数理解析研究所講究録 1290 巻 2002 年 104-107

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Relatively Simple Chain Complexes

城西大学

・理学部

山崎

正之 (Masayuki Yamasaki )Faculty of Science,
Josai UniversityIntroduction
The purpose of this short note is to give acharacterization of geometric module chain complexes

representing the trivial element in the relative$n$,$\epsilon)$of apair$(X, \mathrm{Y})$$p_{X}^{-1}(\epsilon)$-controlled Whitehead group $Wh(X,$$\mathrm{Y};p_{X}$,of metric spaces. These groups were introduced in [5]. Here $px$ : $Warrow X$ isagiven control map, and$n$is the restriction on the dimensions of chain complexes. For different

are in general different abelian groups, but they are all the same afterstabilization (Propositions 4.6, 4.7, and the comment after 4.7 of [5]).
n’s,$Wh(X, \mathrm{Y};p_{X}, n, \epsilon)$

Main Theorem. There exist a positive constant$\alpha$such that the following holds: For any chain

complex $C$ representing the trivial element of $Wh(X, \mathrm{Y},px, n, \epsilon)$ , there exist -dimensional trivial$n$chain complexes , $T’$ and an -dimensional free$T$and $F\oplus T’$ are$\alpha\epsilon$$n$$\alpha\epsilon$

chain complex $F$ on$p\mathrm{x}|\mathrm{Y}^{\alpha\epsilon}$such that CAT-simple isomorphic. Actually $\alpha=500$ works.

In the first section, we review some facts from [5] and give the proof of the main theorem.
The main ingredient of the proof is the restriction operation of simple isomorphisms describedin [3]. In the second section we discuss how this can be used in the theroy of controlled L-theory.

1. Proof of the Main TheoremWe first review the definition of the controlled Whitehead group of apair. Letof metric spaces, and $px$ : $Warrow X$ be acontinuous map.
alence classes of -dimensional free$n$$\epsilon$$Wh(X, \mathrm{Y};p_{X}, n, \epsilon)$$(X, \mathrm{Y})$be apairis the set of equiv-chain complexes on $px$ which are strongly$\epsilon$

contractible

over $X$ -Y. The equivalence relation is generated by -stable -simple equivalences away from. Here the term “stable” means that we allow taking sums with trivial chain complexes$n$$40\epsilon$$\mathrm{Y}^{20\epsilon}$. . . $arrow 0arrow FF\underline{1}arrow 0arrow\ldots$ ,105
the term “away from$\mathrm{Y}^{20\epsilon}$”means that we allow taking sums with free chain complexes lying

over the specified set, the prefix indicates that all the complexes involoved should be strictly-dimensional( $i.e$ . $C_{i}=0$ for $i<0$ and $i>n$ ), and the prefix 406 indicates that everything$n$$n$

involved should have appropriate size control. The following is immediate from 4.1 of [5].
Proposition 1. If $[C]=0\in Wh(X, \mathrm{Y},p_{X},$n,$\epsilon)$on$p_{X}$, free$D’\oplus T’$$86\epsilon$

chain completes $D$ , $D’$ on, then there exist trivial chain complexes T, $T’$

$p_{X}|\mathrm{Y}^{20\epsilon}$, and an$86\mathrm{e}$equivalence: $C\oplus D\oplus Tarrow$$f$.

For each , the$i$$86\mathrm{e}$isomorphism$f_{i}$:$C_{i}\oplus D_{i}\oplus T_{i}arrow D_{i}’\oplus T_{i}’$is the composition of an$86\mathrm{e}$deformation$C_{i}\oplus D_{i}\oplus T_{i}=G_{0}\mathit{9}arrow G_{1}1arrow g_{2}\cdotsarrow G_{m}\mathit{9}m=D_{i}’\oplus T_{i}’$Each$g_{j}$is either(1) an elementary automorphism of the form
(2) ageometric isomorphism$\mathbb{Z}[S]arrow \mathbb{Z}[S’]$$86\mathrm{e}$deformation

above, then define$G_{0}arrow g_{1}’G_{1}arrow g_{2}^{J}$by the matrix$g_{j}’$

if$g_{j}$...

