed by Siebenmann [74], [76]). Specifically, the Whitehead torsion of a chain equivalence / : E' >E of chain complex bands in A^,^" 1 ] is given by r ( / ) = * + ( £ ) - *+(£?) = * " ( £ ) - $ " ( £ ' ) € Wh{l\[z,z-i\) , since there are defined commutative squares of chain equivalences
q± E'
J
-
>E
with In particular, if E is a contractible finite chain complex in A[z, z~l] the fibering obstructions are given by the torsion $+(£) = $-(£) = r(E)e Wh(f\[z,z-1]) . If A has an involution then the n-dual of a chain complex band E in A[z, z~x\ is a chain complex band En~* with algebraic fibering obstructions given by
$±(£n-*) = {-)n~l^{Ey e wh^foz-1]). For a pair (B, A C B) of additive categories with involution the dual of the mixed invariant [C,h] G Wh(f\[z,z~1]) of a self chain equivalence h : C >C of an A-finitely dominated chain complex C in B is given by
[CM* = Ti-zhiCfoz-^-^Cfoz-1))' = ri-z-'h* :C'[z,z-1]-^C*[z,z-1)) If (i£, 0) is an n-dimensional symmetric Poincare complex in f\[z, z~l\ with E a chain complex band then (rE,i'6) is an A-finitely dominated (n —l)-dimensional symmetric Poincare complex in Fi(A). The 'covering translation' IE
defines a self chain equivalence
;x
> zx
20. T H E ALGEBRAIC FIBERING OBSTRUCTION
163
such that
r(0* = (-)nr(0eWh(f\),
[i!£]* = (-)n-1\i'
The torsion T(E,6)
=
is such that T{E,6)
=
$+{E)
= *+(£) + (-)»*+(£;)• + \i'-E, C 1 ] G Wh(f\[z, z-1)) , BT(E,9)
=
[
E
]
+
^
€ Niio(A) . The algebraic mapping torus T(£) is an n-dimensional symmetric Poincare complex in Fi(f\)[z,z~1] with a preferred AfzjZ'^-finite structure, with respect to which
r(T(0) = [yEtC1] = i-rtfEtC1]'
ewhifkfoz-1]).
Any chain equivalence E ~ T{h) to the algebraic mapping torus of a self chain equivalence h : F >F of an A-finitely dominated chain complex F can be lifted to a chain equivalence vE ~ F , and F supports an (n — 1)-dimensional symmetric Poincare structure preserved by ft, such that there is defined a homotopy equivalence of homotopy A-finite n-dimensional symmetric Poincare complexes (E,0) ~ T(h:(F^)—^(F^)). Thus$+(£) = $ - ( £ ) = 0 G W/i(A[^,z- 1 ])ifand only if(£,0) is simple homotopy equivalent to the algebraic mapping torus T(h) of a simple self homotopy equivalence h : (F, <j)) >(F, ) of an (n — 1)-dimensional symmetric Poincare complex (F, >) in A, in which case (E, 0) (but not necessarily (F, $)) is simple. For a simple n-dimensional symmetric Poincare complex (E,6) in
AM"1]
T(E,$)
= &+(E)+ (-)"$-(E)*
= 0eWh{f\[z,z-1}) ,
and N+T(E,0)
= N+$+(E) + (-)n(N- *+(£))• = OGNT
so the two Nil-components of $ + ( £ ) are related by = (-)n-\N-$+(E))*
€ NTlo(A) .
164
LOWER K- AND L-THEORY
Since
there is redundancy in the two algebraic fibering conditions $+(£) = *-(JE) =
OeWh(F\[z,z-1]),
and $ + ( £ ) = 0 € WT^A^,*" 1 ]) if and only if (E,0) is simple homotopy equivalent to the algebraic mapping torus T(h) of a simple self homotopy equivalence h : (F, >) >(F,<j>) of an (n — l)-dimensional symmetric Poincare complex (F, >) in A. For any such (F, <^) the Tate Z2-cohomology class of the torsion
is the image of the Wh2-invariant T2(E,0) £
Hn(l2;Wh2(f\[z,z-1]))
under the split surjection induced by the W/i2-analogue B2 : Wh2(f\[z,z-1]) l
oiB: Wh(f\[z,z- \)
>Wh(l\)
^ o (Po(A))
r{F,) =
B2T2(E,6)eH"-\Z2;Wh(f\)).
