\SURGERY SPACES OF CRYSTALLOGRAPHIC GROUPS/

by

Masayuki Yamasaki

Dissertation submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

in

Mathematics

APPROVED:

F. S. Quinn, Chairman

_ _..___.__,_.--~----- '- .a-: I C3<1 ---'---'---r----R. A. McCoy J. T. Arnold

R. Olin R. L. Snider

AUGUST, 1982

Blacksburg, Virginia

Acknowledgements

I would like to thank my thesis adviser, Professor F. S. Quinn,

for suggesting this problem to me and for providing encouragement

and many valuable comments. I would also like to express my

gratitude to for his earlier direction and

instruction and for providing encouragement from Japan.

ii

Table of Contents

page

O. Introduction.......... . . . . . . . . • . . . . . . . . . . . . . . . . . . . . • . . . . . • 1

1. Preliminaries

1.1 Chain complexes .............•....•.......•.......... 10 1. 2 Geometric modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . • . . 23

2. Glueing and splitting

2.1 Glueing of quadratic pairs and triads ............... 27 2.2 Splitting lemma for quadratic Poincare

complexes and pairs. . . . . . . . . . . . . . . . . . . . . . . . • • . . . . . . . 30 2.3 Splitting lemma for geometric quadratic

Poincare complexes and pairs ...•.................... 39 2.4 Glueing and splitting over a triangulation

of a manifold ............•.......................... 50

3. Surgery spaces and assembly

3.1 Surgery spaces .........•..........................•. 56 3.2 Homology theory and assembly ........................ 61

4. Crystallograp~ic groups

4.1 Preliminaries on crystallographic groups ...........• 76 4.2 Induction theory., ................................. , 82 4. 3 Induction theorems .............. , ................... 88 4. 4 Ca1cula ti on of surgery groups ....................... 9 5

Bibliography ..........................................•...... 101

Vita .•.............. , . . . . . • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

iii

0. Introduction

Classification of topological spaces is one of the main

problems in topology. Although classification of 1- and 2-

dimensional manifolds up to homeomorphism is possible, the problem

becomes extremely harder for higher dimensional manifolds. The

classification problem of manifolds up to homeomorphism can be

devided into two problems:

1. classification up to homotopy, and

2. classification up to homeomorphism in a given homotopy type.

The first problem is harder than the classification problem of

finitely presented groups, because any such group can be realized

as the fundamental group of some compact manifold. In this sense,

classification is impossible.

Let us introduce some notation. ([W], [KS]) Let X be a

finite Poincare complex (i.e. a finite CW complex satisfying

Poincar~ duality) with formal dimension m such that ax is a

manifold (possibly empty). A homotopy-topological structure

on X is represented by a pair (Mm, f) where K11 is a compact

manifold and f is a homotopy equivalence: Mm~ X which

maps 3M onto 3X homeomorphically. Two such structures (M, f)

(M', f') are equivalent if there exist an h-cobordism (W;M,M')

from M to M' rel boundary (i.e. which is a product cobordism

1

2

from ClM to a:M') and a homotopy equivalence F : W ·--?> X x I

such that F(x) = (f(x),O) for x EM and F(x) = (f'(x),l) for

x E: M' . .$ (X) denotes· the set of the equivalence classes of

structures on X.

Notice that if m ~ 6, (lX = ¢ and the Whitehead group Wh(rr1X)

= 0, then the vanishing of ~(X) implies the uniqueness of the

homotopy-topological structure on X up to homeomorphism. Thus

the computation of .J(X) is important in solving the second

problem.

Surgery is a technique devised for studying these problems.

Let X be a topological space, M a manifold, and f: M ~ X be

a map. For simplicity let us assume M is smooth and closed. The

homotopy group rr(f) of f measures how near f is to a homotopy

equivalence.· If X is reasonably nice, then the vanishing of

rr(f) implies that f is a homotopy equivalence. Consider an

example. Let M be a 2-dimensional torus T2 and X be a 2-

dimensional sphere s2 . Take a small disk D2 in T2 , and let P

be a point on s2 .

homeomorphism from

Since s2- P is an open disk, there is a

int(D2) ( interior ) onto s2- P. By

sending T2- int(D2) to P, we get a (degree one) continuous map

f: T2 ~ s2 . Obviously this is not a homotopy equivalence.

The loop a(see Fig.O) represents a non-trivial element of the

fundamental group n 1T2 , but f sends a to a trivial loop in s2 ;

N

Figure 0.

3

f

s2

holes

= 2-sphere

t t

00 2-disks

2 Surgery on T .

4

i.e., a represents a non-trivial element of n2(f). One can

"kill" this element by performing surgery on M (and f) . It goes

as follows. See Fig.O. a has a thin neighborhood homeomorphic

1 l D1 . 2 11 N to an annu us S x in T ; ca it . Delete int(N) from T2 ;

then the result has two holes, so attach two 2-disks to each hole.

