appear to in Topology and Its Applications
Controlled K-Theory
Andrew Ranicki and Masayuki Yamasaki1 Department of Mathematics, University of Edinburgh, Edinburgh, Scotland, UK Department of Mathematics, Josai University, Sakado, Saitama, Japan Abstract The controlled niteness obstruction and torsion are dened using controlled algebra, giving a more algebraic proof of the topological invariance of torsion and the homotopy niteness of compact ANRs. Introduction The Wall niteness obstruction and Whitehead torsion are the traditional applications of algebraic K-theory to topology, relating geometric niteness properties of spaces to the algebraic properties of modules over the fundamental group ring. The methods of controlled algebra developed by Connell and Hollingsworth [8], Chapman [6] and Quinn [15], [16] use a more rened version of algebraic K-theory in which the algebra is parametrized by a metric space. For any > 0 there is a notion of -controlled K-theory, in which the size of any operation is restricted to be at most (some multiple of) in the metric space. In fact, only the controlled Whitehead group of automorphisms was dened directly, with the controlled reduced projective class group obtained from it by a version of the splitting theorem of Bass, Heller and Swan [2] which embeds the reduced projective class group K0 (Z[]) of a group ring Z[] as a direct summand in the Whitehead group W h( Z). In this paper we develop the controlled algebra of projections, dene the K0 groups directly, and relate the controlled K0 and W h-groups to each other by various exactness properties. The algebraic methods are used to give a self-contained treatment of the following results: 1. A homeomorphism of nite CW complexes is simple. This is the topological invariance of Whitehead torsion, originally proved by Chapman [3]. 2. Every compact ANR has the homotopy type of a nite CW complex. This is the Borsuk conjecture, originally proved by West [24].
1
Partially supported by Grant-in-Aid for Scientic Research, and Josai University Overseas Research Fellowship AMS (MOS) Subj. Class.: Primary 57Q10, 57Q12 Secondary 19J05, 19J10 Keywords: torsion niteness obstruction 1 controlled K-theory
3. The results of Ferry [10] and Chapman [4], [5] generalizing 1. and 2., by which an -domination (resp. -homotopy equivalence) for suciently small implies the vanishing of the ordinary Wall niteness obstruction (resp. Whitehead torsion). A P L homeomorphism is a simple homotopy equivalence; the combinatorial invariance of Whitehead torsion is proved by Milnor [15] using an induction argument, in which the key ingredient is the computation W h({1}) = 0 of Higman [13]. Similarly, the key ingredient in any proof of 1. is the computation W h(Zn ) = 0 of Bass, Heller and Swan [2]. The failure of the Hauptvermutung shows that a homeomorphism of nite CW complexes is not in general homotopic to a P L homeomorphism. However, a homeomorphism has zero controlled torsion and hence also zero Whitehead torsion. The proofs of 2. and 3. are closely related to the proof of 1: the connection between the three results has long been recognized. In our proofs we have attempted to minimize the geometry and maximize the algebra. It should be noted that the squeezing arguments of Ferry and Pedersen provide an alternative algebraic method of proof of 1.,2.,3., using the bounded algebraic K-theory of Pedersen and Weibel. In this approach the controlled niteness obstruction of a controlled nite domination over a nite-dimensional space X is identied with the bounded torsion of a bounded homotopy equivalence over the open cone O(X), and a controlled torsion is identied with a bounded K2 -invariant. See Ferry, Hambleton and Pedersen [11,7] for a brief account. In subsequent work we shall extend the methods to controlled L-theory, giving a similarly direct proof of the topological invariance of the rational Pontrjagin classes, originally obtained by Novikov. The plan of the paper is as follows. In 1 we rst review the category of geometric modules and geometric morphisms due to Quinn, and then discuss projective modules and projective module chain complexes in this category. In 2 we introduce control into these. In 3 we dene the controlled projective class groups K0 (X, pX , n, ) using n-dimensional -controlled projective complexes on a control map pX : M X. In 4 we dene similarly the controlled torsion groups W h(X, pX , n, ) using n-dimensional -controlled contractible free complexes on a control map pX : M X. In 5 the K0 and W h-groups are related by the stable-exact sequence of a pair (X, Y X). The excision and Mayer-Vietoris properties of the controlled K-groups are developed more generally in 6. The controlled analogue of the decomposition W h( Z) = W h() K0 (Z[]) 2Nil0 (Z[]) of Bass [1] is obtained in 7. This is used in 8 to establish a Vietoris-type property of controlled K-theory invariants: if suciently controlled they vanish in a less controlled K-group. Controlled niteness and torsion invariants are dened in 9. The 2
topological invariance and niteness results 1., 2. and 3. are proved in 10. Finally, an appendix gives a brief account of controlled lower K-theory. The referee asked if there is a categorical approach to our stable isomorphisms and stably exact sequences, and even went so far as to suggest an appropriate category. An object in this category should be a system of abelian groups {A | (0, )}, with A mapping to A if < , and a morphism f : {A } {B } should be a collection of homomorphisms f : A Bk making some obvious diagrams commute. Here k is a constant independent of , but depending on f . A category for controlled algebra is certainly desirable. (For bounded algebra there are the categories of Pedersen and Weibel, as well as Anderson and Munkholm). Regrettably, we have not been able to provide such a categorical treatment in this paper. The second-named author would like to thank the Mathematics Department of Edinburgh University for its hospitality in the academic year 1990-91. Both authors would like to thank Bruce Hughes for conversations in the course of that year, and Masaharu Morimoto and the referee for carefully reading the various drafts and making valuable suggestions. 1. Geometric modules. Basically our treatment of geometric modules and geometric morphisms follows the original work by Quinn [18]. Some notation and terminology is from the work of Connolly and Koniewski [9,4] and some is new. z Let M be a topological space, and let |S| be a set together with a function S : |S|M. In the following we identify a function with its graph; thus, S also represents a subset of |S| M . The rst (resp. second) component of an element s S |S| M will be denoted |s| |S| (resp. [s] M ). The function S maps |s| to [s]. The projection |S| M M induces a bijection from the graph S to |S|. Denition. The free Z-module on the graph S is called the geometric module on M generated by S, and is denoted Z[S]. A geometric module Z[S] is said to be nitely generated (f.g.) if |S| (and hence S) is a nite set. The direct sum A Z[S ] of a family {Z[S ]}A of geometric modules on M indexed by a set A is dened as follows: First make a copy S of each S by: S : |S | = |S | {} |S | M. The |S |s are disjoint subsets of A |S | A; the disjoint union of |S |s will be denoted A |S |. The S s dene a unique function A S : A |S | M . Now A Z[S ] is dened to be Z[ A S ]. In this paper, we shall pretend that the |S |s are mutually disjoint, writing A Z[S ] = Z[ A S ] without mentioning taking copies S of S from now on. 3
S
Examples. (1) When |S| is empty, Z[S] is denoted 0. (2) Let M be a CW complex and x an integer n 0. Let |S| be the set of the n-cells of M . For each n-cell e |S|, let e : D n M be the characteristic map for e. The correspondence S : |S| M ; e e (O) denes a geometric module Z[S] on M , where O denotes the center of the n-disk D n . As an abstract abelian group, it is the group of Z-coecient cellular n-chains of M . Denitions. Let Z[S] and Z[T ] be geometric modules on M . Consider triples (s, , t) consisting of elements s S, t T and a path : [0, ] M ( 0) such that (0) = [s] and ( ) = [t]. Such a triple (s, , t) will be called a path from s to t. A geometric morphism f : Z[S] Z[T ] is a formal linear combination m (s , : [0, ] M, t )
of paths from generators of Z[S] to generators of Z[T ], with integer coecients. Here is some index set, and the number of paths starting from each generator is required to be nite. Two geometric morphisms f = m (s , , t ) and f = m (s , , t ) from Z[S] to Z[T ] are equal (f = f ) if there exists a bijection : such that m() = m and (s() , () , t() ) = (s , , t ) (for all ),
after deleting terms with zero coecients. The beginning and the end points of the paths in a geometric morphism f = m (s , , t ) : Z[S]Z[T ] determine a Z-module homomorphism: |f | : Z[S] Z[T ] ; s
s =s
m t .
Examples. (1) A geometric morphism with no term is called the zero geometric morphism, and is denoted 0. |0| is the zero homomorphism. (2) Let Z[S] be a geometric module on M and dene a one-point path cs : {0} M by cs (0) = [s], for s S. The geometric morphism 1(s, cs , s) : Z[S] Z[S]
sS
is called the identity geometric morphism on Z[S], and is denoted 1Z[S] or simply 1. For a geometric morphism f : Z[S]Z[T ], the equalities f 1Z[S] = f = 1Z[T ] f hold. |1Z[S] | is the ordinary identity homomorphism on Z[S]. (3) The geometric morphism 0(s, , t) is equal to 0, for any path (s, , t). (4) The geometric morphisms 2(s, , t) + 3(s, , t) and 5(s, , t) are not equal, because the numbers of terms with non-zero coecients are dierent. 4
Denitions. The sum of two geometric morphisms is dened by formally combining the two linear combinations. The integer multiplication of a geometric morphism is dened by termwise integer multiplication. The dierence f g of f and g is dened by f + (1)g. The composition gf of two consecutive geometric morphisms f=
m (s , , t ) : Z[S]Z[T ],
g=
n (t , , u ) : Z[T ]Z[U ]
is dened to be n m (s , , u ),
,,t =t
where : [0, + ] M is the composite path (x) = (x) (x ) if 0 x , if x + ,
of two paths : [0, ] M , : [0, ] M with ( ) = (0). If f = m (s , , t ) ( A), then f =
A A,
m (s , , t )
is called the direct sum of the family {f }A of geometric morphisms. We shall often use matrices to present a geometric morphism between direct sums of geometric modules. Let
n m
f=
m (s , , t ) :
j=1
Z[Sj ]
i=1
Z[Ti ]
be a geometric morphism. Dene geometric morphisms fij : Z[Sj ] Z[Ti ] (1 i m, 1 j n) by: fij = m (s , , t ).
,s Sj ,t Ti
f is completely determined by fij s; f is equal to the sum i,j fij if we regard fij s as geometric morphisms between Z[ Sj ] and Z[ Ti ] via the inclusions Sj 1jn Sj , Ti 1im Ti . An m n matrix (fij )1im,1jn will be used to express f using n n n fij s. For example, the direct sum i=1 fi : i=1 Z[Si ] i=1 Z[Si ] can be written as a diagonal matrix with entries f1 , . . . , fn . 5
Suppose : M M is a covering map. Given a geometric module Z[S] on M , one can form the pullback geometric module Z[S] on M by the standard pullback construction S : |S| = {(|s|, m) |S| M | S(|s|) = (m)} M ; (|s|, m) m . The induced covering S : |S| |S| ; (|s|, m) |s| determines a covering S : SS of the graph S: S : S |S| |S| S. Next we dene the pullback of a geometric morphism with respect to . Let Z[S] and Z[T ] be the pullbacks of Z[S] and Z[T ] with respect to : M M . Let (s, , t) be a path in M from s S to t T . If s S is mapped to s by ) in M from s to some element t T such S : S S, there is a unique path (, , t s that the composition is equal to . Such a path is called a lift of (s, , t). Now, for a geometric morphism f = m (s , , t ) : Z[S] Z[T ], dene its pullback geometric morphism f : Z[S]Z[T ] by: f=
S ( )=s s S
m ( , , ), s t
where ( , , t ) is the lift of (s , , t ) starting from s . It is easily checked that s gf = g f . Suppose is a regular covering with the group of covering translations . Then S : SS is also a regular -covering. acts freely on S, and Z[S] is freely generated as a Z[]-module by any complete set of orbit representatives of S. If f : Z[S] Z[T ] is a geometric morphism, the Z-module homomorphism |f | is actually a Z[]-module homomorphism between the based free Z[]-modules Z[S] and Z[T ]. For a xed : M M assembly is a functor {(f.g.) geometric modules on M and geometric morphisms} {(f.g.) free Z[]-modules and homomorphisms} ; Z[S] Z[S] . We shall be particularly concerned with the case when : M M is the universal cover of M (assuming M is connected and locally 1-connected). A geometric module Z[S] determines a Z[1 (M )]-module Z[S], which will be called the assembly of Z[S]. Similarly, a geometric morphism f : Z[S] Z[T ] determines a Z[1 (M )] module homomorphism |f | : Z[S] Z[T ], which will be called the assembly of f . 6
Denition. Two paths (s, : [0, ] M, t), (s , : [0, ] M, t ) are homotopic if s = s , t = t , and there exist a non-negative continuous function (y) (0 y 1) and a continuous map h : {(x, y) R2 |0 x (y), 0 y 1}M such that (0) = , (1) = , h(x, 0) = (x) (x [0, ]), h(x, 1) = (x) (x [0, ]), h(0, y) = [s], h( (y), y) = [t] M (y [0, 1]). A homotopy () of a geometric morphism is a nite sequence of the following operations: 1. homotopies of the paths, 2. combining two terms m(s, , t) + n(s, , t) into (m + n)(s, , t), and its inverse. For example, if a path (s, , t) is homotopic to (s, , t), then (s, , t) (s, , t) is homotopic to the zero geometric morphism: (s, , t) (s, , t)
operation 1
(s, , t) (s, , t)
operation 2
0(s, , t) = 0.