There exists a simple homotopy equivalence (E,9) ~ T(h:(F,4)—*(F,4>)) with r(h:F
>F) = r(F,<j>) = 0 € Wh(t\)
if and only if
B2T2(E,6)
=
n 1
0Gff - (Z
The W/12-invariant is implicit in the way Wall [84, §12B] used fibering obstruction theory to identify the codimension 1 splitting obstruction groups LS* in the two-sided non-seperating case with i?*+1(Z2; Wh{n)). Indeed, the groups M* of [84,Thm. 12.6] are the higher L-groups decorated by im(ij : Wh2(7r) >Wh2{'K x Z)). The fibering Wh2-invariant is closely related to the pseudo-isotopy Wh2-invariant of Hatcher and Wagoner [33], and appears explicitly in the work of Kinsey [41] on the uniqueness of fibre bundle structures over S 1 . If X is an n-dimensional geometric Poincare complex which is a CW band then the torsion of T(£) with respect to the preferred finite struc-
20. THE ALGEBRAIC FIBERING OBSTRUCTION
165
ture is such that
r(T(0) = rir-.TiC^nO) = [XiC1] = (-) n [*,CT, T(X) = T(T(0) + *+(x) + (-)n$+(xy
A 'candidate for fibering' in the sense of Siebenmann [76] is a compact n-dimensional manifold X with a finitely dominated infinite cyclic cover X. Again, only the case £* = 1 : TT = fti(X) >Tr\(X) is considered here, so that TTI(X) = TT X Z. For n ^ 4 X is a CW band, since it is a CW complex via the handlebody decomposition, but in any case X has a preferred intrinsic finite structure. Any finite CW complex Y in the preferred simple homotopy type of X is a CW band which is a simple n-dimensional geometric Poincare complex. The algebraic fibering obstruction of X is defined by $+(X) = $+(C(X)) G Wh(n x Z) , interpreting C(X) as C(Y) if n = 4. $ + ( X ) is the total fibering obstruction of [76] and Farrell [22], such that $+(X) = 0 if (and for n > 6 only if) X fibers over S 1 , i.e. is homeomorphic to the mapping torus T(h) of a self-homeomorphism h : F >F of a compact (n — 1)-dimensional manifold F , with the infinite cyclic cover X Z-homeomorphic to T(h) = FxR. The total fibering obstruction $ + ( X ) is related to the original two-stage fibering obstruction theory of Farrell [21] by the geometrically significant split exact sequence i,
0
> Wh(7r) —> Wh(n x Z) K 0 (Z[TT])
© I«io(Z[7r]) ® Nilo(Z[7r]) —> 0 .
The images
= -[X-] = -\i'-C(X)/CN+C(X)+]
€ ^o(ZW)
are the primary obstructions of [21], being the components of the splitting obstruction of Farrell and Hsiang [23] (cf. 10.9) ; if these vanish and n > 6 there exists a fundamental domain (V; 17, £[/") for X which is an ft-cobordism, with the torsion r ( F ; (7, (U) G W/^TT) the secondary obstruction, such that
(7r x Z) K 0 (Z[TT]) © ffilo(Z[TT]) © NT1O(Z[7T]))
(n x Z)) .
166
LOWER K- AND L-THEORY
= 0 if and only if (V; U, (U) is an s-cobordism, which by the s-cobordism theorem is homeomorphic to a product U x ([0,1]; {0}, {1}). Since X is simple
T(X) = ^+(x) + (-)n^+(xy + [x,c1] = o BT(X) = [x+] + (-)»[x-]* = oa»(z[f]). The identity is a generalization of the end obstruction duality theorem of Siebenmann [73, §11]. Since X is a manifold r2(X) = 0 € Wh2{K x Z), so $ + ( X ) G Wh(7r x Z) is the only algebraic obstruction to X being simple homotopy equivalent to the mapping torus T(h) of a simple self homotopy equivalence h : F >F of a simple (n — 1)-dimensional geometric Poincare complex F. Siebenmann, Guillou and Hahl [77,5.13] developed a version of the fibering obstruction theory for Hilbert cube manifolds. In the corrected version of this theory (Chapman and Siebenmann [19, p.208]) there are two independent obstructions, which are precisely the two algebraic fibering obstructions 3>+(i£), $~(J5) of the associated chain complex band E. It is possible to extend the algebraic fibering obstruction theory to fibrations over manifolds other than S 1 , which in the non-simply-connected case necessarily involves both lower K- and lower L-theory. For the geometric fibering obstruction theory see Casson [17], Quinn [55], Burghelea, Lashof and Rothenberg [14] and Levitt [43]. For fibrations over Tn part of the obstruction concerns the expression of a non-compact manifold as a product M x Rn - see Bryant and Pacheco [13] for the corresponding obstruction theory. There is also a connection with the geometric theory developed by Hughes, Taylor and Williams [37] for relating manifold approximation fibrations and bundles.