The result is a 2-sphere. One also needs to modify the map f

by sending the new attached disks to P. The new map is a homotopy

equivalence. In general, in order that we can carry out th~s

surgery operation, we have to make some assumptions on f: M --7

X. First of all X has to be a Poincare complex, and f has to be

of degree one. But this is not sufficient. Suppose we want to

do surgery on an element a E TI (f) • r+l ' then we need to have an

embedding Sr m-r M111 r O) the g: x D ~ such that g(S x represents

element 3a. For this purpose we assume that X has a bundle v over

it, and there exists a stable trivialization F of TM 9 f*(v),

where TM denotes the tangent bundle of M.

Wall (W] defined an abelian group Lm(n 1X,w) depending only

on the group n 1x, the homomorphism w: n1x ~ {±1}, and the

value of m modulo 4, and showed that each triple (M,f,F) corre-

sponds to an element of Lm(n1X,w), called the surgery obstruction,

which vanishes if we can perform surgery on (M,f) several times

and get a (simple-)homotopy equivalence. If m ~ 5, then the

converse is also true. See (W] for the detail.

Surgery can be applied for uniqueness argument. Suppose

5

we have two homotopy equivalences f: M ~ X and f': M' ~ X

and they are bordant; i.e., there exist a manifold N whose boundary

is the disjoint union of M and M' and a map g: N-----+ X such that

g\M = f and g\M' = f'. This happens if both f and f' are obtained

by doing surgery to the same map. Take a product map (N;M,M")

---7> (Xx I;X x O,X x 1), and attempt a surgery on this. If we

can make this inr.c a homotopy equivalence without touching M and

M' , then we obtain an h-cobordism.

Now let us go back to the classification problem 2. Recall

the Sullivan-Wall surgery exact sequence [.WJ:

[x x I rel () 'G/TOP.J --~ Lrn+l (TI 1 X) _, .9(X)

where m dim X ~ S. Farrell and Hsiang proved the following in

tFH2].

Theorem Let Mn be a closed aspherical manifold whose fundamental

group is virtually nilpotent. Then .$(Mn) = 0 for n ~ 4.

This is proved by showing that the function [X rel Cl,G/TOPJ

~ Lm(TI1X) is a bijection. As a corollary they obtained:

Theorem Let Nn be a closed connected flat Riemannian manifold

6

where n f. 3,4 and let Mn be an aspherical manifold such that

n n n n 1T 1 (M ) is isomorphic to 1T 1 (N ) , then N and M are homeomorphic.

We would like to prove a similar result for some topological

spaces which are almost manifolds but have singularities. Let

r be a crystallographic group acting on IR.n and consider the

orbit space JRn/r. When r is torsion free, JRn/f is an n-dimen-

sional aspherical manifold, and Farrell and Hsiang's result

will apply to this. If f has torsion, JRn/f is a stratified space,

i.e., a nice union of manifolds. The following is our conjecture:

Conjecture If a stratified space is homotopy equivalent to JRn/r

in some nice way, then it is homeomorphic to JRn/r.

This thesis is the first step toward this conjecture. As

with the Farrell-Hsiang theorem this conjecture is approached by

showing that functions in appropriate "stratified" exact sequences

are bijections. Our main result is a partial computation of

one of the terms in one of these exact sequences, specifically

the surgery group of the group r. Let Wr be a free contractible f-space. Then the map p:

( JRn x Wr) Jr --7' JRn /r has point inverses p -l (x) = Wr/r x, which

are classifying spaces for "isotropy" subgroups r . Quinn has x

defined ~-spectra JL(X) whose homotopy groups are the surgery

7

obstruction groups Li(TI 1 X) [Ql]. This functor can be applied

fibrewise to obtain a "sheaf'' of spectra JL(p) ___,, 1Rn /f, with

fibre over x, TI.(p -l(x)). Next Quinn has defined IQ3] homology

groups with spectral sheaf coefficients H.,.~( 1Rn/f; TI.(p)). For

technical reasons we use a definition of TI. using the Poincar~

chain complex of Ranicki [R2,3,4]. The homotopy groups are

the limits L~00 of Ranicki 's lower L-theory L~-m) (RU which 1 1

may differ from L. possibly by 2-torsion. The following is our 1

main theorem.

Theorem (4.4. 1) If a crystallographic group f has no 2-torsion,

there is a natural isomorphism:

The map a is essentially Quinn's "assembly" map.