$(\begin{array}{ll}1 h’0 1\end{array})$deleting paths whose starting points are in$\mathbb{Z}[S_{1}]\oplus \mathbb{Z}[S_{2}]arrow \mathbb{Z}[S_{1}]\oplus \mathbb{Z}[S_{2}]$$S$

and $S’$ .
as follows. Firstly, if$arrow g_{\acute{m}}G_{m}$, where:$h’$. Then$g_{j}’=1$

defines ageometric isomorphism of $D$ with ageometric submodule of$D_{i}$and the corresponding submodule to get an$\overline{f}_{i}$where$E_{i}$

is the submodule ofbasis elements of$C_{i}\oplus T\%$$\overline{f_{i-1}}\circ(dc\oplus d_{T})\circ\overline{f_{i}}^{-1}$is a

is of type (1)is obtained from$\mathbb{Z}[S_{2}]arrow \mathbb{Z}[S_{1}]$$p_{X}^{-1}(\mathrm{Y}^{(20\dagger 86\mathrm{x}2)\epsilon})$$g_{j}$

over$\mathrm{Y}^{192\epsilon}$

is of type (2), then let $g_{j}’=g_{j}$ . Then the new deformation is geometric overWe can delete$173\mathrm{e}$$C_{i}\oplus T_{i}arrow E_{i}$there, since

$f$

is an

outside of$\mathrm{S}6\mathrm{e}$$\mathrm{Y}^{(20+86\mathrm{x}4)\epsilon}$$d_{E}^{2}$is$\mathrm{Y}^{(20+86\mathrm{x}4)\epsilon}$$d_{E}$$E$. Secondly,$\mathrm{Y}^{106\epsilon}$

lying overand$\mathrm{Y}^{106\epsilon}$.

-simple isomorphism$(86\cross 3+2)\epsilon$, and it is$172\mathrm{e}$: $E_{i}arrow E_{i-1}$ by the$173\mathrm{c}$morphismhomotopic to 0, and therefore $(E, d_{E})$$\mathrm{Y}^{(20+86\cross 3)\epsilon}$; therefore,$d_{E}$

is equal tohomotopic to the boundary map of $T’$

chain map. Replace the boundary map. Nowbygenerated by the basis elements corresponding to the. We define the boundary map. The composition$86\epsilon$$D_{i}’\oplus T_{i}’$$h$,chain complex. Note that $f_{i}’=f_{i}$ outside of$f_{i-1}\circ(d_{C}\oplus d_{T})\circ f_{i}$outside of$D_{i}’\oplus T_{i}’$

:

, or

made up of paths with coefficient dbl which give a

one-t0-0ne correspondence of the basis elements
Make anew:$(\begin{array}{ll}1 h0 1\end{array})$$d_{E}$

by the boundary map of $T’$splitg as the sum of afree chain complex

and atrivial chain complex $7”’=\{E_{i}(X-\mathrm{Y}^{(20+86\mathrm{x}5)\epsilon})\}$ , and$\overline{f}$isomorphism between $C\oplus T$ and $F\oplus T’$ . This completes the proof.

$F=\{E_{i}(\mathrm{Y}^{(20+86\mathrm{x}5)\epsilon})\}$

is a$\square$$(1 +86\mathrm{x}3)\epsilon$-simple

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2. Controlled L-theory
In [2], Pedersen, Quinn, and Ranicki established the controlled surgery exact sequence when thelocal fndamental group of the control map is trivial. One of the key ingredients was the splittingof quadratic Poincare complexes. In this section we discuss an obstruction for splitting in thegeneral case.

Fix acontrol map $pB$ on ametric space $B$ and fix $n\geq 3$ . Let $(D, \psi)$ be an n-dimensional

quadratic Poincare complex on$p_{B}$of radius $<\delta([4][6])$ , and let $W$ be asubset ofconstruct pairs $(Carrow D’)$ , $(Carrow D’)$ such that $D’$ and $D’$ lie over $B-W$ andand the union $D’\cup cD’$ is equivalent to D.