The assemblies of homotopic geometric morphisms are the same homomorphisms. In fact, Quinn [18] has shown that, when M is connected and locally 1-connected, the assembly map Z[S] Z[S] with respect to the universal cover M of M denes a natural equivalence between the category of geometric modules on M and homotopy classes of morphisms and the category of based free Z[1 (M )]-modules (with basis specied up to the action of 1 (M )). Let : M N be a continuous map. For a geometric module A = Z[S] on M , its direct image A is dened to be the geometric module Z[S : |S| M N ] on N . Taking a direct image corresponds to changing the coecient ring of the assembly from Z[1 (M )] to Z[1 (N )]. For an element s = (|s|, [s]) of the graph S, s will denote the element (|s|, [s]) of the graph of S : |S| N . If f = m (s , , t ) : A B is a geometric morphism between geometric modules A, B on M , then f : A B will denote the geometric morphism m (s , : [0, ] M N, t ). If f g, then f g. Next we introduce chain complexes in the category of the geometric modules on M . Chain complexes will play a key role in this paper. Denition. A chain complex on M is a sequence of morphisms of geometric modules on M {C, d} : . . . Cr+1 Cr Cr1 . . . such that dr dr+1 0. 7
dr+1 dr
Homotopies d2 0 are used in the denition of chain complexes instead of the strict equalities, because boundary morphisms arising from CW complexes satisfy only d2 0. See [17] for a detailed discussion. We shall actually need to work with chain complexes in the category of projective modules. Denitions. A geometric morphism p : A A from a geometric module A on M to itself is a projection if p2 p. A projective module on M is a pair (A, p) consisting of a geometric module A on M and a projection p : A A. (A, p) is nitely generated if A is nitely generated. The direct sum i (Ai , pi ) of projective modules (Ai , pi ) is dened by ( i Ai , i pi ). A morphism f : (A, p) (B, q) between two projective modules is a geometric morphism f : A B satisfying qf f and f p f . The morphism dened by the zero geometric morphism is called the zero morphism and is denoted 0. For example, if (A, p) is a projective module on M , then the geometric morphism p : A A denes a morphism from (A, p) to itself. This morphism p serves as the identity morphism up to homotopy. Thus projective modules on M and the homotopy classes of morphisms form a category. A morphism f : (A, p) (B, q) is an isomorphism if there exists a morphism g : (B, q) (A, p) such that gf p and f g q; g is called the inverse of f . A projective module of the form (A, 1) can be identied with the geometric module A and is called a free module. The morphisms between free modules (A, 1) and (B, 1) are exactly the geometric morphisms between A and B. Denition. A projective chain complex on M is a sequence of morphisms of projective modules on M {(C, p), d} : . . . (Cr+1 , pr+1 ) (Cr , pr ) (Cr1 , pr1 ) . . . such that dr dr+1 0. When there is no ambiguity, we often omit the boundary morphisms d from notation. A projective chain complex (C, p) is n-dimensional if Cr = 0 for r < 0 and for r > n. If all pr s are 1, then it is called a free chain complex and can be identied with the chain complex C : . . . Cr+1 Cr Cr1 . . . of geometric modules. The direct sum of two projective chain complexes (C, p), (D, q) is dened by (C, p) (D, q) : (Cr , pr ) (Dr , qr ) (Cr1 , pr1 ) (Dr1 , qr1 ) . A projective chain complex (C, p) is nitely generated (f.g.) if Cr is nitely generated for every r. 8
dC dD dr+1 dr
Denitions. (1) A chain map f : (C, p) (D, q) between projective chain complexes is a collection f = {fr } of morphisms fr : (Cr , pr ) (Dr , qr ) such that dr fr fr1 dr . (2) A chain homotopy h : f g between chain maps f, g : (C, p) (D, q) is a collection h = {hr } of morphisms hr : (Cr , pr ) (Dr+1 , qr+1 ) such that dr+1 hr + hr1 dr gr fr . (3) A chain map f : (C, p) (D, q) is a chain equivalence if there exists a chain map g : (D, q) (C, p), called a chain homotopy inverse, such that gf p and f g q. (4) Two projective chain complexes (C, p) and (D, q) are chain equivalent, (C, p) (D, q), if there exists a chain equivalence between them. (5) A projective chain complex (C, p) is contractible if it is chain equivalent to the zero chain complex. In this case, a chain homotopy h : 0 p : (C, p) (C, p) is called a chain contraction. (6) A chain map f : (C, p) (D, q) is an isomorphism, f : (C, p) (D, q), if there = exists a chain map g : (D, q) (C, p), called an inverse of f , such that gf p and f g q. Thus each fr is an isomorphism of projective modules. (7) The algebraic mapping cone C(f ) of a chain map f : (C, p) (D, q) is a projective chain complex dened by dC(f ) = dD 0 ()r1 f dC : C(f )r = (Dr , qr ) (Cr1 , pr1 )
C(f )r1 = (Dr1 , qr1 ) (Cr2 , pr2 ) . (A chain map f is a chain equivalence if and only if C(f ) is contractible. See 2.4 below.) Remark. Let {(C, p), d} be a projective chain complex. The morphisms dr : (Cr , pr ) (Cr1 , pr1 ) are, by denition, geometric morphisms dr : Cr Cr1 . Thus we have a free chain complex C = {Cr , dr } : . . . Cr+1 Cr Cr1 . . . . The geometric morphisms {pr } form a chain map from C to itself. 2. Controlled algebra. Now we introduce geometric control. A continuous map pX : M X to a metric space is called a control map. If M is given a specic control map pX , we say Z[S] is a geometric module on pX . Suppose W is a subspace of X. The restriction of pX to the subset p1 (W ) of M will be denoted by pW : p1 (W ) W . For 0, X X the closed neighborhood of W in X is denoted by W . Obviously, (W ) W + . For > 0, W denotes the set {x W |d(x, X W ) } W . 9
dr+1 dr
Given a control map pX , we use the following convention for radii of geometric morphisms and homotopies. A geometric morphism f has radius if the image of the path : [0, ] M is contained in p1 ({pX (0)} {pX ( )} ) for each path (s, , t) X appearing with non-zero coecient in f . A homotopy of geometric morphisms is an homotopy ( ) if 1. in operation 1, each homotopy (even the constant one) of a path (s, , t) has image in p1 ({pX (0)} {pX ( )} ), and X 2. in operation 2, each path (s, , t) in the combined terms (or split term) has image in p1 ({pX (0)} {pX ( )} ). X In other words, the morphism is required to have radius at every stage of the homotopy operations 1 and 2. Proposition 2.1. Assuming that the relevant operations on geometric morphisms are possible, the following hold true: (1) If f f and f f , then f max{ ,} f . (2) If f f and g g , then mf + ng max{ ,} mf + ng for any m, n Z. (3) The composite gf of a geometric morphism f of radius and a geometric morphism g of radius has radius + . (4) If f f and g g , then gf + g f . Proof: Immediate from the denition. Let pX : M X be a control map for M . In the following denition, all geometric modules are on pX . Denitions. A projection p : A A is an projection if p2 p. A projective module (A, p) is an projective module if p is an projection. A morphism f : (A, p) (B, q) between projective modules is an morphism if f has radius and satises: qf f , f p f . An morphism f : (A, p) (B, q) is an isomorphism if there exists an morphism g : (B, q) (A, p) such that gf 2 p and f g 2 q. Remarks. (1) In the denition of morphisms and isomorphisms, we do not require (A, p) and (B, q) to be projective modules for any particular > 0 so that the denition is as simple as possible. There is an extra advantage that the zero morphism is always a 0 morphism. (2) The choice of coecients of in the denition above looks rather arbitrary. Here is an explanation: First of all, we want an projection p : A A to be an morphism between (A, p) and itself. Secondly, sizes should behave nicely with respect to composition. (See 2.2.) A dierent denition of projections will force a possibly dierent denition of morphisms and isomorphisms. One possibility is to use p2 2 p, but this forces us to use 3 in the denition of isomorphisms, which is not so desirable. Anyway this is not a crucial problem. 10
Proposition 2.2. The composition gf : (A, p) (C, r) of a morphism (resp. isomorphism) f : (A, p) (B, q) and an morphism (resp. isomorphism) g : (B, q) (C, r) is a + morphism (resp. isomorphism). Proof : Obviously gf has radius + , and r(gf ) = (rg)f + gf, (gf )p = g(f p) + gf.
So, gf is a + morphism. If further f is a isomorphism with inverse f 1 and g is an isomorphism with inverse g 1 , then (f 1 g 1 )(gf ) 2+2 f 1 qf 2 f 1 f 2 p , and similarly (gf )(f 1 g 1 ) 2+2 r. Denition. A projective chain complex (C, p) on M is an projective chain complex on pX if 1. each (Cr , pr ) is an projective module, 2. each dr : (Cr , pr ) (Cr1 , pr1 ) is an morphism, and 3. dr dr+1 2 0 for each r. A free projective chain complex will be called a free chain complex. Denitions. (1) A chain map f : (C, p) (D, q) is an chain map if fr : (Cr , pr ) (Dr , qr ) are morphisms such that dr fr fr1 dr . (2) A chain homotopy h : f g between chain maps f, g : (C, p) (D, q) is an chain homotopy, h : f g, if the hr s are morphisms such that dr+1 hr +hr1 dr 2 gr f r . (3) An chain map f : (C, p) (D, q) is an chain equivalence if there exists an chain map g : (D, q) (C, p), called an chain homotopy inverse, such that gf p and f g q. (4) Two projective chain complexes (C, p) and (D, q) are chain equivalent, (C, p) (D, q), if there exists an chain equivalence between them. (5) A projective chain complex (C, p) is contractible if it is chain equivalent to the zero chain complex. In this case, an chain homotopy h : 0 p : (C, p) (C, p) is called an chain contraction. (6) An chain map f : (C, p) (D, q) is an isomorphism, (C, p) (D, q) if there = exists an chain map g : (D, q) (C, p), called an inverse, such that gf 2 p and f g 2 q. Thus each fr is an isomorphism of projective modules. An isomorphism of projective chain complexes is always an chain equivalence. For projective chain complexes of dimension 0, the converse is also true. The identity chain map p = {pr } on an projective chain complex (C, p) is an isomorphism. 11
Proposition 2.3. (1) The composition f f of an chain map f : (C, p) (D, q) and an chain map f : (D, q) (E, r) is an + chain map. (2) The composition f f of an isomorphism f : (C, p) (D, q) and an isomorphism f : (D, q) (E, r) is an + isomorphism. (3) The composition f f of an chain equivalence f : (C, p) (D, q) and an chain equivalence f : (D, q) (E, r) is an + chain equivalence. Proof : (1) and (2) are obvious. We prove (3): Let g and g be chain homotopy inverses of f and f with chain homotopies h : gf p, k : f g q and chain homotopies h : g f q, k : f g r. Then, d(f kg + k ) + (f kg + k )d 2 2 and similarly, d(h + gh f ) + (h + gh f )d 2
+2 +2 +2 +2
f (dk + kd)g + (r f g )
+2
f (q f g)g + (r f g ) 2 r (f f )(gg ) ,
f g f f gg + r f g
p (gg )(f f ).
Proposition 2.4. Let f : (C, p) (D, q) be an chain map. If the algebraic mapping cone C(f ) is contractible, then f is a 2 chain equivalence. If f is an chain equivalence, then C(f ) is 3 contractible. Proof : Given an chain contraction, : 0 morphisms dened by = k ()r g ? h q p : C(f ) C(f ), let g, h, k be the
: C(f )r = (Dr , qr ) (Cr1 , pr1 ) C(f )r+1 = (Dr+1 , qr+1 ) (Cr , pr ) .
Then g : (D, q) (C, p) is a chain homotopy inverse of f with chain homotopies h : gf p : (C, p) (C, p), k : f g q : (D, q) (D, q). Although the radius of g is , g is a 2 chain map because we only have dg 2 gd. Therefore f is a 2 chain equivalence. Conversely, suppose that f is an chain equivalence with chain homotopy inverse g : (D, q) (C, p) and chain homotopies h : gf k : fg k + (f h kf )g ()r g p : (C, p) (C, p) q : (D, q) (D, q) .
A 3 chain contraction of C(f ) is given by: = ()r (f h kf )h h :
C(f )r = (Dr , qr ) (Cr1 , pr1 ) C(f )r+1 = (Dr+1 , qr+1 ) (Cr , pr ) .
12
3. Controlled niteness obstruction. We start with a brief review of the projective class group and niteness obstruction in the uncontrolled case and then go on to deal with the controlled analogues. Given a ring A and an integer n 0, let K0 (A, n) be the quotient of the Grothendieck group of n-dimensional f.g. projective A-module chain complexes by the subgroup of f.g. free A-module chain complexes. For n = 0 this is the reduced projective class group of A K0 (A, 0) = K0 (A) , the quotient of the Grothendieck group of f.g. projective A-modules by the subgroup of f.g. free A-modules. The reduced projective class of an n-dimensional f.g. projective A-module chain complex P
n
[P ] =
r=0
()r [Pr ] K0 (A)
is a chain homotopy invariant, such that [P ] = 0 if and only if P is chain equivalent to a nite f.g. free A-module chain complex. The reduced projective class denes isomorphisms K0 (A, n) K0 (A) ; [P ] [P ] . The singular chain complex of the universal cover M of a nitely dominated space M is chain equivalent to a nite f.g. projective Z[1 (M )]-module chain complex C(M ). The niteness obstruction of Wall [23] is the reduced projective class [M ] = [C(M )] K0 (Z[1 (M )]) , such that [M ] = 0 if and only if M is homotopy equivalent to a nite CW complex. We use projective chain complexes to dene a controlled analogue analogue of the projective class groups. To obtain a correct analogue, we have to use chain complexes that are nitely generated. Denition. Two nitely generated projective chain complexes (C, p) and (C , p ) on pX are n-stable chain equivalent if there exists an chain equivalence between (C, p) (E, 1) and (C , p ) (E , 1) for some nitely generated n-dimensional free chain complexes (E, 1), (E , 1) on pX . For a xed > 0, n-stable chain equivalence is not an equivalence relation. If (C, p), (C , p ) are n-stable chain equivalent and (C , p ), (C , p ) are also n-stable chain equivalent, then (C, p) and (C , p ) are only n-stable 2 chain equivalent. Denition. K0 (X, pX , n, ) is the set of equivalence classes [C, p] of nitely generated n-dimensional projective chain complexes on pX . The equivalence relation is generated by n-stable chain equivalence. K0 (X, pX , 0, ) will be denoted K0 (X, pX , ). 13
We shall also need an analogue which uses projective chain complexes that are not necessarily nitely generated. Such an object arises naturally when we take a pullback of a nitely generated projective chain complex via an innite-sheeted covering map. Denition. A geometric module on a product space M N is said to be M -locally nite if, for any y N , there is a neighbourhood U of y in N such that M U contains only nitely many basis elements; a projective module (A, p) on M N is said to be M -locally nite if A is M -locally nite; a projective chain complex (C, p) on M N is M -locally nite if each (Cr , pr ) is M -locally nite. For M -locally nite geometric modules, we only consider control maps of the form qX = pX 1N : M N X N, where pX : M X is a given control map for M , N is a metric space, and X N is given a maximum product metric. Denition. Two M -locally nite projective chain complexes (C, p) and (C , p ) on qX are M -locally nitely n-stable chain equivalent if there exists an chain equivalence between (C, p) (E, 1) and (C , p ) (E , 1) for some M -locally nite n-dimensional free chain complexes (E, 1), (E , 1) on qX .
M Denition. K0 (X N, qX , n, ) is the set of equivalence classes [C, p] of M -locally nite n-dimensional projective chain complexes on qX . The equivalence relation is M generated by M -locally nitely n-stable chain equivalence. K0 (X, pX , 0, ) will be M denoted K0 (X, pX , ).