The assembly map is defined in §3.2. The outline of the proof

is like. that in [FH2], and we use an induction on the size of r. If r r ' 'll f t 11 h" T'' then lRn/r . = · ~ or some crys a . ograp ic group . ,

is a fibre bundle over s1 with fibre 1Rn-l/r', and the theorem

is proved by a standard homology property on the left side. the

splitting theorem on the right, and the induction hypothesis.

Otherwise, the structure theorem for crystallographic: groups of

[FH2J implies that there exists a surjection f ~ f"' s onto

8

some finite group and that we can use a hyperelementary induction

with respect to the maximal hype.relementary subgroups {H} of f" s

(4.1.2). More precisely, we have a commutative diagram:

n Hj (JR /CH; JL(pH))---7

aH l where CH is the preimage of H in r. Since the two rows are exact,

the map a is proved to be an isomorphism if the next two columns

are isomorphisms. But we cannot do this directly, because, for

example the size of CH may not be smaller than that of r. We have

to use the proof of 5-lemma applied to a single element. For -oo n

example, pick an element y of Lj ((JR xc Wr)/r) and represent

it by a geometric quadratic Poincare complex on p, y, with radius

r measured on JRn/r. We want to show that the restriction image

yH of y in each

If the size of CH is strictly smaller than the size of f, then

by induction hypothesis this is the case. If not, then 4.1.2

· i· h h · h · k" 1Rn/CH __,. JRm/r-imp ies t at t ere is a s rin ing map a : ,

for some crystallographic group r- of rank m ~ 1, and that we can

make the radius of the restriction image yH of y arbitrarily

small on 1Rm/r-, by choosing a very large integers. Now the method

of [Q3] is applied to characterize things of small radius as

9

exactly elements of the sheaf homology groups (3.2.2 and 3.2.3).

So yH comes· from an element of Hj ( JRm ;r-; lL(o•,pH)). Lastly this

is proved to be isomorphic to Hj (]Rn /CH; lL(pH)) using the

induction hypothesis.

1. Preliminaries

1.1. Chain complexes

In 1.1, we fix the notation concerning chain complexes and

review some basic facts. The same notation and sign convention as

in tR3J will be used.

Let R denote a ring with involution. An R-module chain complex

C is a sequence of R-modules and R-module homomorphisms

d d d c . . . --~ cr+l --+ c --+ c -r r-1

(rE:Zl)

2 such that d = 0. See [SJ or [R3J for the definitions of chain maps,

chain homotopies, chain equivalences, and chain contraction. A

finitely generated projective R-module chain complex C is strictly

n-dimensional if C = 0 for r > n and r < 0. A finite complex of r

finitely generated projective R-module is n-dimensional if it is

chain equivalent to a strictly n-dimensional chain complex.

The algebraic mapping cone C(f) of a chain map f: C ~~ D is

the R-module chain complex defined by

C(f)r = Dr El1 Cr-1

___...... C(f)r-1 = Dr-1 ro Cr-2·

The algebraic ~apping cylinder M(f) of f is the R-module chain

complex defined by

10

11

dM(f) =(:OD :J : M(f) r = Dr lb Cr-l $ Cr ~ M(f) r-l

A triad of R-module chain complexes

consists of R-module chain maps

f: C ~ D, f': C' -----> D', g: C ~ C', ·h: D ~ D'

and an R-module chain homotopy

k: f'g ~hf: C ~ D' (i.e., dk + kd f I g - hf),

1.1.1. Lermna A triad .J (given above) of finite dimensional R-module

chain complexes induces an R-module chain map from the algebraic

mapping cone of g to that of h, which is an R-module chain equivalence

if both f and f' are R-module chain equivalences.

Proof. The map is given by a matrix

12

C(g) r c I tl7 c 1 ----7 c (h) = DI $ D 1 . r r- r r r-

The second part is obtained by applying 5-lemrna to the following

commutative diagram:

__, H (C) --7-H (C') _,,. H (C(g)) _,. Hr-1 (C) ----7 Hr-1 ( C') ___:;. r r r l: 1~ l l ~ l ~

= =

~H (D) ~ H (D') ~H (C(h)) ~ Hr-1 (D) --'i>H 1(D') ---7 ••• r r r r-

where the two rows are exact.