$C$$W^{\gamma}$$B$. One canrespectively

is Ranicki’s algebraic boundary of $D’$ . As itstands, it has two flaws:(1) It may lie all over $B-W$ .

(2) It may be non-trivial in degrees$n\mathrm{a}\mathrm{n}\mathrm{d}-1$.

The second flaw is easy to remedy. Homologically,$C$is $(n-1)$ -dimensional, and one canuse the usual folding argument to make it into astrictly $(n-1)$ -dimensional complex. Here
we need $n\geq 3$ . The first flaw is harder to remedy, but is strongly contractible away from
$V=W^{\gamma}\cap(B-W)$ , and defines an element $4\in Wh(B-W, V;p, n-1, \gamma)$ , where $p=p_{B}|(B-W)$ .
$C$

Recall from [5] that there is aconstant $\kappa>1$ and ahomomorphism$\partial:Wh(B-W, V;p, n-1, \gamma)arrow\tilde{K}_{0}(V^{\kappa\gamma},p|V^{\kappa\gamma}, n-1, \kappa\gamma)$The image$\partial(\xi)$chain complex

is the obstruction for splitting. Roughly speaking$P$lying over$V^{\kappa\gamma}$, and this represents

equivalent to afree chain complex lying over$V^{\kappa\gamma}$$\partial(\xi)$$C$.

is equivalent to aprojective. If this element is 0, then$C$

is, and we get the desired splitting.

When the local fundamental group of the control map is trivial, the absolute $Wh$ groups and$\overline{K}_{0}$groups are stably trivial (see 8.1 and 8.2 of [5] when$\mathrm{p}=1$). Therefore there is no obstruction.

In fact, since the sequence. . . $arrow Wh(B-W;p, n-1, \gamma)arrow Wh(B-W, V;p, n-1, \gamma)arrow\tilde{K}_{0}(V^{\kappa\gamma}, p|V^{\kappa\gamma}, n-1, \kappa\gamma)$
is stably exact (5.3 of [5]), the relative Whitehead group also vanishes stably, and hence $[C]=$$\mathrm{O}\in Wh(B-W, V^{\lambda\gamma}; p, n-1, \lambda\gamma)$

for some $\lambda>0$ . By the main theorem, there exist $(n-1)-$

dimensional trivial chain complexes$T$$F$on$p_{B}|V^{(1+\alpha)\lambda\gamma}$

and $T’$ and an $(n-1)$ -dimensional freesuch that $C\oplus T$ and $F\oplus T’$ are$\alpha\lambda\gamma$$\alpha\lambda\gamma$chain complex-simple isomorphic. This $F$ can be usedas the center piece for the splitting. This is essentially the proof for splitting in [2]. And fromthis the stability (squeezing) of controlled -theory follows
$L$

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References[1] E. K. Pedersen, Controlled algebraic $K$ -theory, asurvey, Geometry and topology:Aarhus(1998), 351 –368, Contemp. Math., 258, Amer.math. Soc, Providence, RI, 2000.

[2] E. Pedersen, F. Quinn and A. Ranicki, Controlled surgery with trivial local fundamentalgroups, (preprint).
[3] F. Quinn, Ends of maps I., Annals of Maths. 110, 275 –331 (1979).

[4] A. Ranicki and M. Yamasaki, Symmetric and quadratic complexes with geometric control,

Proc. of TGRC-KOSEF vo1.3,139–152 (1993), available electronically on WWW fromhttp:$//\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}$.Josai. ac.

$\mathrm{j}\mathrm{p}/\sim \mathrm{y}\mathrm{a}\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{a}\mathrm{k}\mathrm{i}/$.

[5] A. Ranicki and M. Yamasaki, Controlled $K$ -theory, Topology Appl. 61, 1–59 (1995).
[6] M. Yamasaki,$L$-groups of crystallographic groups, Invent. Math. 88, 571-602 (1987).

Dept. of Mathematics
Josai University
Sakado, Saitama 350-0295, Japanyamasaki@math.Josai. ac.jp