Important Notice. In the rest of this section we mainly discuss K0 (X, pX , n, ) and all the chain complexes are assumed to be nitely generated unless explicitly stated otherwise. But the argument carries over to the M -locally nite case without any modication. Proposition 3.1. Direct sum induces an abelian group structure on K0 (X, pX , n, ). Further, if [C, p] = [C , p ] K0 (X, pX , n, ), then there is a 3 chain equivalence (C, p) (E, 1) (C , p ) (F, 1) for some n-dimensional free chain complexes (E, 1), (F, 1) on pX . In particular, (C, p) and (C , p ) are n-stable 3 chain equivalent. Proof : We shall show the existence of inverses. Note that if (A, p) is an projective module, then (A, 1 p) is also an projective module and the direct sum (A, p) (A, 1p) is isomorphic to (A, 1); the morphism (p, 1p) : (A, p)(A, 1p) (A, 1) gives a desired isomorphism with an inverse t (p, 1p) : (A, 1) (A, p)(A, 1p). 14
Suppose {(C, p), dC } is an n-dimensional projective chain complex. The direct sum {(C, p), dC } {(C, 1 p), 0} with an n-dimensional projective chain complex: {(C, 1 p), 0} : . . . 0 (Cn , 1 pn ) . . . (C0 , 1 p0 ) 0 is isomorphic to the free chain complex:
dC dC 0 0
{(C, 1), dC } : . . . 0 (Cn , 1) . . . (C0 , 1) 0. Thus [(C, 1 p), 0] is the inverse of [(C, p), dC ]. Next suppose [(C, p), d] = [(C , p ), d ]. We use the cancellation of inverses argument originally employed by Chapman to prove a similar result for controlled Whitehead groups [6, 3.5]. There is a sequence of n-dimensional projective chain complexes {(C, p), d} = {(C (1) , p(1) ), d}, {(C (2), p(2) ), d}, . . . , {(C (m) , p(m) ), d} = {(C , p ), d }, where {(C (k) , p(k) ), d} {(E (k) , 1), d} {(C (k+1) , p(k+1) ), d} {(F (k) , 1), d} for some n-dimensional free chain complexes {(E (k) , 1), d}, {(F (k) , 1), d} on pX . The following composition gives the desired 3 chain equivalence:
m1 m
{(C, p), d}
k=1 m1
{(E {(E
k=1 (1)
(k)
, 1), d}
k=1 m
{(C (k) , 1), d} {(C (k) , 1 p(k) ), 0} {(C (k) , p(k) ), d}
{(C, p), d} = = {(C, p), d} {(C
(k)
, 1), d}
k=1 (1)
m1
,1 p
m
), 0}
k=1
{(C (k) , p(k) ), d} {(E (k) , 1), d}
k=2
{(C (k) , 1 p(k) ), 0} {(C (m) , p(m) ), d}
m1 (1)
{(C, p), d} {(C
(1)
,1p
m
), 0}
k=1
{(C (k+1) , p(k+1) ), d} {(F (k) , 1), d}
k=2 m
{(C (k) , 1 p(k) ), 0} {(C , p ), d }
m1 (k)
=
k=1
{(C
(k)
,p
(k)
), d} {(C
m
,1 p
(k)
), 0}
k=1
{(F (k) , 1), d} {(C , p ), d }
m1
{(C , p ), d } =
k=1
{(C (k) , 1), d}
k=1
{(F (k) , 1), d} . 15
Remark. By the construction of the additive inverse, one can conclude that the class [C, p] K0 (X, pX , n, ) depends only on the projective modules (Ci , pi ) and not on the boundary morphisms. Next we discuss maps between control maps which induce homomorphisms of controlled K0 -groups. Let pX : M X and pX : M X be control maps. A map from pX to pX is a pair of continuous maps = ( : M M , : X X ) satisfying pX = pX . Let , be positive numbers and k be a positive integer. Consider the following condition on : C(, , k) : if x, y X, and d(x, y) k then d((x), (y)) k . Suppose that satises the conditions C(, , 1) and C(, , 2) and apply to chain complexes: if (C, p) is projective chain complex on pX , then (C, p) = ( C, p) is an projective chain complex on pX , and if two projective chain complexes (C, p) and (C , p ) on pX are n-stable chain equivalent, then (C, p) and (C , p ) are n-stable chain equivalent. Therefore induces a homomorphism : K0 (X, pX , n, ) K0 (X , pX , n, ). The equality () = is easily veried. Note that the condition C(, , 1) does not imply the other condition C(, , 2). Also note that if X is compact, then for any > 0, there exists a > 0 satisfying these conditions. Inclusion maps are typical examples of maps which satisfy C( , , k) for every positive number and every positive integer k. Let i : Y X be an inclusion map and : p1 (Y ) M be the corresponding inclusion map. Then (, i) is a map X from pY to pX , where pY : p1 (Y ) Y is the restriction of pX , and it induces a X homomorphism (, i) : K0 (Y, pY , n, ) K0 (X, pX , n, ) for every > 0 and every n 0. This homomorphism will be called the homomorphism induced by i and will be denoted i . More generally, if , then (, i) also satises C(, , k) for every k 1 and induces a homomorphism (, i) : K0 (Y, pY , n, ) K0 (X, pX , n, ) . This homomorphism will be called a stabilization map. It is also commonly called the relaxation of control map. M In the case of K0 , we only consider maps between control maps of the form = ( 1M : M N M N , 1X : X N X N )
from pX 1N to pX 1N , where : N N is a continuous map. Stabilizations are dened similarly. 16
Given an > 0, we are interested not in the group K0 (X, pX , n, ) itself but in the image of K0 (X, pX , n, ) in it for suciently small > 0. Such an image tends to get stable as gets smaller; e.g., see 8.2. Below we shall see that an analogue of the isomorphism K0 (A, n) K0 (A, 0) holds only stably in the controlled setting. = Let n > 0 be an integer. We shall study the relationship between K0 (X, pX , n, ) and K0 (X, pX , ). There is a homomorphism : K0 (X, pX , ) K0 (X, pX , n, ) obtained by viewing a 0-dimensional chain complex as an n-dimensional one. We shall see that this is onto. Proposition 3.2. An n-dimensional projective chain complex (C, p) is chain equivalent to an n-dimensional projective chain complex (D, q) such that qr = 1 : Dr Dr for r > 0 and (D0 , q0 ) = ( r:even (Cr , pr )) ( r:odd (Cr , 1 pr )). Proof : Let (Dr , qr ) = ir (Ci , 1) for r > 0 and let (D0 , q0 ) be as in the statement above. Dene the boundary morphism dD by: d 0 0 0 ... 0 0 0 ... 1 pr pr+1 0 0 ... 0 (dD )r = 0 0 1 pr+2 0 ... : 0 0 pr+3 . . . 0 . . . . .. . . . . . . . . . if r > 1 (Cr1 , 1) (Cr , 1) (Cn , 1), Dr (C0 , p0 ) (C1 , 1 p1 ) (Cn , pn ), if r = 1 and n is even, (C0 , p0 ) (C1 , 1 p1 ) (Cn , 1 pn ), if r = 1 and n is odd.
Then (D, q) = {(Dr , qr ), dD } is an n-dimensional projective chain complex. The following chain maps give the desired chain equivalence and its chain homotopy inverse: pr = t ( pr 0 . . . 0 ) : (Cr , pr )(Dr , qr ) pr = ( pr 0 . . . 0 ) : (Dr , qr )(Cr , pr ).
Corollary 3.3. The homomorphism : K0 (X, pX , ) K0 (X, pX , n, ) is onto. Proof : For an element [C, p] K0 (X, pX , n, ), let (D, q) be as in 3.2. Then the sum of (C, p) and the 0-dimensional free chain complex 00(D0 , 1)0 17
is
chain equivalent to the sum of the 0-dimensional
projective chain complex
00(D0 , q0 )0 and the n-dimensional free chain complex
dD dD
(D2 , 1) (D1 , 1) (D0 , 1)0 . Here dD : (D1 , 1) (D0 , 1) denotes the morphism dened by the same geometric morphism which was used to dene the morphism dD : (D1 , 1) (D0 , q0 ). An chain equivalence is given by pr : (Cr , pr ) (Dr , 1) 0 p0 q0 1 q0 (r > 0), (r = 0).
: (C0 , p0 ) (D0 , 1) (D0 , q0 ) (D0 , 1)
Proposition 3.4. This correspondence (C, p) (D0 , q0 ) denes a well-dened homomorphism : K0 (X, pX , n, ) K0 (X, pX , 9 ) . Proof : First suppose (C, p) is chain equivalent to 0. Let be an chain contraction of (C, p). Dene a 3 chain contraction by: = d. This has a larger radius but the identity ( )2 6 0 holds. (Cf. Whitehead [25,(6.2)].) Using this one can show that d+ : (Cr , pr ) (Cr , pr )
r:even r:odd
d+ :
r:odd
(Cr , pr )
r:even
(Cr , pr )
are 3 inverses of each other. Therefore r:odd (Cr , pr ) and r:even (Cr , pr ) represent the same element in K0 (X, pX , 3 ), and [D0 , q0 ] = 0 K0 (X, pX , 3 ). Next suppose f : (C, p) (C , p ) is an chain equivalence. By 2.4 its algebraic mapping cone C(f ) is 3 contractible. By the argument above, r (1)r [C(f )r , pr r pr1 ] = 0 K0 (X, pX , 9 ). But this element is the same as r (1) [Cr , pr ] r r (1) [Cr , pr ]. Since direct sum with free chain complexes corresponds to direct sum with free modules, this nishes the proof. The two homomorphisms and above are stable inverses: the following diagrams commute. 18
K0 (X, pX , )
K0 (X, pX , n, )
K0 (X, pX , 9 )
K0 (X, pX , n, )
K0 (X, pX , 9 )
K0 (X, pX , n, )
K0 (X, pX , 9 )
K0 (X, pX , n, 9 )
The following is a corollary to 3.2. Corollary 3.5. Let n > 0. If [C, p] = 0 K0 (X, pX , n, ), then (C, p) is 60 chain equivalent to an n-dimensional free 30 chain complex on pX . Proof : Let (D, q) be as in 3.2; (C, p) is chain equivalent to (D, q). By the previous proposition, [D0 , q0 ] = 0 K0 (X, pX , 9 ). By 3.1, there is a free module (F, 1) such that (D0 , q0 ) (F, 1) is 27 isomorphic to some free module (G, 1). The inclusion map of (D, q) into the sum (D , q ) of (D, q) and the 1-dimensional free chain complex 0(F, 1) (F, 1)0 is an chain equivalence. Let f : (D0 , q0 ) = (D0 , q0 ) (F, 1) (G, 1) be a 27 isomorphism of projective modules. If we replace the boundary map dD : (D1 , 1) (D0 , q0 ) of (D , q ) by f dD : (D1 , 1) (G, 1), then we get a free 28 chain complex (D , 1) with Dr = Dr (r > 0) and D0 = G. The isomorphisms 1 : Dr Dr f : D0 D0 (r>0)
1
dene a 28 chain map from (D , q ) to (D , 1), and its inverse is a 55 chain map (f 1 (f dD ) (227+1) dD ). Thus we get a 55 isomorphism between (D , q ) and (D , 1). Composing these we get a 57 chain equivalence from (C, p) to an ndimensional free 28 chain complex.
4. Controlled Whitehead torsion. We start with a brief review of Whitehead torsion in the uncontrolled case, and then go on to deal with the controlled analogues. Given a group and an integer n 1 let W h(, n) be the quotient of the Grothendieck group of n-dimensional contractible based f.g. free Z[]-module chain complexes by the subgroup of the elementary complexes. For n = 1 this is the Whitehead group of W h(, 1) = W h() , 19
a quotient of the Grothendieck group of isomorphisms of based f.g. free Z[]-modules. The torsion of a contractible based f.g. free Z[]-module chain complex C is dened by (C) = (d + : Codd Ceven ) W h() using any chain contraction : 0 1 : CC. (It is usually more convenient if we 2 further require = 0. A corresponding requirement in the controlled case also helps size estimation. But this is not really necessary in the uncontrolled setting. See [7, p.52].) Torsion denes isomorphisms W h(, n) W h() ; [C] (C) . The Whitehead torsion of a homotopy equivalence f : LM of nite CW complexes (f ) = (f : C(L)C(M )) W h(1 (M )) is such that f is simple ( (f ) = 0) if and only if f is homotopic to a deformation, that is a composite of elementary expansions and collapses - see Milnor [15] and Cohen [7] for detailed expositions. We x the control map pX : M X to a metric space X and an integer n 1. Given a subspace Y X and > 0, we dene the relative controlled Whitehead group W h(X, Y, pX , n, ), the controlled analogue of W h(, ). In 5 this will be related to the controlled projective class groups K0 of 3 by a stably-exact sequence. The controlled Whitehead groups W h(X, pX , ) = W h(X, , pX , 1, ) were previously dened by Quinn [18]. Important Notice. As in the previous section, all the modules and chain complexes will be assumed to be nitely generated. But the argument carries over to the M -locally nite case without any modication: one can dene the M -locally nite controlled (relative) Whitehead groups W hM (X N, Y N, pX 1N , n, ) using M -locally nite chain complexes and can prove analogous results. Denition. A geometric morphism f : Z[S] Z[S] is elementary if Z[S] is the direct sum of two geometric modules Z[S1 ] and Z[S2 ] and f = 1 h 0 1 : Z[S1 ] Z[S2 ] Z[S1 ] Z[S2 ],
for some morphism h : Z[S2 ] Z[S1 ]. Such an f is an isomorphism and its inverse 1 h f 1 = is also elementary. 0 1 An isomorphism f : Z[S] Z[S ] between geometric modules of the same rank is geometric if there is a bijection : S S such that f has no paths from s S to 20
s S unless s = (s) and there is exactly one path from s to (s) for each s S, whose coecient is 1. Its inverse f 1 is obtained by reversing the orientation of the paths. A deformation is a sequence D : Z[S1 ] Z[S2 ] . . . Z[Sm+1 ] of elementary automorphisms and geometric isomorphisms. D is an deformation 1 1 if all composite geometric morphisms fj fj1 fi , fi1 fi+1 fj have radius . (If fm fm1 f1 has radius , then all the composites fj fj1 fi have radius 2, and similarly for the inverses.) When D is an deformation, the composite isomorphism f = fm fm1 f1 is called an -simple isomorphism. The composite 1 1 1 f 1 = f1 f2 fm gives an inverse of f . The composite of an -simple isomorphism and a -simple isomorphism is an ( + )-simple isomorphism. Denition. A free chain complex of the form: ... 0 0 A A 0 0 ... is called an elementary trivial chain complex. A free chain complex T is trivial if it is the direct sum of elementary trivial chain complexes. A trivial chain complex is 0 contractible. An chain map f = {fr } : C D between free chain complexes on pX 1 is an -simple isomorphism if each fr is an -simple isomorphism and f 1 = {fr } is an chain map. We use the notation: f : C , D. The composite of an -simple = isomorphism and a -simple isomorphism (of chain complexes) is an ( + )-simple isomorphism. Let n be an integer. Two free chain complexes C and C on pX are n-stable -simple equivalent if there exists an -simple isomorphism between C T and C T for some n-dimensional trivial chain complexes T and T on pX . Warning. Do not confuse n-stable -simple equivalences (of free chain complexes) with n-stable chain equivalences (of projective chain complexes) dened in the previous section. Any two free chain complexes of dimension n are always n-stable chain equivalent because they are free. Let Y be a subspace of X. The restriction to Y of a geometric module A = Z[S] on pX is the geometric module on pY generated by the elements (|s|, [s]) of S such that [s] p1 (Y ) and is denoted A(Y ); i.e., X A(Y ) = Z[S|S 1 p1 (Y ) : S 1 p1 (Y )p1 (Y )]. X X X (Recall that pY is a restriction of pX .) The restriction to Y of a geometric morphism f= m (s , , t ) : A B is dened to be: f |Y =
[s ]p1 (Y X ) 1 f1 f2 fm
m (s , , t ) : A B.
21
Note that f and f |Y have the same domain. Of course one can also restrict the domain and the target: If f has radius , then f |Y determines a geometric morphism from A(Y ) to B(Y ), which will be also denoted f |Y . Suppose f , g : A B be two geometric morphisms. When f |Y = g|Y , we write f = g over Y . When f |Y g|Y , we write f g over Y . Consider an elementary geometric morphism f = 1 h 0 1 : Z[S1 ] Z[S2 ] Z[S1 ] Z[S2 ].
If we replace h by the morphism h|(X Y ), then we get another elementary geometric morphism f[Y ] which coincides with f over X Y and is the identity over Y . f[Y ] is called the localization of f away from Y . Note that (f[Y ] )1 = (f 1 )[Y ] . For a deformation D = (f1 , f2 , . . . , fm ), dene the localization of D away from Y by (f1 , f2 , . . . , fm ), where fj = (fj )[Y ] if fj is elementary and fj = fj if fj is geometric. The composite fm f1 is called the localization away from Y of the simple isomorphism f = fm f1 , and is denoted f[Y ] . If f is an -simple isomorphism, then f[Y ] is also an -simple isomorphism, coincides with f over X Y , and is geometric over Y . Denition. (1) Let f, g : C D be chain maps between free chain complexes on pX . A collection h = {hr } of morphisms is an chain homotopy over Y between f and g, h : f Y g, if dh and hd both have radius 2 and dh + hd 2 g f over Y . (2) An chain map f : C D is an chain equivalence over Y if there exist an chain map g : D C and chain homotopies over Y :gf Y p and f g Y q. (3) An chain homotopy h : 0 Y 1 : C C over Y is called an chain contraction over Y , and C is said to be contractible over Y . (4) A strong chain contraction over Y of C, will mean an chain contraction of C over Y which satises the additional condition r+1 r 2 0 over Y . If such a exists, we say C is strongly contractible over Y (or strongly contractible if Y = X). This extra condition can be achieved in the following way. Suppose is an chain contraction of C over Y . Then = d is a strong 3 chain contraction of C over Y 3 . (We used this construction in the proof of 3.4.) (5) Two free chain complexes C and C on pX are said to be n-stable -simple equivalent away from Y if there exist n-dimensional free chain complexes D and D on pY such that C D and C D are n-stable -simple equivalent. We use the notation: n,Y C C . For example, an n-stable -simple equivalence away from the empty subset is the same as an n-stable -simple equivalence. For a xed > 0, n-stable -simple n,Y equivalence away from Y is not an equivalence relation; in general, C C and C
m,Z
C imply C
max{n,m},Y Z +
C . 22
Denition. Let Y be a subspace of X. W h(X, Y, pX , n, ) is dened to be the set of equivalence classes of n-dimensional free chain complexes on pX which are strongly contractible over X Y . The equivalence relation is generated by n-stable 40 -simple equivalences away from Y 20 . If Y is the empty set, it will be omitted from the notation, and if n = 1, then n is omitted; e.g., W h(X, pX , n, ) = W h(X, , pX , n, ) and W h(X, Y, pX , ) = W h(X, Y, pX , 1, ), etc. Remark. Y 20 is used in this denition instead of Y so that any element of W h(X, Y, pX , n, ) has an additive inverse. See 4.1 below. 4.1 also says that the equivalence relation generated by n-stable 40 -simple equivalences away from Y 20 implies n-stable 86 -simple equivalence away from Y 20 . (Recall that n-stable 40 simple equivalences away from Y 20 are not equivalence relations in general.) The proof of 4.1 is rather long and occupies the next several pages. Proposition 4.1. Direct sum induces an abelian group structure on W h(X, Y, pX , n, ). Further if [C] = [C ] W h(X, Y, pX , n, ), then C and C are n-stable 86 -simple equivalent away from Y 20 . We need to show the existence of additive inverses. The next lemma shows that the suspension C of C (or the suspension of anything which looks the same as C over X Y ) gives the additive inverse of [C] at least when dim C < n. Lemma 4.2. Suppose C = {Cr , dr } (resp. C = {Cr , dr }) is a free ( resp. ) chain complex of dimension m ( resp. m ) on pX . Let Y be a subspace of X and assume that 1. C has an chain contraction over X Y , 2. Cr (X Y ) = Cr (X Y ) for all r, and 3. dr |X Y = dr |X Y : Cr (X Y ) Cr1 (X Y ) for all r. Let = max{ , }, and n = max{m + 1, m }. Then there is a (6 + )-simple isomorphism from C C to the direct sum of an n-dimensional free 4 + chain complex on pY 11 +2 and an n-dimensional trivial chain complex on pX . In particular, C C is n-stable (6 + )-simple equivalent to 0 away from Y 11 +2 . Proof : Dene morphisms dr : Cr Cr1 and r : Cr Cr+1 by : dr = r = dr 0 r 0 over X Y over Y over X Y over Y . 23
Consider the following 2 -simple isomorphism and its inverse: fr = ()r 1 0 0 1 1 0 r1 () d 1 1 ()r 0 1 ()r 1 0 0 1 = ()r 1
()r1 d d + 1 ()r (1 d ) d ()r+1 1
: (C C)r = Cr Cr1 Cr Cr1 ,
1 fr
=
1 ()r+1 0 1
1 0 ()r d 1
=
: Cr Cr1 (C C)r = Cr Cr1 , and dene a new chain complex C = {Cr , dr } by Cr = Cr Cr1 ,
1 dr = fr1 (dr dr1 )fr =
d + (d + d)d d + d + dd d(d + d)d d d(d + d) dd
.