-1 The suspension SC (resp. desuspension S C) of an R-module

chain complex C is the R-module chain complex defined by

dsc = de: (SC) = c 1 ~ (SC) 1 = c 2 r r- r- r-

( resp. c ) . r

A diagram of R-module chain maps

f g A ...----- B ___ _,. C

induces a triad

g B -------~c

defined by

C( (J B

(:A \o

13

--7Ae C)

(-)r-lf

A ©B ~C r r-1 r

This push-out triad is natural in the following sense. Suppose we

have a diagram which commutes up to chain homotopy:

14

f g A B C

l :1_/ rB l ~ l re

A' B' ·-7' C'

f' g'

Then it induces a diagram which commutes strictly:

i j A -7 A \J C c B

1 I l rA l r re

A' A' v C' (-- C' i' B' j I

where

l:A (-)r-lh 0

) l

r = rB 0

(-)r h re 2

: (_A \JBC) r = A .::n B a. C --7 (.A' V C') = A' 11' B' ""' C' . r w r-1 ~ r B' r r w r-1 v r

Next we consider the dual situation. A diagram of R-module

chain maps:

i j A ---~ D ~------ C

induces a triad:

15

g c

fl~~-> lj A D

i

defined by

j): A fB C -~ D)

(:)rj) de

0

0

: (A t'\DC)r Ar 61 Dr+l i3J Cr--> (A (\DC)r-1

f (-1 0 O): (A (\DC) A r r

g (O 0 1) : (A ·qDC) c r r

h (O (-)r+l 0) : (A !lDC) r ) Dr+l.

This "pullback" triad is also natural; i.e., a chain-homotopy

conunutative diagram:

i j A D c

1 \:1~ I h/ 1 rA rD / re

..... A' --4 D' C'

i' j I

induces a strictly commutative diagram:

16

f g A~----- --- ----~c

rA r re I

¥ >JI I

!\D 'C' v A' ---A' C'

f I g'

where

( rA 0

(:) rh \ r ~ (-) :+lh1 rD

re 2) 0

"Push-outs" and "pullbacks" are universal:

1.1.2. Lemma Atriad

g B -------'>C

A -----~ D i

induces chain maps

A ffiB $ C --?D r r-1 r r

17

which make the following diagrams commutative:

t (1 0 0) t(O 0 -1) A ~A 1. C

B c

L A D c

i j

f g A B c

t A A ,\DC c

(-1 0 0) (O 0 l)

"Push-out" and "pullback" are essentially inverses of each

other. First begin with a diagram

i j A ·--------:> D --------- c

then the pullback triad

A i

( h = (0 O) )

induces a chain equivalence

18

(i 0 l 0 -j) A 'v' c A f'\DC

D.

The inclusion map

t(O 0 1 0 0) D A v A (\DC

c

gives a chain homotopy inverse. Next begin with a diagram

f g A .c----------- B -------~ C;

then the push~out triad

B -------~c

A ------~ Av C B

induces a chain equivalence

t(-f 0 1 0 g): B -~

and the projection map

(O 0 l 0 0):

gives a chain homotopy inverse.

, 0 .r \

h = \ (-) )

\ o I

Let C be an R-module chain complex, C* denotes the R-module

chain complex defined by

-r where C is the dual of C -r

chain complex defined by

19

-~(Ci:) r-1

c-r+l '

n-* And C denotes the R-module

The generator T e LZ 2 acts on HomR (Ci:, C), the R-module chain complex

of R-module chain maps from C* to C, as follows:

Now let us recall the definition of quadratic complexes and

pairs by A. Ranicki.

1.1.3. Definition An n-dimensional quadratic complex over~ (C,l)J)

is an n-dimensional R-module chain complex C together with an

element ljJ 0 Qn(C). Such a complex is Poincar~ if (l+T)l)J0 :

Cn-* ~ C is a chain equivalence. A ma..e_ (resp. homotopy equiv-

alence) of n-dimensional quadratic complexes over R

f: (C, l)J) __ _____,. (C'' iµ')

is an R-module chain map (resp. chain equivalence) f: C ---7 C' I

such that f%(l)J) = l)J' ~ Q (C). o n

An (n+l)-dim_ensional quadratic pair over R (f: C -~ D, (ol)J, ~1))

is a chain map f from an n-dimensional chain complex C to an (n+l)-

dimensional chain complex D together with an element ( ol)J, l)J) C:

Qn+l(f). Such a pair is Poincare if the R-module chain map D n+ 1-":

20

__ _,.> C(f) defined (up to chain homotopy) by

(Cl+T)ol/!0 )

Dn+ 1-r __ _,. C (f)

(l+T)lj! f"' . r 0

D. ~ C l r r-

is a chain equivalence, in which case the boundary n-dimensional

quadratic complex (C, iµ) is Poincar~. A _!'.1.ap (resp. homotopy

equivalence) of (n+l)-dimensional quadratic pairs over R (g,h;k):

(f: c ~ D, (ol/J,l/!) <= Qn+l (f)) -T (f': c' ---:> D', (olj!', l/!') "' Qn+l (f'))

is a chain complex triad of the type

f c _____ _,D

g

C' f I

such that g: C --7 C' and h: D ~ D' are R-module chain maps

(resp. chain equivalences) and

( Olj! I ' l/J I ) E Qn+ 1 ( f I ) •

See Ranicki's (R2,3J

Qn+l(f), f%, and (g,h;k)%.

a collection

such that

for the precise definition of Q (C), n

An element l/! €: Q (C) is represented by n

s >, 0 }

21

dljJ + (-)r11, d>'~ + (-)n-s-l(lj! + (-)s+JT1lj! ) 0 s ·· 'Vs · s+ l s+ 1 -

n-r-s-l : c __ _,~ c r

( s :;; O)

and if f: C ----7 Dis a chain map, f%ljJ is represented by {flj!sf*}.