REFERENCES
167
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168
LOWER K- AND L-THEORY
[21] F. T. Farrell The obstruction to fibering a manifold over a circle Indiana Univ. J. 21, 315-346 (1971) [22] The obstruction to fibering a manifold over a circle Proc. 1970 ICM Nice, Vol.2, 69-72 (1971) [23] and W. C. Hsiang A formula for Ki(Ra[T\) Proc. Symp. A. M. S. 17, 192-218 (1970) [24] Manifolds with TTI = G xQ T Amer. J.Math.95, 813-845 (1973) [25] and L. E. Jones A topological analogue of Mostow's rigidity theorem Journal of A. M. S. 2, 257-370 (1989) [26] and J. Wagoner Infinite matrices in algebraic K-theory and topology Comm. Math. Helv. 47, 474-501 (1972) [27] S. Ferry A simple homotopy approach to finiteness obstruction theory Proc. 1980 Dubrovnik Shape Theory Conf., Springer Lecture Notes 870, 73-81 (1981) [28] and E.Pedersen Epsilon surgery I. Math.Gott. 17 (1990) [29] S.Gersten On the spectrum of algebraic K-theory Bull. A. M. S. 78, 216-219 (1972) [30] I. Hambleton and I. Madsen Actions of finite groups on Rn+* with fixed set Rk Can. J. Math. 38, 781-860 (1986) [31] and E. Pedersen Bounded surgery and dihedral group actions on spheres J. of A. M. S. 4, 105-126 (1991) [32] , A. Ranicki and L. Taylor Round L-theory J. Pure and Appl. Alg. 47, 131-154 (1987) [33] A. Hatcher and J. Wagoner Pseudo-isotopies of compact manifolds Asterisque 6 (1973) [34] G. Higman The units of group rings Proc. London Math. Soc. (2) 46, 231-248 (1940) [35] B. Hughes Bounded homotopy equivalences of Hilbert cube manifolds Trans. A. M. S. 287, 621-643 (1985) [36] and A. Ranicki Ends of complexes to appear [37] , L. Taylor and B. Williams Bundle theories for topological manifolds Trans. A. M. S. 319, 1-65 (1990) [38] M.Karoubi Foncteurs derives et K-theorie Seminaire HeidelbergSaarbriicken-Strasbourg sur la if-theorie, Springer Lecture Notes 136, 107-186 (1970) [39] K-theory, An Introduction Springer (1978) [40] C. Kearton Some non-fibred 3-knots Bull. L. M. S. 15, 365-367 (1983) [41] C. Kinsey Pseudoisotopies and submersions of a compact manifold to the circle Topology 26, 67-78 (1987)
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[42] R. Lashof and M. Rothenberg G-smoothing theory Proc. Symp. Pure Maths. XXXII, Part I, 211-266 (1978) [43] N. Levitt A necessary and sufficient condition for fibering a manifold Topology 14, 229-236 (1975) [44] W. Luck and A.Ranicki Chain homotopy projections J.of Algebra 120, 361-391 (1989) [45] I. Madsen and M. Rothenberg On the classification of G-spheres I. Equivariant transversality Acta Math. 160, 65-104 (1988) [46] M. Mather Counting homotopy types of manifolds Topology 4, 93-94 (1965) [47] R. J.Milgram and A.Ranicki The L-theory of Laurent polynomial extensions and genus 0 function fields J.reine u. angew. Math. 406, 121-166 (1990) [48] S.P. Novikov The algebraic construction and properties of hermitian analogues of K-theory for rings with involution, from the point of view of the hamiltonian formalism. Some applications to differential topology and the theory of characteristic classes Izv. Akad. Nauk SSSR, ser. mat. 34, 253-288, 478-500 (1970) [49] E.Pedersen On the K-i-functors J.of Algebra 90, 461-475 (1984) [50] K-i-invariants of chain complexes Proc. 1982 Leningrad Topology Conference, Springer Lecture Notes 1060, 174-186 (1984) [51] On the bounded and thin h-cobordism theorem parametrized by Rk Proc. Symp. on Transformation Groups, Poznan 1985, Springer Lecture Notes 1217, 306-320 (1986) [52] and A.Ranicki Protective surgery theory Topology 19, 239-254 (1980) [53] and C. Weibel A nonconnective delooping of algebraic K-theory Proc. 1983 Rutgers Topology Conference, Springer Lecture Notes 1126, 166-181 (1985) [54] K-theory homology of spaces Proc. 1986 Arcata Algebraic Topology, Springer Lecture Notes 1370, 346-361 (1989) [55] F. Quinn A geometric formulation of surgery Topology of manifolds, Proc. 1969 Georgia Topology Conf., Markham Press, 500-511 (1970) [56] — Ends of Maps II. Inv.Math.68, 353-424 (1982) [57] A. Ranicki Algebraic L-theory II. Laurent extensions Proc. London Math.Soc.(3) 27, 126-158 (1973) [58] Algebraic L-theory III. Twisted Laurent extensions Proc. 1972