Then dr dr+1 8 +2 0, and dr fr 6 + fr1 (dr dr1 ). Since d + d 2 1 over X Y 2 , we have 0 1 over X Y 3 . dr 4 0 0 Modify dr over X Y 3 by a 3 + homotopy to get a 4 + chain complex r , dr }: C = {C 0 1 over X Y 3 dr = 0 0 r d over Y 3 . f = {fr } can be thought of as a (6 + )-simple isomorphism from C C to C. C is a direct sum of its restrictions C(Y 11 C(X Y
+2
) = {Cr (Y 11
+2
), dr |Y 11
+2
},
+2
11 +2
) = {Cr (X Y
11 +2
), dr |X Y 11
}.
C(X Y 11 +2 ) is a trivial complex, and C(Y 11 on pY 11 +2 .
+2
) is a free 4 + chain complex
To get an n-dimensional inverse when n > 1, we use the following folding argument. A dual argument (folding up from the bottom dimension) was used in Yamasaki [26]. 24
Lemma 4.3. Let Y be a subspace of X and C be an n-dimensional free chain complex (n > 1) on pX with a strong chain contraction over X Y . Then C is n-stable 16 -simple equivalent away from Y 17 to the (n 1)-dimensional free chain complex:
{C, d } : . . . 0 Cn1 Cn2 Cn Cn3 . . . C0 0 d (d 0) d d
which has a strong
chain contraction over X Y .
Proof : Let i, j (resp. r, q) be inclusion maps (resp. projections) of Cn (Y ), Cn (XY ) into Cn (resp. Cn to Cn (Y ), Cn (X Y )). By assumption, we have a homotopy: dj 2 j : Cn (X Y ) Cn . Consider the following 3 chain complex C : . . . 0 Cn (Y ) Cn1 Cn2 Cn (X Y ) Cn3 . . . C0 0 where = dn i dn jqdn i. Then the following diagram gives an n-stable 4 equivalence between C and C : 0 Cn fn d Cn1 Cn (X Y ) fn1 0 Cn (Y ) Cn (X Y ) where fn = fn1 = fn2 = 1 qdi 0 1 : Cn = Cn (Y ) Cn (X Y ) Cn (Y ) Cn (X Y ) 1 q 0 1 : Cn1 Cn (X Y ) Cn1 Cn (X Y ) Cn1 Cn (X Y ) d 1 Cn2 Cn (X Y ) fn2 Cn2 Cn (X Y ) Cn3 ... d Cn3 ...
d q (d 0) d d
1
1 dj 0 1 1 0 0 1
: Cn2 Cn (X Y ) Cn2 Cn (X Y ) .
Note the following: 25
1. C has a strong
chain contraction over X Y dened by: d ) , n3 = n3 0 , r = r (r < n 3),
n2 = ( n2
2. Cr (X Y ) = Cr (X Y ) for all r, 3. dr |X Y = dr |X Y : Cr (X Y ) Cr (X Y ) for all r. By 4.2, C C is n-stable 9 -simple equivalent to 0 away from Y 17 . Also note that C C is n-stable 7 -simple equivalent to 0 away from Y 13 , again by 4.2. Therefore C and C are n-stable 16 -simple equivalent away from Y 17 : C
n,Y 13 7
C C C
n,Y 17 9
C.
Corollary 4.4. Let n > 1. Then [C] is the additive inverse of [C] in W h(X, Y, pX , n, ). In fact there is a n-stable 23 -simple equivalence away from Y 17 between C C and 0. Proof : C C =
n,Y 17 16
C C
n,Y 13 7
0.
The existence of inverses when n = 1 is a special case (C1 = C0 , C0 = C1 , d = , = ) of the next lemma.
d
Lemma 4.2 . Let C : 0 C1 C0 0 be a 1-dimensional free
d
chain complex
and C : 0 C1 C0 0 be a 1-dimensional free chain complex. Assume 1. C has an chain contraction over X Y . 2. C1 (X Y ) = C0 (X Y ), C0 (X Y ) = C1 (X Y ) 3. d |X Y |X Y : C1 (X Y ) C0 (X Y ), and let = max{ , }. Then the direct sum C C is (5 +)-simple isomorphic to the direct sum of a 1-dimensional free 3 + chain complex on pY 5 + and a 1-dimensional trivial chain complex on pX . In particular C C is 1-stable (5 +)-simple equivalent to 0 away from Y 5 + . Proof : Dene morphisms d : C0 C0 , : C1 C1 , : C0 C0 by d, , over X Y and by 0, 0, 0 over Y . Dene a 1-dimensional free 3 + chain complex E by: E1 = C 1 C 1 , E0 = C 0 C 0 d d d + dd d d d over Y 2 dE = d d d 1 over X Y 2 . 26
Dene an -simple isomorphism f1 : C1 C1 E1 and a 2 -simple isomorphism f0 : C0 C0 E0 by f1 = 0 1 1 , f0 = 1 0 d 1 1 0 1 .
1 A direct calculation shows that dE 3 + f0 (d d )f1 , and one can check that f = {fr } : C C E is a (5 + )-simple isomorphism. E is a direct sum of the free 3 + chain complex E(Y 5 + ) on pY 5 + and the trivial chain complex E(X Y 5 + ).
This completes the proof of the existence of additive inverses. Next, suppose that [C] = [C ] W h(X, Y, pX , n, ). One can argue as in the previous section using Chapmans trick to show that C and C are n-stable 86 -simple equivalent away from Y 20 : By denition there are elements [C (i) ] W h(X, Y, pX , n, ) such that C = C (0)
n,Y 20 40
C (1)
n,Y 20 40
...
n,Y 20 40
C (m) = C .
By 4.4 and 4.2 , there are elements [D (i) ] W h(X, Y, pX , n, ) such that C (i) D (i) Then C
n,Y 17 23 n,Y 20 40 n,Y 17 23
0
for i = 0, 1, . . . , m.
C (C (0) D (0) ) . . . (C (m1) D (m1) ) (C (m) D (m) ) C (C (1) D (0) ) . . . (C (m) D (m1) ) (C (m) D (m) ) (C D (0) ) (C (1) D (1) ) . . . (C (m) D (m) ) C (m) C. C . This ends the proof of 4.1.
=
n,Y 20 40
Therefore C
n,Y 20 86
The next proposition gives a sucient condition for two chain complexes to represent the same class in the relative controlled Whitehead group. Proposition 4.5. Suppose [C, d] and [C , d ] are elements of W h(X, Y, pX , n, ). If Cr (X Y ) = Cr (X Y ) and dr |X Y = dr |X Y for every r , then [C] = [C ]. Proof : We rst consider the case n > 1. Let be a strong X Y of C. Dene : Cr Cr+1 (r Z) by = 0 over X Y over Y . 27 chain contraction over
Then is a strong chain contraction over X Y 2 of C . Let C and C be the (n1)dimensional chain complexes obtained by applying 4.3 to C, and C , respectively. Then n,Y 17 n,Y 19 C C , C C , 16 16 and by 4.2 we have C C Composing the equivalences: C
n,Y 17 16 n,Y 13 7
0 , C C
n,Y 13 7
0.
C C C
n,Y 19 16
C,
we get an n-stable 32 -simple equivalence away from Y 19 between C and C . The case n = 1 is similar; use the additive inverse of C (Lemma 4.2 ) instead of C above. Suppose pX : M X and pX : M X are control maps and Y , Y are subspaces of X and X . If a map = (, ) : pX pX satises (Y ) Y and the conditions C(, , 1), C(, , 2), C(, , 20), C(, , 40), then it induces a homomorphism : W h(X, Y, pX , ) W h(X , Y , pX , ). The equality ( ) = holds. As in the case of K0 , an inclusion map i : (A, B) (X, Y ) induces a homomorphism i : W h(A, B, pA , n, ) W h(X, Y, pX , n, ) . And more generally, if , there is a stabilization map W h(A, B, pA , n, ) W h(X, Y, pX , n, ) . The groups W h(X, Y, pX , n, ) and W h(X, Y, pX , ) are only stably isomorphic. If m < n, there is a canonical homomorphism : W h(X, Y, pX , m, ) W h(X, Y, pX , n, ) that sends [C] to [C]. Fix an integer n > 1. Proposition 4.6. The map : W h(X, Y, pX , ) W h(X, Y, pX , n, ) is onto: if C is an n-dimensional free chain complex with a strong chain contraction over X Y , then the 1-dimensional free chain complex C : Codd = C1 C3 . . . Ceven = C0 C2 . . . with the strong chain contraction d + : Ceven Codd over X Y represents the same element as C in W h(X, Y, pX , n, ). Proof : Lemma 4.3 says that any element (C) W h(X, Y, pX , n, ) comes from an element (C) W h(X, Y, pX , n 1, ). So we can repeatedly use 4.3 to show the surjectivity.
d+
28
Proposition 4.7. This correspondence [C] [C ] denes a well-dened homomorphism : W h(X, Y, pX , n, ) W h(X, Y (90n+250) , pX , (90n + 250) ) . Proof : We rst show that the class [C ] is independent of the choice of . If is another strong chain contraction over X Y of C, there is a homotopy: (1 + )(d + ) 3 (d + )(1 + ) : Codd Ceven over X Y .
Here the two morphisms 1 + : Codd Codd and 1 + : Ceven Ceven are n -simple isomorphisms; in fact, they can be written as products: (1 + |C1 )(1 + |C3 ) (1 + |C0 )(1 + |C2 ) respectively. Furthermore we have n homotopies (1 + )(1 + )1 n 1, (To see this, use the identity (1 + )1 = 1 + ( )2 + (1)k ( )k , where k =
n 2
(1 + )1 (1 + ) n 1.
.) Dene an (n + 3) morphism : Codd Ceven by: = (1 + )(d + )(1 + )1 .
Then , viewed as a 1-dimensional chain complex, represents an element of W h(X, Y n , pX , (n + 3) ); an (n + 3) chain contraction over X Y n is given by h = (1 + )(d + )(1 + )1 . The identity above implies that C and are n -simple isomorphic; therefore, [C ] = [] W h(X, Y (n+1) , pX , (n + 3) ). Next we compare and C . It turns out that (n+3) d + over X Y (n+1) .
Modify by a homotopy over X Y (n+1) to get representing the same class as and satisfying the strict identity: = d+ over X Y (n+1) . 29
By 4.5, and C represent the same class in W h(X, Y n , pX , (n + 3) ). Therefore C and C represent the same class in W h(X, Y n , pX , (n + 3) ). Next, suppose we are given two elements [C], [C ] W h(X, Y, pX , n, ) and assume that there is a 40 -simple isomorphism f : C = C D C = C D , where D and D are n-dimensional free 40 chain complexes on Y 20 . If (resp. ) is a strong chain contraction of C (resp. C ) over X Y , then = 0(resp. 20 = 0) is a strong chain contraction over X Y of the free 40 chain (resp. C), and = f f 1 is a strong 81 chain contraction over X complex C Y 60 of C. And f induces a 121 -simple isomorphism between the 1-dimensional chain complexes C and C . C and C (resp. C and C ) represent the same element in W h(X, Y 20 , pX , 40 ). Finally C and C represent the same element in W h(X, (Y 60 )81n , pX , 81(n + 3) ), by the argument in the preceding paragraph. Therefore [C ] = [C ] in W h(X, Y (81n+60) , pX , (81n + 243) ). Adding a trivial complex to C corresponds to adding a trivial complex to C , so it does not change the class of C . Thus is well-dened. It is obviously a homomorphism. The homomorphisms , are stable inverses as in the previous section, and we have similar commutative diagrams. If f : C D is an chain equivalence between n-dimensional free chain complexes on pX , then C(f ) is strongly 9 contractible. We dene the torsion (f ) of f by (f ) = [C(f )] W h(X, pX , n + 1, 9 ) . (When n = 0, i.e. f is an isomorphism, f is identied with C(f ), and its torsion (f ) is dened in W h(X, pX , ).) Proposition 4.8. If f : C D and g : D E are n-dimensional free chain complexes on pX , then When n = 0, the equality holds in W h(X, pX , 2 ). Proof: Dene a trivial chain complex {T, dT } by (dT )r = Then a 1 0 0 0 0 0 1 0 : Tr = Dr Dr1 Dr1 Dr2 = Tr1 . is given by: 0 1 0 00 1 0 0 0 1 0 0 ()r k f 1 0 1 chain equivalences between
(gf ) = (g) + (f ) W h(X, pX , n + 1, 18 ) .
5 -simple isomorphism C(f ) C(g) C(gf ) T 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 00 1 0 0 1 0 0 0 1 0 g 0 1 r r 0 0 () () 1 d 0 0 1 0 f 0 30
0 0 1 0
: Dr Cr1 Er Dr1 Dr Cr1 Er Dr1 , where f 1 is an chain homotopy inverse of f and k is an chain homotopy f f 1 (Cf. Ranicki [20, 4.2 i)].) 1.
5. Relative K-theory. The homology groups H (X, Y ) of a pair of spaces (X, Y X) are such that there is dened an exact sequence . . . Hn (Y ) Hn (X) Hn (X, Y ) Hn1 (Y ) . . . . Analogously, for any group homomorphism , there is an exact sequence W h() W h() W h(, ) K0 (Z[]) K0 (Z[]). The relative group W h(, ) is a quotient of the Grothendieck group of triples (C, D, f ) with C a nite f.g. projective Z[]-module chain complex, D a nite based f.g. free Z[]-module chain complex, and f : Z[] Z[] C D a chain equivalence. We shall now use the algebraic theory of chain homotopy dominations [17] to obtain a stableexact sequence of the type W h(Y ) W h(X) W h(X, Y ) K0 (Y ) K0 (X) relating controlled niteness obstruction and torsion groups, using the relative groups W h(X, Y, pX , n, ) of 4. Remark. If we replace pX : M X, Y X, and nitely generated (f.g.) below by qX = pX 1N : M N X N , X N X N , and M -locally nite, M respectively, then we obtain an analogous result for W hM and K0 . Denition. An domination (D, f, g, h) of a free chain complex C on pX is a free chain complex D on pX together with chain maps f : C D, g : D C, and an chain homotopy h : gf 1 : C C. C is said to be dominated by D. Proposition 5.1. Let C be a free chain complex on pX and let Y be a subspace of X. If (C, 1) is chain equivalent to a projective chain complex (D, r) on pY , then C is dominated by the free chain complex D obtained from (D, r) by forgetting the projection r. Conversely, if C is dominated by an n-dimensional (f.g.) free chain complex on pY , then (C, 1) is (2n + 5) chain equivalent to an n-dimensional (f.g.) (n + 4) projective chain complex on pY (n+4) . Proof : Let f : (C, 1) (D, r), g : (D, r) (C, 1) be inverse chain equivalences with chain homotopies h : gf 1 : (C, 1) (C, 1) , k : f g 31 1 : (D, r) (D, r) .