An element (oljJ,ljJ) e Qn+l (f) is represented by a collection

such that

(d(6•1' ) + (-)r(6•1' )d* + (-)n-s(6• 1' + (-)s+lT6ljJ ) + (-)nflj! f* '¥ s '¥ s '¥ s+ 1 s+ l s '

dljJ + (-)rljJ d* + (-)n-s-l(ljJ + (-)s+lTljJ )) s s s+l s+l co

0 Z HomR(Dn-r-s,Dr) e Ho~(Cn-r-s-l,Cr) r=-oo

and (g,h;k)%(o~J,ljJ) is represented by

+ (-)n+pkTl/! k* g\jJ g'") I s ) o}. s+ 1 ' s

(s ~ 0),

1.1.4. Definition A cobordism of n-dimensional quadratic Poincare

complexes over R (C,ljJ), (C' ,ljJ') is an (n+l)-dimensional quadratic

Poincare pair over R

((f f')))

with boundary (C ffi C', ljJ ~ -ljJ').

22

1.1.5. Proposition (Ranicki) Cobordism is an equivalence relation

on the set of n-dimensional quadratic Poincare complexes over R,

such that homotopy equivalent Poincare complexes are cobordant.

The cobordism classes define an abelian group, the n-dimensional

quadratic L-group of R, L (R) (n ~ O), with addition and inverses n

by

( c, iµ) + (C I , iµ ' ) (C + C', tµ + tµ'), -(C,iµ) (C,-tµ).

23

1.2. Geometric modules

We define geometric modules and their homomorphisms, following

Connell and Hollingsworth [CH] and Quinn~2,3]. We start with the

constant coefficient case.

1.2.1. Definition Suppose Xis a metric space, and Risa ring.

l) If {x }, a e A, is a collection of points of X, then the a

geo~et ric R-m~dul_~ with basis {x } is the free module R ( {x } ) . a a

2) If h: M --7 M' is an R-module homomorphism of geometric modules

with bases {x }, {y0 } respectively, then the underlying sei:_ i_unc-a µ

tion is obtained by ~(xa) = {yB [ yB has non-zero coefficient in

h(x )j. a

3) h has radius r if h(x) S x r(= the r-neighborhood of x ). ----- - a u. a 4) A homomorphism with radius r, h: M --> M', is an r-isomorphism

(with support C <:::. X) if there exists a homomorphism g: M' ---7> M

with radius r such that hg = J and gh = 1 ( hg = 1 on C and gh = l

on C) .

Next we consider the non-constant coefficient case.. Although

a more general definition is possible, we restrict ourselves to the

following special case.

1.2.2. Definition Let p: E -----7>X be a continuous map. where Xis

a metric space, q: E ---7> E a covering space of E ( E may not he

connected), and f the group of covering transformations of q. Let

24

{x } be a subset of E. A geom~tric Zlf=-~odule M on p generated by Cl

-1 is a free abelian group generated by the points o( q ({x }) Cl

together with the Zlf-module structure induced by the action of r.

Although E is not mentioned explicitly, it is part of the information,

and we always fix a lifting x of x in E. Cl Cl

Let Mand M' be geometric modules on p generated by {xa}, {y6}

respectively. A Zlf-module homomorphism h: M --? M' has rad~us r

-1 -1 if, for each pair v and v' of points in q ({xa}) and q ({y6:> such that v' has a non-zero coefficient in h(v) written as a linear

combination of points in q- 1({y6}) with integral coefficients,

there exists an arc connecting v and v' inside (pq)- 1(pq(v)r).

Geometric rhain complexes, chain maps, chain homotopies, chain

equivalences. etc. on p (with support C ~ X) are defined in the

obvious way. For example, a geometric c:haii:! comple~ on p (with

support C) is a sequence of homomorphisms of geometric modules on p:

d. d. 1 -. ~"" M 1---7 Mi ' i-1 ~ Mi-2 ---7'" ...

such that d. 1 d. 0 ( di-ldi/c = O) for each i. The radius of a 1- 1

chain complex is sup.(radius of d.). An r-chain equivalence_ is 1 1 ------

not just a chain equivalence of radius r. We also require that

there is a chain homotopy inverse with radius r and that the two

chain homotopies involved have radius r.