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[59] [60] [61] [62] [63] [64] [65]
[66]
[67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77]
LOWER K- AND L-THEORY
Battelle Seattle /^-theory Conf., Vol. Ill, Springer Lecture Notes 343, 412-463 (1973) Algebraic L-theory IV. Polynomial extension rings Comm. Math. Helv. 49, 137-167 (1974) The total surgery obstruction Proc. 1978 Arhus Topology Conf., Springer Lecture Notes 763, 275-316 (1979) The algebraic theory of surgery I. Foundations Proc. London Math.Soc.(3) 40, 87-192 (1980) The algebraic theory of surgery II. Applications to topology Proc. London Math. Soc. (3) 40, 193-283 (1980) Exact sequences in the algebraic theory of surgery Mathematical Notes 26, Princeton (1981) The algebraic theory of finiteness obstruction Math. Scand. 57, 105-126 (1985) The algebraic theory of torsion I. Foundations Proc. 1983 Rutgers Topology Conference, Springer Lecture Notes 1126, 199-237 (1985) Algebraic and geometric splittings of the K- and L-groups of polynomial extensions Proc. Symp. on Transformation Groups, Poznan 1985, Springer Lecture Notes 1217, 321-364, (1986) The algebraic theory of torsion II. Products ivT-theory 1, 115-170 (1987) Additive L-theory if-theory 3, 163-195 (1989) Algebraic L-theory and topological manifolds Cambridge Tracts in Mathematics and Mathematical Physics, CUP (1992) and M. Yamasaki Controlled K-theory preprint Controlled L-theory to appear J.Shaneson Wall's surgery obstruction groups for G x Z Ann. of Maths. 90, 296-334 (1969) L. Siebenmann The obstruction to finding the boundary of an open manifold Princeton Ph.D.thesis (1965) A torsion invariant for bands Notices AMS 68T-G7, 811 (1968) Infinite simple homotopy types Indag. Math. 32, 479-495 (1970) A total Whitehead torsion obstruction to fibering over the circle Comm. Math. Helv. 45, 1-48 (1972) , L. Guillou and H. Hahl Les voisinages ouverts reguliers: criteres homotopiques dfexistence Ann. scient. Ec. Norm. Sup. (4) 7, 431462 (1974)
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[78] S.B. Slosman Submanifolds of codimension one and simple homotopy type Mat. Zam. 12, 167-175 (1972) [79] J. Stallings Onfibering certain 3-manifolds Proc. 1961 Georgia Conference on the Topology of 3-manifolds, Prentice-Hall, 95-100 (1962) [80] J.A. Svennson Lower equivariant K-theory Math. Scand. 60, 179201 (1987) [81] W. Vogell Algebraic K-theory of spaces, with bounded control Ada, Math. 165, 161-187 (1990) [82] F. Waldhausen Algebraic K-theory of generalized free products Ann. of Maths. 108, 135-256 (1978) [83] C. T. C. Wall Finiteness conditions for CW complexes Ann. of Maths. 81, 56-69 (1965) [84] Surgery on compact manifolds Academic Press (1971) [85] S.Weinberger On fibering four- and five-manifolds Israel J. Math. 59, 1-7 (1987) [86] The topological classification of stratified spaces to appear [87] M.Weiss and B.Williams Automorphisms of manifolds and algebraic K-theory I. if-theory 1, 575-626 (1988) [88] J.H. C.Whit ehead Simple homotopy types Amer. J. Math. 72, 1-57 (1950) [89] M. Yamasaki L-groups of crystallographic groups Inv. Math. 88, 571602 (1987)
172
LOWER K- AND L-THEORY