Then (D, f, g, h) is the desired domination. Next, suppose (D, f, g, h) is a domination of C, where D is an n-dimensional free chain complex on pY . Dene an innite (n + 2) chain complex {C , d } on pY (n+4) by Ci = D 0 D 1 . . . D i , fg d f hg 1 fg f h2 g f hg d = f h3 g f h2 g . . . . . . 2i1 2i2 f h g f h g 0 0 0 0 . . . d
0 d fg f hg . . . 2i3 f h g
0 0 d 1 fg . . . 2i4 f h g
... 0 ... 0 ... 0 ... 0 . .. . . . . . . 1 fg
: C2i = D0 D1 . . . D2i C2i1 = D0 D1 . . . D2i1 , 1 fg d 0 0 ... 0 0 f hg fg d 0 ... 0 0 f h2 g f hg 1 fg d ... 0 0 d = f h3 g f h2 g f hg fg ... 0 0 . . . . . . .. . . . . . . . . . . . . . f h2i g f h2i1 g f h2i2 g f h2i3 g . . . 1 f g d : C2i+1 = D0 D1 . . . D2i+1 C2i = D0 D1 . . . D2i . By a direct calculation, one can check that d may be useful:
2
(n+4) 0. Here the following formulae
h2k gf h2k1 gf h + h2k2 gf h2 . . . + gf h2k (2k+2) h2k dh2k+1 h2k+1 d , h2k+1 gf h2k gf h + h2k1 gf h2 . . . gf h2k+1 (2k+3) dh2k+2 + h2k+2 d . These can be obtained by the substitution gf 2 1 dh hd. The chain map f : C C and the (n + 1) chain map g : C C dened by 0 . . f = . : Ci C i = D 0 . . . D i 0 f g = ( hi g hi1 g . . . hg
(d f 2 f d) ,
g ) : Ci = D0 . . . Di Ci (dg (n+2) g d ) 32
are inverse (n + 1) chain equivalences, as there are dened chain homotopies h : gf k : fg with 0 k = 1 0 1 .. . .. .. .. . . . .. 1 0 . 0 1 0 = gf 1 : C C (dh + hd 2 1 g f )
1 : C C
(d k + k d (n+2) 1 f g )
: Ci = D0 D1 . . . Di Ci+1 = D0 D1 . . . Di+1 . Let E be the n-skeleton of C : E : . . . 0 Cn Cn1 . . . C1 C0 , and dene q = {qi : Ei Ei } by qi = 1 : E i = C i E i = C i qn = where (0 i n 1) if n is even if n is odd ,
d d
p : En = Cn E n = Cn 1 p : E n = Cn E n = Cn
fg f hg p = f h2 g . . .
d 1 fg f hg . . .
0 d fg . . .
... ... . . . : Cn = D0 . . .Dn Cn = D0 . . .Dn . .. .
In other words, qn = 1 dn+1 : Cn+1 = Cn Cn . Since di = p (resp. di = 1 p) 2 if i is even (resp. odd) and i > n, the (n + 4) homotopy d (n+4) 0 implies the (n + 4) homotopy p2 (n+4) p. Therefore q is an (n + 4) projection. Furthermore dn qn (n+4) dn , because 0 (n+4) dn dn+1 = dn (1 qn ) = dn dn qn . 33
Thus (E, q) is an n-dimensional (n + 4) projective chain complex on pY (n+4) . The chain maps I : (C , 1) (E, q), J : (E, q) (C , 1) dened by 1 : Ci E i = Ci I = qn : Ci E i = C i 0 : Ci E i = 0 1 : Ei = Ci C i qn : Ei = C i C i J = 0 : Ei = 0 C i if 0 i n 1 if i = n if i > n if 0 i n 1 if i = n if i > n
are inverse (n + 4) chain equivalences. In fact IJ (n+2) q : (E, q) (E, q) K : JI where K = 1 : (C , 1) (C , 1)
(d K + Kd (n+2) 1 J I) if 0 i n 1 if i n.
0 : Ci Ci+1 1 : Ci = Cn Ci+1 = Cn
Therefore (C, 1) and (E, q) are (2n + 5) chain equivalent. Remark. This is a controlled version of Proposition 3.1 of Ranicki [19]. The rst author would like to thank Erik Pedersen for correcting the error of sign in the formulae for d and p in [19]. Proposition 5.2. If a free chain complex (C, 1) on pX is dominated by a free chain complex on pY for some Y X, then C is contractible over X Y . Conversely, if C is an n-dimensional (f.g.) free chain complex on pX which is contractible over X Y , then C is 3 dominated by an n-dimensional (f.g.) free chain complex on pY (n+2) . Proof : Let (D, f, g, h) be an domination of C. Since the radius of f is , f restricts to 0 on X Y . Therefore h is an chain contraction of C over X Y . Next let be an chain contraction of C over X Y . For each integer r, dene a geometric module Dr by Dr = Cr (Y (nr+2) ) . The restriction of the boundary morphism dr : Cr Cr1 to Y (nr+2) can be viewed as a morphism dr : Dr Dr1 , because dr has radius . And obviously dr dr+1 2 0 : Dr+1 Dr1 . Therefore {Dr , dr } is a subcomplex of C; i.e., the inclusion map i : D C is an chain map. By assumption 1 d d : Cr Cr 34
is 2 homotopic to a morphism Fr : Cr Cr which is 0 over X Y . Fr has radius 2 , so it denes a morphism fr : Cr Dr such that F = if . f = {fr } is a 3 chain map, because i(df f d) = dF F d 3 d(1 d d) (1 d d)d 3 dd + dd 3 0. Since we have a 2 homotopy d + d 2 1 if , (D, f, i, ) is a 3 domination of C. Let n > 0. The inclusion maps i : Y X and j : (X, ) (X, Y ) induce stabilization maps: W h(Y, pY , n, ) W h(X, pX , n, ) W h(X, Y, pX , n, ), K0 (Y, pY , n, ) K0 (X, pX , n, ) . We construct a connecting homomorphism : W h(X, Y, pX , n, ) K0 (W, pW , n, ) for any subspace W of X containing Y Kn and any number greater than or equal to Kn , where Kn = 12n + 70:if C is an n-dimensional f.g. free chain complex on pX which is strongly contractible over X Y , then ([C]) = [E, q] K0 (W, pW , n, ) , where (E, q) is any n-dimensional (3n + 12) projective chain complex on pY (4n+14) that is (6n + 15) chain equivalent to (C, 1). Such a projective chain complex (E, q) exists, because C is 3 dominated by an n-dimensional f.g. free chain complex D on pY (n+2) by the proposition above, and then by 5.1 (C, 1) is (6n+15) chain equivalent to an n-dimensional f.g. (3n + 12) projective chain complex (E, q) on pY (4n+14) . We show that is well-dened. Suppose C is another n-dimensional f.g. free chain complex on pX representing the same element as C in W h(X, Y, pX , n, ) and suppose (C , 1) is (6n + 15) chain equivalent to an n-dimensional f.g. (3n + 12) projective chain complex (E , q ) on pY (4n+14) . Without loss of generality, we may assume that there is a 40 -simple isomorphism C D T 40 =
, i i j
C D T
where D, D are n-dimensional f.g. free 40 chain complexes on pY 20 and T , T are ndimensional f.g. free trivial chain complexes on pX . In particular, (C, 1) (D, 1) and (C , 1)(D , 1) are 40 chain equivalent. Therefore (E, q)(D, 1) and (E , q )(D , 1) are (12n + 70) chain equivalent. Therefore (E, q) and (E , q ) represent the same element in K0 (W, pW , n, ). 35
Remark. [C] depends only on the behaviour of C near Y . More precisely, let (C, d), (C, d) be n-dimensional free chain complexes on pX with strong chain contractions , over X Y respectively, and suppose 1. Cr (Y (2n+5) ) = Cr (Y (2n+5) ) (2n+4) 2. dr |Y = dr |Y (2n+4) (2n+4) r |Y (2n+4) 3. r |Y = for all r. Then the construction above yields the same (E, q) for C and C. Further need not be contractible all over X Y for (E, q) to be dened. more, note that C Thus, in order to compute ([C]), we may replace C by another n-dimensional free complex C which satises 1 and 2 above and use an chain contraction over (2n+4) Y Y which satises 3 above. Now, for W Y Kn and
i
Kn , we have a sequence
j
W h(Y, pY , n, ) W h(X, pX , n, ) W h(X, Y, pX , n, ) K0 (W, pW , n, ) K0 (X, pX , n, ) , where Kn = 12n + 70. It is easy to verify that the compositions j i , j , and i are 0. Theorem 5.3. Fix an integer n 1. There exists a constant Ln ( 1) which depends only on n such that the followings hold: (1) Suppose Y Y Ln and Ln . Then the stabilization map W h(X, pX , n, ) W h(X, pX , n, ) maps the kernel of j : W h(X, pX , n, ) W h(X, Y, pX , n, ) , into the image of i : W h(Y , pY , n, ) W h(X, pX , n, ) . (2) Suppose Z W Ln and Ln . Then the stabilization map W h(X, Y, pX , n, ) W h(X, Z, pX , n, ) maps the kernel of : W h(X, Y, pX , n, ) K0 (W, pW , n, ) , into the image of j : W h(X, pX , n, ) W h(X, Z, pX , n, ) . (3) Suppose Z W Ln and Ln , and also assume V Z Kn , Kn so that : W h(X, Z, pX , n, ) K0 (V, pV , n, ) is dened. Then the stabilization map K0 (W, pW , n, ) K0 (V, pV , n, ) maps the kernel of i : K0 (W, pW , n, ) K0 (X, pX , n, ) , into the image of . We shall use the following lemma. 36
i
Lemma 5.4. Let C and D be free chain complexes. (1) If two chain maps f, f : C D are chain homotopic, then there is a 2-simple isomorphism between C(f ) and C(f ) (2) There is a 2-simple isomorphism from C(1C : C C) to a free trivial chain complex. Proof : (1) Let h : f is given by: 1 0 ()r h 1 f be a chain homotopy. A desired 2-simple isomorphism
: C(f )r = Dr Cr1 C(f )r = Dr Cr1 .
(2) Dene a free trivial chain complex T = {Tr , dr } by Tr = Cr Cr1 , dr = 0 0 1 0 : Cr Cr1 Cr1 Cr2 .
The required 2-simple isomorphism from C(1C ) to T is given by: ()r 1 ()r1 dC 0 1 : C(1C )r = Cr Cr1 Tr = Cr Cr1 .
Proof of 5.3: We show that Ln = 27000(9n + 34) has the desired properties. (1) Suppose [C] W h(X, pX , n, ) is an element of the kernel of j . By 4.1 there exists an 86 -simple isomorphism f : C DT D T for some n-dimensional f.g. free 86 chain complexes D, D on pY 20 and some ndimensional f.g. free trivial complexes T , T on pX . Let i : D C D T and j : D D T denote the inclusion maps, and q : D T D denote the projection map. The map i is the direct sum of the chain equivalence 0 : 0 C, the 86 chain equivalence 1 : D D, and the 0 chain equivalence 0 : 0 T , and hence is an 86 chain equivalence. Similarly, q is an 86 chain equivalence. Therefore the composite g = qf i : D D is a 386 chain equivalence, and C(g) is 900 contractible. C(g) T is equal to C(jg : D D T ). Since T is 0 contractible, there is a 0 chain homotopy jq 1, and it induces an 86 chain homotopy jg = (jq)f i f i. By 5.4 C(jg) is 2 86 -simple isomorphic to C(f i). The 86 chain map dened by f 1 : C(f i)r = (C D T )r Dr1 (D T )r Dr1 = C(i)r 37
is a 2 86 -simple isomorphism from C(f i) to C(i), because its inverse is a 2 86 chain map. And C(i) is equal to C T C(1D : D D). Finally C(1D ) is 2 86 -simple isomorphic to a trivial chain complex T . By composing C T T 192 =
,
C T C(1D ) = C(i) 192 =
,
C(f i) 192 =
,
C(jg) = C(g) T ,
we get a 600 -simple isomorphism between C T T and C(g) T . C(g) represents an element of W h(Y , pY , n + 1, 2700 ). (C(g) has a strong 2700 chain contraction , and the 5400 homotopy 2 0 takes place over Y .) By 4.3, this element comes from some element [C] W h(Y , pY , n, 2700 ). C and C may not represent the same element in W h(X, pX , n, 2700 ), but they do represent the same element in W h(X, pX , n + 1, 2700 ): ([C]) = ([C]) W h(X, pX , n + 1, 2700 ), and hence we have ([C]) = ([C]) W h(X, pX , (90(n + 1) + 250) 2700 ). Therefore [C] = ([C]) = ([C]) = [C] W h(X, pX , n, 27000(9n + 34) ). (2) Suppose [C] W h(X, Y, pX , n, ) is an element of the kernel of . By denition, [C] = [E, q] K0 (W, pW , n, ) where (E, q) is an n-dimensional f.g. (3n + 12) projective chain complex on pY (4n+14) that is (6n + 15) chain equivalent to (C, 1). Since [E, q] = 0 in K0 (W, pW , n, ), (E, q) is 60 chain equivalent to an n-dimensional f.g. free 30 chain complex (D, 1) on pW by 3.5. By composing these we obtain a 61 chain equivalence f : D C. C(f ) is an (n+1)-dimensional free 61 chain complex on pX and is 183 contractible; hence it is strongly 549 contractible and determines an element of W h(X, pX , n+1, 549 ). By 4.5, C(f ) and C represents the same element in W h(X, W, pX , n + 1, 549 ). By 4.3 there exists an element [C] W h(X, pX , n, 549 ) which maps to [C(f )]. One can use the homomorphisms and as in (1) to show that C and C represent the same element in W h(X, W 549(90n+340) , pX , n, 549(90n + 340) ) and hence in W h(X, Z, pX , n, ). (3) Suppose [E, q] K0 (W, pW , n, ) is an element of the kernel of i . (E, q) is 60 chain equivalent to an n-dimensional f.g. free 30 chain complex (C, 1) on pX . C is 60 contractible over X W 60 , and hence strongly 180 contractible over X W 240 . Therefore C denes an element in W h(X, Z, pX , n, ) and [C] = [E, q] K0 (V, pV , n, ). 38
6. Excision and the Mayer-Vietoris sequence. Throughout this section assume that X = X+ X is the union of two closed subspaces X+ and X with intersection Y = X+ X . The excision isomorphisms of ordinary homology H (X+ , Y ) H (X, X ) = and the Mayer-Vietoris exact sequence . . . Hn (Y ) Hn (X+ ) Hn (X ) Hn (X) Hn1 (Y ) . . . have various algebraic K-theory analogues. In this section we rst discuss the excision maps of controlled Whitehead groups, and then use them to introduce a MayerVietoris sequence in controlled K-theory of the type W h(Y ) W h(X+ ) W h(X ) W h(X) K0 (Y ) K0 (X+ ) K0 (X ) . As in ordinary homology and the bounded K-theory exact sequences of Ranicki [22] the main ingredient is a chain level Mayer-Vietoris decomposition: a free chain complex C on X can be expressed as a sum C = C+ + C of complexes with C dened on a neighbourhood of X in X, and C+ C dened on a neighbourhood of Y in X. If C is contractible then C+ , C , C+ C are nitely dominated, but not in general contractible. Remarks. (1) The assumption of X+ and X being closed ensures that any path connecting a point in X and a point in X+ passes through Y . More precisely, suppose : [0, s] X is a path with (0) X and (s) X+ , and suppose ([0, s]) {(0)} for some . By the connectivity of the interval, there exists a t [0, s] such that (t) Y . Since {(0)} {(t)}2 , ([0, s]) is contained in Y 2 . This argument will be used in place of the relation X = X Y , which is false in general. (This assumption is not essential. Without this, the argument in this section works if we replace sets of the form X V by (X V ) , etc.) (2) If we replace pX : M X, X+ , X by pX 1N : M N X N , X N+ , X N , respectively, and use M -locally nite chain complexes rather than nitely generated chain complexes, then we obtain an analogous result for W hM and K0 , which will be used in the next section. There is an inclusion-induced homomorphism i : W h(X+ , Y, pX+ , n, ) W h(X, X , pX , n, ) . We construct its stable inverse exc : W h(X, X , pX , n, ) W h(X+ , X+ Y (n+300) , pX+ , n, 90 ) . 39
For a chain complex {C, d} representing an element of W h(X, X , pX , n, ), let {C+ , d+ } be any n-dimensional f.g. free 90 chain complex on pX+ such that 1. (C+ )r (X+ Y (n+180) ) = Cr (X+ Y (n+180) ), and 2. d+ = dC over X+ Y (n+270) . (Such a C+ can be constructed by letting (C+ )r = Cr (X+ Y r ), for example.) If is a strong chain contraction of C over X X , then