25

For the dual w·~ of a geometric module M on p generated by

{xa}• we use the standard dual basis, denoted by {xa*}. x* a (resp.

xit ) corresponds to the same point in E (resp. E) as x (resp. x ) . a a a

In general it is hard to work with geometric modules on p,

but if p behaves like a product (at least locally) then we can

understand geometric modules on p with small radii pretty well.

The following definition describes such a nice p.

1.2.3. Definition 1) If p: E ~x and X ~ Y, then Y is a p-NDR

(neighborhood deformation retract) subset of X if there is a

neighborhood U of Y, and homotopies H: U x I-~ X, iiA: p- 1(U) x I

~ E such that H is identity on U x {O} and Y x I, H(U x {l}) S

Y, HA is the identity on p-l(V) x {O}and p- 1(Y) x I, and the diagram

-1 HA

p (U) x I E

p x 1 l l p

U x I -> x H

conunutes.

2) A stratified system of fibrations on a space X consists of a

map p: E ---7 X and a closed finite filtration of X, X = ~ ? ...

-1 2 x0 , such that each Xj is a p-NDR subset of X, and each p: p (Xj

- X. 1) --7 X. - X. 1 is a fibration. J- J J-

26

1.2.4. Remark If R = ~r, then the involution of R is given by

-1 x r---7 x for x G f. Thus we consider only the "orientable"

case.

2. Glueing and Splitting

2.1. Glueing _Qi quadratic pairs and triads

In this section we review from Ranicki [R2,3] to fix notation.

Consider two n-dimensional quadratic pairs over R whose boundaries

are the inverses of each other:

c' (g': B ---)C', (c5ij;',-lj;))

c II = (g II : B --> c II ' ( cS lj; II ' lj;) ) .

The union of c' and c" along the (n-1)-dimensional quadratic complex

b = (B,lj;) is an n-dimensional quadratic complex (C' UBC", cS<j;'!Jlj; cSlj;")

defined by

(cSw' u o<J;") lj; s

Olj; I

\(-) n-r:~Wsg' *

__ __,, (CI v C") B r

0 0

(-)n-r-sTlj; s+l 0

( - ) s g "lj; s o<J;" s

c I ~ B tf.j C" r r-1 r'

and will be denoted by c' v be". If both c' and c" are Poincare,

then so is c' U b c".

)

To state the splitting lemma for quadratic Poincare pairs, we

need to introduce the notion of "quadratic triad," which is a

quadratic pair with a splitting of its boundary into two pieces.

27

28

More precisely. an (n+l)-dimensiona1 quadratic_ triad over R (:], y;)

is a triad of R-module chain complexes:

g II B C11

J g, I ~> I f II

\J,. ,j,' C' ----- D

f'

such that B is (n-1)-dimensional, C' and C11 are n-dimensional, and

D is (n+l)-dimensional, together with a representative

of an element of the triad Q-group Qn+l(~). Such a quadratic triad

is Poincare if

i) the n-dimensional quadratic pairs over R

c' (g': B--7C', (oijJ',-ijJ))

c" (g": B --7C", (oijJ",ijJ))

are Poincare, and

ii) the (n+l)-dimensional quadratic pair over R

d = ((f' (-)r-lh -£"): c' v c" ~ n, (o~, oiJJ' v oiJJ")) B w is Poincare.

Suppose we have two (n+l)-dirnensional quadratic triads over R:

! g ' B c·

I\ h' 1 I ' -31= g' c \jll (iJJ,ot/J' ,-ol}J· ,oijJ') i~ 1

C' D' f I

29

g" B C"

.12= g! l. ~ l f" I -cl/!, oiJJ', ol/J", ol/J").

c· D" I f. 2

Then their union (J1 V .J2 , IJl1 V '1' 2 ) is an (n+l)-dimensional

quadratic triad over R:

where

g" B ----- C"

g'l ~h 1 j"f"

(l/J, olj;', ol/J", ol/J)

c I DI v D" I

j'f' c·

j I 0 0): D' ~ (D' V D") r c! r

I D' iJ C" $ D" r r-1 r

·11 __ t(Q J 0 -1): D" --7 (DI v D") r c! r

-h"): B ---7 (D' V D") r C ! r

0-ij) =<o-ij}' s

(D' lJ ,D")n+l-r-s c·

0

(-)n+l-r-sTol/J! s+l

S I I <-) f" ol/J. 2 s

D'n+l-r-s ~ C!n-r-s $ D"n+l-r-s

I (D' V D") = D' $ C. f9 D".