Index adjoining quadratic pairs, 116 A-finite structure, 159 algebraically significant injection in ^ ' " ( A ^ , * - 1 ] ) , 88 splitting of Ki3O(f\[z, z" 1 ]), 87 splitting of Ln{I\[z,z-1)), 144 B-contractible, 18 B-isomorphism, 18 b- neighbour hood ,A/&(Y,X), 15 boundary quadratic complex, 114 bounded morphism, 14 object, 14 category based f.g. free, B'(A), 9 based X-graded, GX(A), 13 based Z-graded, Gi(A), 63 bounded X-graded, C*(A), 14 idempotent, Px(A), 14 bounded (X, F)-graded, C x ,y(A), 16 idempotent, Px,y(A), 45 bounded Z-graded, Ci(A), 63 bounded Zm-graded, Cm(A), 101 idempotent, P m (A), 101 countably based, B(A), 70 f.g. projective, P(A), 9 filtered, 13 finite Laurent extension, infinite Laurent extension, G1(f\)[[^-% 74 locally finite X-graded, Fx(A), 118 locally finite Z-graded, Fi(A), 141 ra-fold Laurent extension, A[Zm], 105 neighbourhood, Ny(A), 17 polynomial extension, A[ar], 64 X-graded, G X (A), 12
Z-equivariant R-bounded, C R ( A ) Z , 68
chain complex (A2, Ao)-finitely dominated, 18 band in C X (A), 37 inAfz,*- 1 ], 78 class, [C], 8 finite, 8 homotopy A-finite, 9 n-dimensional, 8 round, 10 suspension, 5C, 8 chain homotopy projection, 9 class at 0, 57 contracted functor, 103 CW complex band, 157 bounded X-graded, 14 X-graded, 14 domain fundamental, 79 subfundamental, 79 domination, A-finite (£),/, y,ft),9 end invariant chain complex, [C]+, 19 Cx+ux- (A)-isomorphism, [f}±, 39 Gx,y(A)-isomorphism, [/]+, 23 negative, [£"]_, 37 positive, [£]+, 37 reduced, [C]+, 19 e-transposition, 153 even symmetric complex, 154 exact sequence, 8 fibering obstruction algebraic, $*(£), 161 geometric, * ± ( X ) , 158 filtration degree, <$(/), 13 finite
173
INDEX
element in N, 76 element in N(£), 77 structure, 157 finitely balanced cobordism, 142 flasque structure, {E,
torsion, Xi(A), 10 Laurent extension ring finite, A[z,z~l), 70 infinite, ^[[z,^- 1 ]], 70 L- group e-quadratic, Ln(A, e), 154 e-symmetric, L n (A,e), 154 intermediate, i f (A) , 118 intermediate lower, i f (A), 148 lower, Li" m) (A), 145 lower e-symmetric, quadratic, L n (A), 114 round, L^(A), 118 S1 -transfer invariant,
Ln{i\[z,z-yNV,
151
Tm-transfer invariant, Ln(f\[Zm])INV, 153 ultimate lower quadratic, L ^ " 0 0 ^ ) , 148 e-symmetric, L ^ ^ ^ A , e), 156 linear morphism
reduced lower, ff_ro(P0(A)), 146 reduced torsion, ifi(A), 47 relative, KX(F), 11 S1 -transfer invariant,
mapping torus, 157 Mayer-Vietoris presentation in f\[z,z~l], 74 finite, 75 torsion, 75 universal chain complex, 77 object, 75 in C X (A), 33 in GiCAM*,*-1]], 76 finite, 76 torsion, 76 sequence, M, 32 mixed torsion-class, 158
Tm-transfer invariant, , 112
nilpotent category Nil(A), 81
K- groups class, K0(A), 8 isomorphism torsion, K[SO(I\), 46 lower, tf-m(A), 103 reduced class, # o (Po(A)), 8 reduced isomorphism torsion,
#r(A), 47
Kiikfrz-1])1™, 108
174
LOWER K- AND L-THEORY
class group, Nilo(A), 81 endomorphism, 81 reduced class group, Nilo(A), 81
union, fundamental cobordisms, 142 quadratic pair, 116
open cone, O(X), 53 protective class chain complex, 9 chain homotopy projection, 9 proper eventually Lipschitz map, 15 (/-fold transfer, in d ( A ) , ql, 107 in tf-theory, ql, 108 in L-theory, q-, 151 Q-fold transfer C m (A), Q\ 111 Q-groups e-quadratic, Q*(C, e), 153 €-symmetric, Q*(C, e), 153 quadratic, Q*(C), 113 quadratic Poincare complex, 113 Poincare pair, 114 quotient object, 20 restriction, r , 73 round quadratic complex, 117 sign, e(L,Af), 46 splitting chain homotopy projection, 9 quadratic complex, 116 Poincare, 117 stable canonical structure, 48 isomorphism, 10 subobject, 15 support, 17 torsion, chain equivalence, r ( / ) , 47 x-based, 55 transposition, 113
Whitehead group, Wh{Co(x){f\\ 57 z-i]), 78 (-equivariant morphism, 71