+ =
0
over X+ Y (n+270) over X+ Y (n+270)
is a strong 90 chain contraction of C+ over X+ Y (n+300) . Thus C+ represents an element of W h(X+ , X+ Y (n+300) , pX+ , n, 90 ). This element is independent of the choice of C+ by 4.5. We claim that this correspondence [C] [C+ ] denes the desired well-dened homomorphism exc. It suces to show that, given 40 -simple isomorphic complexes C and C , one can choose C+ and C+ which are 3600 -simple isomorphic to each other. Let f : C C be the 40 -simple isomorphism and dene C+ by (C+ )r = Cr (X+ Y (r+40) ). Consider the localization g = f[X Y (n+160) ] . As the subspaces X are closed in X, g is geometric over X Y (n+80) and g = f over X+ Y (n+200) . Let C be the chain complex obtained by replacing the boundary map of C with gdg 1 , where d denotes the boundary map of C. C and C are the same (up to 81 homotopy of boundary maps) over X+ Y (n+250) . The 120 -simple isomorphism g : C C (of radius 40 ) restricts to a 40 -simple isomorphism from C+ onto a geometric module subcomplex C+ of C . Then Cr (X+ Y (n+80) ) = (C+ )r (X+ Y (n+80) ), and dC+ = dC 81 dC over X+ Y (n+250) . Therefore C+ and C+ have the desired properties (up to homotopy), and exc is well-dened. The homomorphisms i and exc are stable inverses; i.e., we have commutative diagrams:
W h(X+ , Y, pX+ , n, )
i
W h(X, X , pX , n, )
W h(X+ , X+ Y (n+300) , pX+ , n, 90 )
exc
W h(X, X , pX , n, )
40
W h(X+ , X+ Y (n+300) , pX+ , n, 90 )
exc
W h(X, X , pX , n, )
W h(X+ , X+ Y (n+300) , pX+ , n, 90 )
i
W h(X, X
(n+300)
, pX , n, 90 )
where the vertical arrows are the stabilization maps. Now let n = 90Kn + n + 300 = 1081n + 6600, and let W be any closed subset of X containing Y n and be any number greater than or equal to n . We dene a homomorphism + : W h(X, pX , n, ) K0 (W, pW , n, ) by the composition: W h(X, pX , n, ) W h(X, X , pX , n, ) W h(X+ , X+ Y (n+300) , pX+ , n, 90 ) K0 (X+ W, pX+ W , n, ) K0 (W, pW , n, ) . (Note that W (Y (n+300) )Kn 90 and Kn 90 .) Similarly, dene a homomorphism : W h(X, pX , n, ) K0 (W, pW , n, ) by exchanging the roles of X+ and X . Proposition 6.1. + + = 0 . Proof : + + factors as follows: W h(X, pX , n, ) W h(X, pX , n, 90 ) W h(X, Y (n+300) , pX , n, 90 ) K0 (W, pW , n, ) and the composition of the last two maps is 0. Let us summarize the situation: X = X X+ , Y = X X+ , W Y n , n . Given these data, we have the Mayer-Vietoris sequence:
i i+ exc
W h(Y, pY , n, ) W h(X , pX , n, ) W h(X+ , pX+ , n, ) W h(X, pX , n, ) K0 (W, pW , n, )
i i+ (j j+ ) + =
K0 (X W, pX W , n, ) K0 (X+ W, pX+ W , n, ) where i s and j s are the stabilization maps induced by the inclusion maps. The compositions (j j+ ) i , + (j j+ ) are obviously 0, and i + is also 0 i+ i+ by 6.1. This sequence is stably exact in the following sense. (The ps will denote appropriate restrictions of pX .) 41
Theorem 6.2. Fix an integer n 1. There exists a constant Mn ( 1) which depends only on n such that the followings hold: (1) Suppose Y Y Mn and Mn . Then the stabilization map W h(X , p, n, ) W h(X+ , p, n, ) W h(X Y , p, n, ) W h(X Y , p, n, ) maps the kernel of (j j+ ) : W h(X , p, n, ) W h(X+ , p, n, ) W h(X, pX , n, ) into the image of i i+ : W h(Y , p, n, ) W h(X Y , p, n, ) W h(X+ Y , p, n, ).
(2) Suppose Z W Mn and Mn . Then the stabilization map W h(X, pX , n, ) W h(X, pX , n, ) maps the kernel of + : W h(X, pX , n, ) K0 (W, pW , n, ) into the image of (j j+ ) : W h(X Z, p, n, ) W h(X+ Z, p, n, ) W h(X, pX , n, ). (3) Suppose Mn , and also assume that V W n , n so that the map + : W h(X, pX , n, ) K0 (V, pV , n, ) associated with the two subsets X W is dened. Then the stabilization map K0 (W, pW , n, ) K0 (V, pV , n, ) maps the kernel of i i+ : K0 (W, pW , n, ) K0 (X W, p, n, ) K0 (X+ W, p, n, )
into the image of + . Proof : We show that Mn = max{1200, Ln + L2 , 46(Ln + Kn Ln )} has the desired n properties, where Kn and Ln are as in the previous section. (1) Let ([C ], [C+ ]) W h(X , p, n, )W h(X+ , p, n, ) be an element of the kernel of (j j+ ). By 4.1, there is an 86 -simple isomorphism f : C C+ T T for some trivial complexes T , T . Replacing C and C+ by C T (X ) and C+ T (X X ) respectively, we may assume that f is an 86 -simple isomorphism between C C+ and T . Let f denote the localization of f away from X Y 286 , then f is geometric over X Y 86 and f = f over X+ Y 386 . Replace the boundary map of T by f (dC dC+ )(f )1 . This produces a 200 chain complex E whose boundary 42
map is 172 homotopic to dT over X+ Y 400 . Therefore f denes a 259 -simple isomorphism from C C+ to the direct sum of a 200 chain complex on X Y 800 and a trivial complex. Since f is geometric over X Y 86 and has radius 86 , we can discard the paths in f starting from the basis elements of C to obtain a 259 -simple isomorphism g : C+ D+ T , where D+ is a 200 chain complex on pY 800 and T is the trivial complex T (X+ Y 600 ). D+ is strongly 1200 contractible and denes the same element as C+ in W h(X+ Y 800 , p, n, 1200 ). The 345 -simple isomorphism f (1C g 1 ) : C D+ T C C+ T is homotopic to the identity on T , so we can discard this portion to obtain a 345 -simple isomorphism form C D+ to the trivial complex T (X Y 600 ). Therefore [D+ ] = [C ] in W h(X Y 800 , p, n, 1200 ). Thus ([C ], [C+ ]) W h(X Y , p, n, ) i W h(X+ Y , p, n, ) is the image of [D+ ] W h(Y , p, n, ) by . i+ (2) Suppose [C] W h(X, pX , n, ) is an element of the kernel of + . Let C+ be as in the denition of the excision map. Then [C+ ] in the second row of the following diagram is in the kernel of : [C+ ] W h(X+ , X+ Y (n+300) , p, n, 90 )
K0 (X+ W, p, n, )
[C+ ]
[C+ ] W h(X+ W, Y (n+300) , p, n, 90 )
K0 (W, p, n, )
+ [C] = 0
where the vertical maps are induced by inclusion maps. By 5.3(2), there exists an element [C+ ] W h(X+ W, p, n, ) such that [C+ ] = [C+ ] in W h(X+ W, (X+ W ) W , p, n, ) and hence also in W h(X, W , p, n, ), where = Ln . As = + , there exists an element [C ] W h(X W, p, n, ) such that [C ] = [C ] in W h(X W, (X W ) W , p, n, ), where {C , d } is an n-dimensional free 90 chain complex on pX such that (C )r (X Y (n+180) ) = Cr (X Y (n+180) ), and d = dC over X Y (n+270) . By 4.5, [C] = [C ] + [C+ ] in W h(X, W , p, n, ). Therefore [C] = [C ] + [C+ ] in W h(X, W , p, n, ). Apply 5.3(1) to [C] [C ] + ] W h(X, p, n, ) to obtain an element [D] W h(W +Ln , p, n, Ln ) which [C maps via i to [C] [C ] [C+ ] W h(X, p, n, Ln ). As Z W +Ln and Ln , ] + [D], [C+ ]) denes an element of W h(X Z, p, n, ) W h(X+ Z, p, n, ), ([C and (j j+ ) maps this element to [C] W h(X, p, n, ). (3) Suppose [E, q] K0 (W, pW , n, ) is an element of the kernel of 43 i . By i+
5.3(3), there exist elements [C+ ] W h(X+ W, (X+ W ) W Ln , p, n, Ln ) [C ] W h(X W, (X W ) W Ln , p, n, Ln ) such that [C+ ] = [E, q] K0 ((X+ W ) W , p, n, ) [C ] = [E, q] K0 ((X W ) W , p, n, ) where W = (W Ln )Kn Ln and = (1 + Kn )Ln . By denition, [C+ ] (resp. [C ]) is represented by a projective chain complex (E+ , q+ ) (resp. (E , q )) on pW which is chain equivalent to (C+ , 1) (resp. (C , 1)). By applying 3.1 to [E+ , q+ ] = [E , q ] K0 (W , p, n, ), we obtain free chain complexes F , G on pW such that (E+ , q+ ) (F, 1) 3 (E , q ) (G, 1) . Therefore, there is a 5 chain equivalence f : C G C+ F . C(f ) is an (n + 1)-dimensional strongly 45 contractible chain complex. Apply 4.3 to obtain an n-dimensional strongly 45 contractible chain complex {C(f ), d}. By construc tion C(f )r (X+ W ) = (C+ )r (X+ W ) and dr = dC+ over X+ (W )45 . As (n+180) (n+270) 45 W W and W (W ) , the excision map exc : W h(X, X W, p, n, ) W h(X+ W, (X+ W ) W (n+300) , p, n, 90) used to dene + : W h(X, pX , n, ) K0 (V, p, n, ) maps [C(f )] to [C+ ]. Therefore + maps [C(f )] W h(X, pX , n, ) to [E, q]. 7. Controlled Whitehead group of M S 1 . In this section we establish a controlled analogue of the split exact sequence of Bass [1, XII] for the Whitehead group of Z 0 W h() W h( Z) K0 (Z[]) Nil0 (Z[]) Nil0 (Z[]) 0 with i! induced by the inclusion i : Z. In the controlled analogue there are no Nil-terms, and the sequence is only stably exact. Geometrically, B sends the torsion (f ) W h( Z) of a homotopy equivalence f : M X S 1 between a compact manifold M and the product of a nite Poincar complex X and S 1 to the e 44
i! BN+ N
Siebenmann end obstruction of one of the two ends M = f 1 (R ) of the innite cyclic cover M = f (X R) of M B (f ) = [M + ] = [M ] K0 (Z[]) . B is split by the injection B : K0 (Z[]) W h( Z) ; [P ] (z : P [z, z 1 ]P [z, z 1 ]) . In the terminology of Ranicki [21] this is the algebraically signicant injection of K0 (Z[]) in W h( Z), to be distinguished from the geometrically signicant injection B : K0 (Z[]) W h( Z) ; [P ] (z : P [z, z 1 ] P [z, z 1 ]) with image the subgroup of transfer invariant elements of W h( Z). Theorem 7.1. Let pX : M X be a control map. For any n > 0, > 0 and 18, there is a commutative diagram B W h(X, p , n, 18) K (X, p , n, )
X 0 X
W h(X, pX , n, )
B
K0 (X, pX , n, n )
where pX denotes the following control map: pX : M S 1 M X , the vertical maps are stabilization maps, and n = 1081n + 6600. n above is the constant which was used when we dened the connecting homomorphism + for the Mayer-Vietoris sequence in the previous section. To prove 7.1, it will be useful to consider a control map of the following form: pX 1 : M X , where R or S 1 . We shall consider S 1 as the quotient R/Z and use the metric induced from that of R. The projection map R R/Z = S 1 will be denoted by . We shall always use the maximum metric for a product of metric spaces. The hypothesis of the following lemma is satised by any simplex in euclidean space Rk , or a Hilbert cube I . In the application in this section, will be an interval [s, s] R. 45
projection pX
Lemma 7.2. Let be a compact metric space and assume that there is a strong deformation retraction {rt }0t1 of to a point v such that d(rt (x), rt (y)) d(x, y) for all x, y and t [0, 1]. Suppose that pX : N X is a control map such that there is a strong deformation retraction {Rt } of N to p1 (X {v}) X which covers the strong deformation retraction {Rt = 1X rt } of X to X {v}, and let pX = pX | : (pX )1 (X {v}) X {v} = X . Then there are isomorphisms: K0 (X , pX , n, ) K0 (X, pX , n, ) = W h(X , pX , n + 1, ) W h(X, pX , n + 1, ) = for every n 0 and > 0.
Proof : We consider the K0 case. The isomorphism is given by: (R1 , R1 ) : K0 (X , pX , n, ) K0 (X, pX , n, ) with the inverse i induced by the inclusion i : X X . The composition (R1 , R1 ) i is obviously the identity map. To prove that i (R1 , R1 ) is the identity, we need to show the equivalence of (E, q) and (R1 ) (E, q) for every element [E, q] K0 (X , pX , n, ), but this is obvious because there exists a sequence 0 = t0 < t1 < < tm = 1 such that (Rti ) (E, q) and (Rti+1 ) (E, q) are isomorphic for each i = 0, , m 1. The isomorphisms are given by tracks of {Rt }ti tti+1 . The proof for W h is similar and is omitted. Proof of 7.1 : We dene the homomorphism B : W h(X, pX , n, ) K0 (X, pX , n, n ) for every n > 0 and > 0. Let C be an n-dimensional strongly contractible f.g. free chain complex on pX : M S 1 X. Let C denote the pullback of C via the map 1 1M : M R M S . C is not nitely generated, but is M -locally nite in the sense of 3. C is strongly contractible measured in X, but not necessarily so when measured via pX 1R : M R X R. Let K be a positive number and consider the linear map K : R R dened by K (x) = x/K. If K is suciently large, then K (C) is an M -locally nite n-dimensional strongly contractible free chain complex on pX 1R , thus it represents an element in W hM (X R, pX 1R , n, ). We dene B([C]) to be the image of this element by the composition:
M W hM (X R, pX 1R , n, ) K0 (X J, pX 1J , n, n ) +
= K0 (X J, pX 1J , n, n ) K0 (X, pX , n, n ) , 46
=
where + is the connecting homomorphism in the Mayer-Vietoris sequence for the triad X (R; (, 0], [0, )) and J is some interval [s, s], and the last map is the isomorphism of 7.2 induced by the retraction M J M . Because of this retraction at the end, the image of [K (C)] in K0 (X, pX , n, n ) is independent of the choice of K used for shrinking for a given C. Suppose [C] = [C ] in W h(X, pX , n, ). If we use a suciently large K, then [K (C)] = [K (C )] in W hM (X R, pX 1R , n, ). Therefore B is well-dened. It is obviously a homomorphism. We shall give an alternative description of B later. Next we dene B : K0 (X, pX , n, ) W h(X, pX , n, 18) for every n > 0 and > 0. Let (A, p) be a projective module on pX , and consider a geometric module D = Z[{P }] on S 1 generated by P = (0) S 1 . Dene a path t : [0, 1] R by t() = (0 1), and let z denote the path (P, t : [0, 1] S 1 , P ) from P to P . Dene a homomorphism B0 : K0 (X, pX , ) W h(X, pX , 2) ; [A, p] [fp = (1 p) 1 + p z : A D A D] . Tensor products of geometric modules and tensor products of geometric morphisms are dened in Yamasaki [26]. For the convenience of the reader, we give a brief review. Let Z[R] and Z[S] be geometric modules on M and N respectively. Their tensor product Z[R] Z[S] is dened to be Z[R S : |R| |S| M N ]. For r = (|r|, [r]) R and s = (|s|, [s]) S, r s will denote the element ((|r|, |s|), ([r], [s])) of R S. If (r, : [0, ] M, r ) is a path from r R to r R and (s, : [0, ] N, s ) is a path from s S to s S , then their tensor product (r, , r ) (s, , s ) is the path (r s, , r s ), where : [0, + ] M N is the following composite path: (x) = ((x), (0)) if 0 x , (( ), (x )) if x + .