C ! r r r-1 r

0

0

30

2.2. Splittin_g_ ~mma ~!:.quadratic Poincare complexes and pairs

The object is to give a sort of inverse to the union operation

of 2.1: given a Poincare pair we find the smallest amount of

information required to show that it is equivalent to a union.

Let c = (f: C -~ D, (ow,w)) be an (n+l)-dimensional quadratic

Poincare pair over R and C' (resp. D') be a subcomplex of C(resp. D).

We assume that (C/C') and (D/D') are projective for each r, and · r r

that the image of C' by f lies in D'. Let ic denote the inclusion

map of C' into C, and Pc denote the projection map of C onto C/C'.

We fix the splitting maps·· (C/C') --?C and q · C--7C' for Jc· r r c· r r

the short exact sequence:

ic C' ---·~C

r r Pc .

(C/C') . r

This gives an identification of C r

with C' m (C/C') , and if we r r

define a chain map pc: (C/C') ---:>(SC') = C' 1 r r r-r-1

by (-) qcdjC'

then the boundary map of C is given by a matrix

c~ ~ (C/C')r ~ c~-l@ (C/C')r-l

under this identification. We define maps iD' PD' jD' qD' and Pn

for D in the same way. We further assume

31

These conditions will be used to prove that B in Lemma 2.2.3 below

is (n-1)-dimensional, etc. The auther is not sure whether these

are actually necessary or not. But anyway, in geometric situation,

these are automatically satisfied. See 2.2.6. n--k

Let C'' denote the chain complex (C/C') .

By the assumption on D', f induces a chain map f' = qDfiC:

C' ---7 D' such that inf' = fie. The algebraic mapping cone C(f')

of f' is a subcomplex of the algebraic mapping cone C(f) of f.

Define D" by (C(f)/C(f'))n+i-~·', which is same as

There is a chain map (inclusion map)

(C/C')n-r-------? D" r

(D/D')n+l-r ~ (C/C')n-r

from C" to D".

2.2.3. _Lemma (Splitting lemma for quadratic Poincare pairs) Let

c, C', D', C", D" be as above. Then there are (n+l)-dimensional

quadratic Poincare triads over R:

I B ------ CA B -------- C"

(jl: l~.~ l ljl1) (:12: I ~~ l '¥ ) I

I ' 2 ~' C' ----- D' c· D

32

such that the (n+l)-dimensional quadratic Poincar.e pair induced by

their union is homotopy equivalent to c.

2.2.4. Corollary (Splitting lemma for quadratic Poincare complexes)

If c = (C,ljJ) is an n-dimensional quadratic Poincare complex over R

and C' is a subcomplex of C such that C/C' is projective and 2.2.l

holds, then there exist two Poincar~ pairs:

(B -~C', (8ljJ',-ljJ)) and (B --~c", (8tJ.!",~))

such that c is homotopy equivalent to their union.

The following iJlustrates the splitting Jemma for pairs.

c

cr-----1 \ \

'·.

D'

,_

B

I c D"

··-~--------B

c"

Proof of Split ting Lemma: By the naturali ty of pull-backs,

we have the following (partly homotopy-)cornmutative diagram:

33

g II c B ________ ___,,, c II

g' c

C'

-------f'' ' c

------ ! i

i D'

--~----"'D" g I

D

g II D

where the unlabeled vertical maps are

and the front and the back squares are pull-back triads; thus

B = C' ()CC", g~ = (-1 0 O), g~ (O O 1) and C ! = D' f'\ DD" ' . '

g~ = (-1 0 0), g~ I I

(O 0 1). The chain map g·: B---------> c· is

given by a matrix

f i C' lr> C .<%:1 c11 Dr' "" Dr+l 9 Dr". r vi r+ l "" r -- QJ

f II

We define an n-dimensional quadratic Poincare structure I I

(o~,-~) on the pair g·: B --~? c· as follows:

34

(

0

ol)J = a s

0

0

(·-) n-r-sTo''' · · 't's-1

0 : l C1n-r-s D'n-r-s tll Dn+l-r-s $ D"n-r-s

I __ _,.. c· D' '11 D $ D" r r r+l r

0

I

(-)nr+rp, \

( _) nr+r+ 1 j , )

o I

--~> c· D' fa D ~ D" r r r+l r

0

n-r-s-1 (-) TljJ 1 s-

0 :i

(s ~ 1)

(s ~ 1)

Bn-1-r-s = C'n-1-r-s $ Cn-r-s $ C"n-1-r-s

n-1-r B

B = C' tJ3 C ~C" r r r+l r

0

--~~ B = C' $ C ~ C" r r r+l r'

35

where iJ I = (pD qDfjC): 0,,n-r = (D/D I) © (C/C I) t D' r+l r r and j I = (jD 0): D,n-r > D r+l ·

I I 2.2.5. Proposition (g·: B ~C, (Ol/J,-l/J)) is an n-dimensional

quadratic Poincar~ complex.