Tensor products of geometric morphisms are dened by bilinearly extending this. In general we have a homotopy (f g )(f g) f f g g instead of a strict equality. Let us go back to the denition of B0 . It is easy to check that fp is a isomorphism measured in X; its inverse is given by (1 p) 1 + p z 1 . If we add a free module (E, 1E ) to (A, p), then fp1E = fp (1E z) represent the same class as fp . Next suppose that g : (A, p) (A , p ) is a isomorphism of projective modules, with inverse g 1 . Dene a isomorphism F : (A D) (A D) (A D) (A D) by (1 p) 1 g 1 1 F = (F 2 2 1) , g1 (1 p ) 1 47
then (1 fp )F 5 F (fp 1) . Now by 4.8, [fp ] = [fp ] in W h(X, pX , 2) and hence B0 is well-dened. The desired B is dened by the composition: K0 (X, pX , n, ) K0 (X, pX , 9) W h(X, pX , 18) W h(X, pX , n, 18) . The commutativity of the diagram of 7.1 is easily veried. We rewrite 6.2(2) using 7.1. Let pX , X+ , X , Y be as in 6. For a given > 0, let W be a closed subspace of X containing Y n and be any number 18n . Let pW be the composition: p1 (W ) S 1 p1 (W ) W . X X Dene + : W h(X, pX , n, ) W h(W, pW , n, ) by the following composition: W h(X, pX , n, ) K0 (W, pW , n, /18) W h(W, pW , n, ) . The following composition is 0: W h(X , pX , n, ) W h(X+ , pX+ , n, ) W h(X, pX , n, ) W h(W, pW , n, ) . Furthermore we have: Corollary 7.3. Fix an integer n 1. There exists a constant Mn ( 1) which depends only on n such that, if Z W Mn n and Mn n , the stabilization map W h(X, pX , n, ) W h(X, pX , n, ) maps the kernel of + : W h(X, pX , n, ) W h(W, pW , n, ) into the image of (j j+ ) : W h(X Z, p, n, ) W h(X+ Z, p, n, ) W h(X, pX , n, ) . Proof : Immediate from 6.2 and 7.1. The same constant Mn as in 6.2 can be used. This will be used in the next section for a stable vanishing result for controlled Whitehead torsion. 48
+ (j j+ ) + B projection pX B0
Our next aim in this section is to study W h(X S 1 , pX 1S 1 , n, ). Dene 1 1 1 1 subspaces S+ , S ( S 1 ) by S+ = ([0, 1/2]), S = ([1/2, 0]) and let P = (0) (as before), Q = (1/2), N = (1/4), S = (1/4). When is suciently small (n < 1/8), one can use 7.2 to rewrite the Mayer-Vietoris sequence for the triad 1 1 X (S 1 ; S , S+ ) as follows: W h(X, pX , n, ) W h(X, pX , n, ) W h(X, pX , n, ) W h(X, pX , n, ) W h(XS 1 , pX 1, n, ) K0 (X{P }, pX , n, n )K0 (X{Q}, pX , n, n ) K0 (X = X {S}, pX , n, n ) K0 (X = X {N }, pX , n, n ) , where i : X = X {S} X S 1 i+ : X = X {N } X S 1 are inclusion maps, and J = 1 1 1 1 , J = 1 1 1 1 .
J (i i+ ) + J
If we further assume that Mn n < 1/8, then this sequence is stably exact. Dene B : W h(X S 1 , pX 1, n, ) K0 (X, pX , n, n ) by composing + with the projection onto the rst direct summand, and consider 0 W h(X, pX , n, ) W h(X S 1 , pX 1, n, ) K0 (X, pX , n, n ) 0. The composition B i+ is zero. The map i+ is injective: the projection pM : M S 1 M induces the left inverse of i+ . Let = Mn n . If n < 1/8, then from the stable exactness of the Mayer-Vietoris sequence above one can deduce that this sequence is also stably exact: (1) the stabilization image of kerB in W h(X S 1 , pX 1, n, ) is contained in the image of i+ : W h(X, pX , n, ) W h(X S 1 , pX 1, n, ) , (2) the stabilization image of K0 (X, pX , n, n ) in K0 (X, pX , n, n ) is contained in the image of B : W h(X S 1 , pX 1, n, ) K0 (X, pX , n, n ) . By the remark preceding 5.3, the following diagram commutes. 49
i+ B
W h(X S 1 , pX 1, n, ) F W h(X, pX , n, ) B
B
K0 (X, pX , n, n )
K0 (X, pX , n, n ),
where F denotes the forget-control-in-S 1 map induced by = (1 : M S 1 M S 1 , projection : X S 1 X). Let [C] W h(X, pX , n, ). For a positive integer k, let C k denote the pullback of C via the k-fold covering M S 1 M S 1 ; (m, ()) (m, (k)) . If k is suciently large, then C k represents an element of W h(X S 1 , pX 1, n, ). Again by the remark preceding 5.3, we have the equality : B([C]) = B ([C k ]). This is the alternative description of B mentioned before. Furthermore, we can use pullback to construct a stable right inverse of B of B . For an integer k 1/, dene: B 0,k : K0 (X, pX , ) W h(X S 1 , pX 1, 8) ; [A, p] [(fp )k ] . Here (fp = (1 p) 1 + p z) is regarded as a 1-dimensional chain complex. If we dene B k by: B k = B 0,k : K0 (X, pX , n, ) W h(X S 1 , pX 1, n, 72) then B B k is equal to the stabilization map. Therefore W h(X S 1 , pX 1, n, ) is stably a direct sum of W h(X, pX , n, ) and K0 (X, pX , n, ). This stable splitting does depend on the integer k. But stably it depends only on k mod 2. Suppose l > k 1/. Stretch a portion of (fp )l along an arc S 1 to match with (fp )k over X for some subarc and then use 5.3(1) to conclude that (fp )l (fp )k lies in the image of W h(X, pX , n, 72Ln ), where Ln is the constant given in 5.3. But this element must be zero, because 1p 0 0 ... p 1p 0 ... 0 p l p 1p ... 0 (pM ) [(fp ) ] = 0 . . . . .. . . . . . . . . . 0 0 0 ... 1p is equal to (pM ) [(fp )k ] if l k (mod 2). Therefore B k : K0 (X, pX , n, ) W h(X S 1 , pX 1, n, 72Ln ) depends only on k mod 2. (If we use the geometrically signicant gp = (1p)1pz of Ranicki [21] instead of fp , then B k is independent of k( 0).) 50
8. The eventual Vietoris theorem. A version of the Vietoris theorem appropriate to a non-connective generalized homology theory h states that if a map p : M K of reasonable spaces (such as polyhedra) has h -acyclic point inverses in dimensions 1 hk (p1 (v){v}) = 0 (v K , k 1) then p is an h -isomorphism in dimensions 1 hk (p) = 0 (k 1) . There is an eventual Vietoris theorem for controlled torsion: if a reasonable control map p : M K is such that W hk (1 (p1 (v))) = 0 (v K , k 1) then for every > 0, n > 0 there exists a > 0 such that the stabilization maps W hk (K, p, n, ) W hk (K, p, n, ) (k 1) are zero. See the appendix. In fact, we shall avoid the overt use of the condition W hk (1 (p1 (v))) = 0 involving the lower W h-groups W hk (k 0) by using the controlled version in 7 of the Bass-Heller-Swan splitting, crossing with the k-tori T k = (S 1 )k and using the stronger hypothesis W h(1 (p1 (v)) Zk ) = 0 for all v K, k 0. As an application we study the forget-control assembly maps. For any control map pX : M X and any > 0, there is a forget-control map: W h(X, pX , n, ) W h(X, pX , n, +) , as an extreme of stabilization maps. If M is connected and locally 1-connected, then the assembly map gives an isomorphism W h(X, pX , n, +) W h(1 (X)). The composite of these is the forget-control assembly map. Forget-control assembly maps for K0 are also dened similarly. The Vietoris theorem for controlled torsion implies that, if we further assume that X is a connected compact metric ANR and n 0, then there exists a > 0 such that the image of the forget-control assembly map W h(X, pX , n, ) W h(1 (M )) 51
=
is contained in the kernel of (pX ) : W h(1 (M )) W h(1 (X)) . Similarly, for n 0, there exists a > 0 such that the image of the forget-control assembly map K0 (X, pX , n, ) K0 (Z[1 (M )]) is contained in the kernel of (pX ) : K0 (Z[1 (M )]) K0 (Z[1 (X)]) . These results were originally obtained by Chapman and Ferry, using more geometric methods. Let K be a nite polyhedron, and suppose that the control map pK : M K has an iterated mapping cylinder structure (Hatcher [12]), and that W h(1 (p1 (v)) Zk ) = 0 (k 0) K for every vertex v K. For each k 0, let pK denote the composition: pK
(k) (k) (k)
: M T k M K .
projection
pK
Then pK also has an iterated mapping cylinder structure induced from that of pK and satises the same Whitehead group condition. Theorem 8.1. Let pK be as above. For any n > 0 and such that the stabilization map
(k) (k)
> 0, there exists a > 0
W h(K, pK , n, ) W h(K, pK , n, ) is the zero map for every k 0. Proof : In the following proof, we do not distinguish a simplicial complex from its underlying polyhedron. For a simplicial complex L, (L) will denote the number of simplices in L. Fix pK and n > 0. We inductively show that there exists a sequence ( ) 1 ( ) 2 ( ) 3 ( ) (> 0) of positive functions such that if L is a subcomplex of K with (L) l, then the stabilization maps W h(L, pL , n, l ( )) W h(L, pL , n, ) 52
(k) (k)
are 0 for all k 0 and all > 0. Here pL is the restriction of pK to L. The theorem is a special case of this. When l = 1, i.e., L is a point {v}, 1 ( ) = works, because W h({v}, p{v} , n, ) = W h(1 (p1 (v) T k )) = 0 K for every > 0. Assume we have constructed 1 , . . . , l1 . Let L be a subcomplex of K with (L) l. Let be a simplex of L that is not a face of any other simplex of L. Then L is the union of L+ = and L = L interior() with intersection L0 = . Since (L0 ) < l and (L ) < l, the stabilization maps W h(L0 , pL0 , n, l1 ( )) W h(L0 , pL0 , n, ) W h(L , pL , n, l1 ( )) W h(L , pL , n, ) are 0 for all > 0 and k 0 by induction hypothesis. Note that this is also true for L+ , because (k) (k) W h(L+ , pL+ , n, ) W h({v}, p{v} , n, ) = 0 = for all 0 and k 0, by 7.2. Now x > 0. Let N denote a regular neighbourhood of L0 in L. Here and in the rest of the proof, a regular neighbourhood of a subcomplex means a star neighbourhood of some iterated barycentric subdivision of the original simplicial structure. This is to ensure that there exists a strong deformation retraction of the regular neighbourhood of the subcomplex which can be covered by a strong deformation retraction of the preimage by pK . Thus one can choose a strong deformation retraction {rt }0t1 of L N to L so that it is covered by a strong deformation retrac tion {t } of p1 (L N ) to p1 (L ). This induces a strong deformation retraction r K K (k) (k) (k) {t = rt 1T k } of (pK )1 (L N ) to (pK )1 (L ). Unlike 7.2, rt may increase the r distance. But, by the compactness of L+ N , there exists a positive number ( ) ( l1 ( )) which makes the following diagram commute for all k 0 W h(L N , p(k) , n, ( )) (1 , r1 ) r W h(L , p(k) , n, l1 ( ))
(k) (k) (k) (k) (k) (k) (k)
(k)
(k)
W h(L N , p(k) , n, ) i W h(L , p(k) , n, )
Since the second row is the zero map, the top row is also the zero map for all k 0. Similarly, there exists a positive number + ( ) such that W h(L+ N , p(k) , n, + ( )) W h(L+ N , p(k) , n, ) 53
is the zero map for all k 0. Let ( ) = min{ + ( ), ( )}, and choose a positive number suciently small so that 1. Mn n ( ), and 2. there exists a smaller regular neighbourhood N of L0 in L such that N Mn n N , where Mn is the constant given in 7.3 and n is the constant dened in 6. As in the case of L N , there exists a positive function 0 () such that W h(N, p(k) , n, 0 ()) W h(N, p(k) , n, ) is the zero map for every > 0 and k 0. Now choose L ( ) > 0 suciently small so that 1. L ( ) < 0 ()/18n , and 2. N L0 n , where is as above. Consider the following commutative diagram. W h(L, pL , n, L ( ))
(k) L ( )
+
W h(N, pN
(k+1)
, n, 0 ())
0 W h(L, pL , n, L ( ))
(k)
+
W h(N, pN
(k+1)
, n, )
W h(L N , p(k) , n, ( )) W h(L+ N , p(k) , n, ( )) 0 W h(L N , p(k) , n, ) W h(L+ N , p(k) , n, ) A simple diagram chase shows that the stabilization map
(k) W h(L, pL , n, ( ))
W h(L, pL , n, )
(k)
W h(L, pL , n, L ( )) W h(L, pL , n, ) is the zero map for all k 0. Since there are only nitely many subcomplexes L with (L) l, we can dene l ( ) to be min{ L ( )| (L) l}. This completes the inductive step and the theorem is proved. Corollary 8.2. Let pK be as above. For any n 0 and > 0, there exists a > 0 such that (k) (k) K0 (K, pK , n, ) K0 (K, pK , n, ) 54
(k)
(k)
is the zero map for every k 0. Proof : When n > 0, this follows immediately from 8.1 and 7.1. The n = 0 case follows from the n = 1 case. The following is an algebraic version of Ferry [10, Cor.3.2]: Corollary 8.3. Let X be a connected compact metric ANR embedded in the Hilbert cube I . For any n 0 and > 0, there exists a > 0 such that the stabilization maps K0 (X, 1X , n, ) K0 (X, 1X , n, ) , W h(X, 1X , n + 1, ) W h(X, 1X , n + 1, ) are both zero. Consequently, there exists a X,n > 0 such that the forget-control assembly maps K0 (X, 1X , n, X,n ) K0 (Z[1 (X)]) , W h(X, 1X , n + 1, X,n ) W h(1 (X)) are both zero. Proof : X has a neighbourhood U with a retraction r : U X. We may assume that U is of the form K I N , where K is a codimension 0 P L submanifold of I N . Let m = n + 1. By the compactness of U there is a > 0 such that (r, r) : 1U 1X induces a homomorphism (r, r) : W h(U, 1U , m, ) W h(X, 1X , m, ) . Since W h(Zk ) = 0 (Bass-Heller-Swan [2]), we can apply 8.1 to 1K : K K; there (k) (k) exists a > 0 such that the homomorphism W h(K, 1K , m, ) W h(K, 1K , m, ) is the zero map for every k 0. Let r : U = K I N K denote the projection and i : K = K (0, 0, . . .) U denote the inclusion map. These induce isomorphisms in W h which are inverses of each other by 7.2. The following diagram commutes: W h(X, 1X , m, )
(k)
i
W h(U, 1U , m, )
(k)
(r , r ) (k) W h(K, 1K , m, ) = 0 = i
W h(X, 1X , m, )
(k)
(r, r)
W h(U, 1U , m, )
(k)
W h(K, 1K , m, )
(k)
where the vertical maps are stabilization maps. Therefore, all the vertical maps are zero maps. Let = 1, k = 0 and let X,n be the corresponding . Since the forget-control map W h(X, 1X , n + 1, X ) W h(1 (X)) factors through W h(X, 1X , n + 1, 1), it is the zero map. The claim for K0 (with a smaller X,n ) follows from the k = 1 case and 7.1. 55
The following is an algebraic version of Chapman [5, Theorem 1 ]: Corollary 8.4. Suppose pX : M X is a control map of a connected locally 1connected space M to a connected compact metric ANR X embedded in I . For any n 0, there exists a > 0 such that the images of the forget-control assembly maps W h(X, pX , n + 1, ) W h(1 (M )) K0 (X, pX , n, ) K0 (Z[1 (M )]) are contained in the kernels of (pX ) : W h(1 (M )) W h(1 (X)) (pX ) : K0 (Z[1 (M )]) K0 (Z[1 (X)]) respectively. Proof : Let X,n be as in 8.3. The claim for Whitehead groups is immediate from the following commutative diagram. (pX , 1X ) W h(X, pX , n + 1, X,n ) W h(X, 1X , n + 1, X,n ) 0 W h(1 (M )) The K0 case is similar. 9. Controlled niteness obstruction and torsion. We shall now use the theory of transverse CW complexes to dene controlled niteness obstruction and torsion using the algebraically dened value groups of 3 and 4. Previously, Chapman [6, 5,7] had dened controlled niteness obstruction and torsion using geometrically dened value groups. The geometric invariants determine the algebraic invariants - we shall not need this, and for our purposes it suces to consider only the algebraic ones, since these assemble to the uncontrolled niteness obstruction and torsion respectively. Let K be a CW complex. K (k) will denote its k-skeleton. A map f : (M k , M ) (K (k) , K (k1) ) from a smooth k-dimensional manifold (possibly with boundary) is said to be transverse to the k-cells if for each open k-cell ek of K, f 1 (ek ) is a disjoint k union of the interiors of nitely many closed k-balls Bi in M such that there exists a k k homeomorphism i : Bi D k to the k-ball D k with ek i = f |Bi for each i. Here k k ek ; D K denotes the characteristic map for the closed k-cell e . Any continuous 56 (pX ) W h(1 (X))
map f : (M k , M ) (K (k) , K (k1) ) is homotopic rel to one that is transverse to the k-cells. A CW complex K is transverse if the attaching maps : S k K (k) of the (k + 1)-cells are all transverse to the k-cells for every k. Any nite CW complex is simple homotopy equivalent to a transverse CW complex. A subdivision (Milnor [15]) of a transverse CW complex is also transverse. A map f : K L between transverse CW complexes is t-cellular if it is cellular and for each cell ek of K, the composition (D k , S k1 ) (K (k) , K (k1) ) (L(k) , L(k1) ) is transverse to the k-cells, where is the characteristic map for ek . Any map can be homotoped to a t-cellular map. A t-cellular map f : K L induces a chain map f% : f C(K) C(L). Here C() denotes the geometric cellular chain complex dened by Quinn [18] and f () denotes the geometric module chain complex obtained by applying f to the modules and morphisms. f Ck (K) is generated by the images of the centers of k-cells of K in L. For each k-cell of K, consider the characteristic map : D k K and take the radial paths in D k starting at the center of D k and ending at the preimages by f of the centers of the k-cells of L. A path is assigned a + sign (resp. a sign) if f is orientation-preserving (resp. orientation-reversing) about its endpoint. The chain map f% is dened by the sum of the images of these paths in L with assigned sign. To see that this actually denes a chain map, let Ui be the preimage of the open k-cells of L via f : (D k , S k1 ) (L(k) , L(k1) ). Make f : D K Ui L(k1) transverse to the centers {v } of the (k 1)-cells of L keeping the boundary xed. Then the preimage of {v } in D k Ui is the disjoint union of circles and arcs. Using these arcs, we can make the paths in f% d df% into pairs of opposite sign so that the paired paths are homotopic in L rel . If two t-cellular maps f , g are homotopic, then there is a diagram f C(K) f% = g% g C(K) that commutes up to chain homotopy. Here the vertical map is the geometric isomorphism given by the homotopy. 57 C(L)
f
If f : K L and g : L M are both t-cellular, then so is the composition gf : K M , and (gf )% g% g (f% ). We shall now dene the controlled torsion of a controlled homotopy equivalence. See Chapman [6, p.2] for the terminology. Let K and L be n-dimensional transverse nite CW complexes, such that L is equipped with a control map pX : L X to a metric space X. The torsion of a p1 ( )X equivalence f : K L will be dened to be an element (f ) W h(X, pX , n+1, 360 ). Let g : L K be a p1 ( )-homotopy inverse of f . Subdivide K and L if necessary, X and assume that the diameter of the image in X of each cell of K and L via pX f and pX is smaller than /10. Transverse CW complexes are saturated in the sense of Quinn [18]. Therefore f is p1 ( /10)-homotopic to a t-cellular map f : K L, X and g is (pX f )1 ( /10)-homotopic to a t-cellular map g : L K. Then f g is t-cellularly p1 (2 )-homotopic to 1L , and g f is t-cellularly (pX f )1 (4 )-homotopic X to 1K . f C(K) and C(L) are both free chain complexes. Consider the induced chain map F = f% : f C(K) C(L) and the composite 7 chain map G : C(L) f g C(L) f C(K) , where the rst map is the geometric 2 isomorphism induced by the homotopy 1 f g . Then F G is 9 chain homotopic to 1 : C(L) C(L). This 9 comes from the size estimate of the trace of each cell of L by the 2 -homotopy pX pX f g . 1 Similarly the (pX f ) (4 )-homotopy gives a 17 chain homotopy GF 1, where : f C(K) f C(K) is the geometric 6 isomorphism induced by the p1 (6 )X homotopy f = f 1L f (g f ) = (f g )f 1L f = f . This homotopy induces a 13 chain homotopy f% f% . Thus GF 20 GF , and GF 37 1. F is a 40 chain equivalence, and its torsion is dened in W h(X, pX , n + 1, 360 ). This class is independent of the choice of f . The forget-control assembly image of this class in W h(1 (X)) is the ordinary Whitehead torsion (f ). Next, we dene the controlled niteness obstruction of a controlled dominated space. Let K and M be n-dimensional transverse CW complexes, and let K M be a p1 ( )-domination of M with respect to a control map pX : M X. Assume that X K is nite, then we may assume that the image of each cell of K by the map pX d has diameter , by subdividing K if necessary. Also assume that the CW decomposition of M is suciently ne so that the image of each cell of M by pX has diameter . 58
u d = f (g% )
Then C(M ) is dominated by d C(K) for some > 0, and hence by 5.1 (C(M ), 1) is (2n + 5) chain equivalent to an n-dimensional (n + 4) projective chain complex. (Alternatively, apply the instant niteness obstruction formula of Lck and Ranicki u [14] to the controlled chain homotopy idempotent induced by ud (ud)2 : K K. If we take this approach, then the assumption above on the CW structure on M is unnecessary.) The reduced projective class of this complex is the controlled niteness obstruction [M ] K0 (X, pX , n, (4n + 10)). The forget-control assembly image in K0 (Z[1 (M )]) is the ordinary Wall niteness obstruction [M ]. 10. The topological invariance and niteness theorems. We shall now use the Vietoris-type theorem of 8 and the controlled torsion and niteness obstruction of 9 to prove that the torsion of a homeomorphism is zero, and that the niteness obstruction of a compact ANR is zero. Theorem 10.1. (Topological Invariance of Torsion) A homeomorphism between nite CW complexes is simple. Theorem 10.2. (Borsuk Conjecture) A compact metric ANR is homotopy equivalent to a nite polyhedron. These were originally proved by Chapman [3] and West [24], respectively. Actually, for these applications the controlled algebra of [16], [17] suces, with geometric morphisms dened without using paths. Proof of 10.1: Let f : K L be a homeomorphism between nite CW complexes. We shall show that the Whitehead torsion (f ) is 0. Without loss of generality, we may assume that L is embedded in the Hilbert cube and that K and L are transverse. Since the Whitehead torsion is combinatorially invariant (Whitehead [25], Milnor [15], Cohen [7]), we may replace K and L by their subdivisions K and L respectively. Approximate f by a t-cellular map f . As in 9 for some > 0, C(f% ) denes an element in W h(L , 1L , ) = W h(L, 1L , ) whose image in W h(1 L) via the forgetcontrol assembly map is the torsion (f ). One can make arbitrarily small by choosing ne subdivisions and a close approximation f . Therefore, (f ) is 0 by 8.3. Proof of 10.2: Without loss of generality, we may assume that X is a subspace of the Hilbert cube. X has a neighbourhood V with a retraction r : V X. If N is suciently large, we can nd a smaller neighbourhood U V of the form K I N , where K is a codimension 0 P L submanifold of I N . Let j : X K denote the composition: j : X U K , 59
inclusion projection
and f : K X denote the composition: f : K = K0 U X . Then f : K X is a nite domination of X: there is a homotopy kt : 1X f j. Let p : K K be a t-cellular approximation of jf and let ht : jf p be a homotopy. Dene p : f C(K) f C(K) by the composition f C(K) f p C(K) f C(K) where the rst map is the geometric isomorphism induced by the homotopy Ht : f
kt f = f (p% ) inclusion r
f jf
f ht
fp .
There is a homotopy Kt : p2 jf jf jf p, and there is induced a chain homotopy between p2 and p , where : f C(K) f C(K) is the geometric isomorphism induced by the homotopy: f fp
Ht p
f pp
f Kt
fp
f .
If N is very large (i.e., I N is very thin), then the homotopy kt is very small. Also the homotopy ht can be assumed to be arbitrarily small. As X is locally contractible, is homotopic () to the identity for suciently large N . Thus we may assume that p is a chain homotopy projection. As in 9 this situation determines an element of K0 (X, 1X , ) for some > 0. Its image in K0 (Z[1 (X)]) via the forget-control assembly map is the ordinary Wall niteness obstruction of X. Since one can make arbitrarily small, the niteness obstruction of X must vanish, by 8.3. Appendix. Controlled lower K-theory. The stably exact sequences in 5 and 6 can be extended to the right by introducing controlled lower K-groups. Denition. For a control map pX : M X and an integer i 0, dene
M Ki (X, pX , n, ) = K0 (X Ri , pX 1Ri , n, ) (n 0)
W h1i (X, Y, pX , n, ) = W hM (X Ri , Y Ri , pX 1Ri , n, ) (n > 0) , using M -locally nite chain complexes (3,4). When i = 0, these are equal to the original controlled K0 - and W h-groups. As in 7, we use the maximum metric for product metric spaces (including Ri ). 60
The M -locally nite version of Mayer-Vietoris sequence for the triad X (R i+1 ; R (, 0], Ri [0, )) reduces to the stable isomorphism :
i
0 W hM (X Ri+1 , pX 1, n, ) K0 (X Ri J, pX 1 1, n, n ) 0 M where n = 1081n + 6600 (as in 6) and J is some interval [s, s]. This is because the terms concerning half innite intervals vanish (Eilenberg swindle). For example, let [D] be an element of W hM (X Ri [0, ), pX 1 1, n, ), D be a complex representing the additive inverse of [D], and t be the translation of M Ri [0, ) Ri by in the positive direction of [0, ), then D t D t2 D t3 D is M -locally nite and represents [D] as well as 0. By 7.2, the projection M Ri J M Ri induces an isomorphism
M K0 (X Ri J, pX 1 1, n, n ) Ki (X, pX , n, n ). =
+
Thus there is a stable isomorphism : W hi (X, pX , n, ) Ki (X, pX , n, n ) with an inverse : Ki (X, pX , n, ) W hi (X, pX , n, n )
2 where n = Mn 2 . (Note that we have already encountered the map with i = 0 in n 7.) This observation permits us to extend the sequences of 5 and 6 to the right as follows :
W h1i (Y, pY , n, ) W h1i (X, pX , n, ) W h1i (X, Y, pX , n, ) W hi (W, pW , n, ) W hi (X, pX , n, ) (i 0, n > 0,
+
> 0, W Y Kn ,
K n n )
W h1i (X0 , p, n, ) W h1i (X , p, n, ) W h1i (X+ , p, n, ) W h1i (X, p, n, ) W hi (W, p, n, ) W hi (X W, p, n, ) W hi (X+ W, p, n, ) (i 0, n > 0, > 0, W Y n , n n ) .
These are stably exact, but the details will be omitted. 61
When X is a point {} and M is connected and locally 1-connected, Ki ({}, M {}, n, ) and W h1i ({}, M {}, n, ) are isomorphic to the ordinary reduced lower K-group Ki (Z[1 (M )]). We can use these controlled lower K-groups to do the stable calculation of 8. Let pK : M K be as in 8.1, a control map of M to a compact polyhedron K with an iterated mapping cylinder structure such that W h1i (1 (p1 (v))) = 0 (v K, i 0) . K Theorem A1. For any n > 0 and > 0, there exists a > 0 such that the stabilization map W h1i (K, pK , n, ) W h1i (K, pK , n, ) is zero for every i 0. See Ranicki [22] for an algebraic treatment of lower K-theory using the bounded algebra of Pedersen and Weibel. References [1] Bass, H. Algebraic K-theory Benjamin (1968) [2] , Heller, A. , and Swan, R. G. The Whitehead group of a polynomial extension Publ. Math. I. H. E. S. 22, 6179 (1964) [3] Chapman, T. A. The topological invariance of Whitehead torsion Amer. J. Math. 96, 488497 (1974) [4] Homotopy conditions which detect simple homotopy equivalences Pac. J. Math. 80, 1346 (1979) [5] Invariance of torsion and the Borsuk conjecture Can. J. Math. 32, 13331341 (1980) [6] Controlled Simple Homotopy Theory and Applications Springer Lecture Notes, vol. 1009 (1983) [7] Cohen, M. M. A Course in Simple-Homotopy Theory Graduate Texts in Math. , vol. 10, Springer (1973) [8] Connell, E. H. and Hollingsworth, J. Geometric groups and Whitehead torsion 62
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[21]
Trans. A. M. S. 140, 161181 (1969) Connolly, F. and Koniewski, T. z Rigidity and crystallographic groups I. Invent. math. 99, 2548 (1990) Ferry, S. The homeomorphism group of a compact Hilbert cube manifold is an ANR Ann. of Math. 106, 101119 (1977) Ferry, S. , Hambleton, I. and Pedersen, E. A survey of bounded surgery theory and applications Proc. MSRI Algebraic Topology Prog. 1989-1990 (to appear) Hatcher, A. E. Higher simple homotopy theory Ann. of Math. 102, 101137 (1975) Higman, G. The units of group rings Proc. London Math. Soc. (2) 46, 231-248 (1940) Lck, W. and Ranicki, A. u Chain homotopy projections J. of Alg. 120, 361391 (1989) Milnor, J. Whitehead torsion Bull. A. M. S. 72, 358426 (1966) Quinn, F. S. Ends of Maps I. Ann. of Math. 110, 275331 (1979) Ends of Maps II. Invent. Math. 68, 353424 (1982) Geometric algebra Proc. 1983 Rutgers Topology Conference, Springer Lecture Notes, vol. 1126, 182198 (1985) Ranicki, A. The algebraic theory of niteness obstruction Math. Scand. 57, 105126 (1985) The algebraic theory of torsion I. Proc. 1983 Rutgers Topology Conference, Springer Lecture Notes, vol. 1126, 199237 (1985) Algebraic and geometric splittings of the K- and L-groups of polynomial 63
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extensions Proc. Symp. on Transformation Groups, Pozna 1985, Springer Lecture n Notes, vol. 1217, 321364 (1986) Lower K- and L-theory L. M. S. Lecture Notes 178, Cambridge University Press (1992) Wall, C. T. C. Finiteness conditions for CW complexes Ann. of Math. 81, 5569 (1965) West, J. E. Mapping Hilbert cube manifolds to ANRs Ann. of Math. 106, 118 (1977) Whitehead, J. H. C. Simple homotopy types Amer. J. Math. 72, 157 (1950) Yamasaki, M. L-groups of crystallographic groups Invent. Math. 88, 571602 (1987)
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