Proof: First of all the chain map

( (l+T) 8l);0

is given by a matrix

0

(-)r(l+T)8l/J g'" 0 D :-P*~--------------:3~---:

~-----------------------~

j'" f"~ q''~ \ C D 0

r-------------1 ( -) nr+r p ~ \ D I I ( )nr+r+l. 1 I - J I L_ _________ !2_..J

0

0

o -f'qc(l+T)l/J0

C-)r+1£c1+T)l/Joqe o

0 0 ~----------------------,

~--=~g-------------~~--J 0

(-)nr+r f. qD Jc

0

0

0

I I

I

which is a chain equivalence made up of three chain equivalences

36

comb.ined together in the w.ay des.crib.ed in Lemma 1,1.l. Next

notice that B is chain equivalent to s-1 C(pc(1+T)t)J0p2); therefore

b.y 2.2.1, H.(B) = 0 for i < 0, and Hn-l+j(B)_ = 0 for _j > 0 i·

owing to the duality of B. The standard "folding" argument(Co] I

shows that B is ( n-1 )-dimensional. Similarly C is n-dimensional.

A direct calculation shows that (olJ},-(j}) is an n-dimensional

quadratic structure. Thus the proposition is proved,

Now we have two (n+l)-dimensional quadratic triads over R:

(fl:

I g'

I B c·

gel~ l C' D'

f I

g II c B -----C

g I D

g! l. ~ l f" C. > D"

g" D

~l = (~, 0, -otjJ, O))

(t/J, o1JJ, 0, O))i,

which turn out to be Poincare; in fact the duality maps of the

(n+l)-dimensional quadratic pairs induced from these can be written

as follows:

(I) (O ( f I

C((f I 0 n+l-1~ -g I))

D ---7>> D'

(_O

37

.-------------· 0 0 : -qDf (.l+T)_tjJO :

--.---------~~--'

r------------, I -q (l+'f)0 11 • 1 , n · 'l'o, --------------1

r--------------1 ( _) nr+r+l p : I D I .&...------------.--

0

(-)nr+r+l f. ) qD JC

D' r'

which is essentially same as the following composition of chain

equivalences:

(II)

pair

(f(l+T)tjJO 1)

(-q (-)nr+r+l0 ) D D

(0 (g" D

0

: C( (g~

r-----------, I p* -j ~'< I L_Q ____ , ___ Q_j

j ef*qfi a

0

0

0 -f")) n+l-* ------? D"

0 0 r--------------------, 1 ( _) nr P* ( _) nr+ 1 j ~'< 1

L------~-----------~-J

0

0

We now claim that the (n+l)-dimensional quadratic Poincare

38

(F f' ©g ! \fl ( -f" 2 (_C' L'BC") C' ~B ~ C" : = -r r r-1 r

' ~ (D' U D") D' iB c· ~ D" (O V 0 ~-0, 0 \) ---,-{) ) ) c' r r r-1 r' ljJ

which is induced by the union of (.)1 ,lfl1 ) and CJ2 ,lfl2 ) is homotopy

equivalent to the original Poincare pair (f: C--? D, (olj.J,lµ)). Here,

(O Vo~) = 0 ~ 0 tB ol)! e 0 $ 0 Ocv s s

D' e D' l @ D r r- r lB D" $ D" r-1 r

(0 v ~O) = 0 & 0 ID lj; $ 0 $ 0 -iJ s s

C' ® C' l ~ C ~ C" (£) C". r r- r r-1 r

It is already shown that the inclusion maps

l 0 0): C ----;i.) C' v C" B

t (O 0 1 0 0): D --~ D' V 1 D" c·

are chain equivalences. The following diagram:

c ) D

t(O 0 1 0 0) l l t(O 0 i C' I.I C" D' l. D"

B ' c·

commutes, and

1 0 O)

39

1 0 O)ljJ (O 0 l 0 O) s (O v!JJ°) s

0 1 0 O)cSljJ (O 0 1 s 0

for each s? 0. Therefore, these two Poincare pairs are.homotopy

equivalent, and the lemma is proved.

2.2.6. Remarks i) If c is strictly (n+l)-dimensional, then the

d · · 2 2 1 d 2 2 2 · f · d h h · of C 'n con itions . . an . . are satis ie w en t e images

and D'n+l by (l+T)l)J0 and (l+T)ol)J0 lie in c 1 0 and D'o respectively.

The vanishing of H0 ((l+T)l)J0) will imply that of H0 (pc(l+T)l)J0p~), etc.

ii) If c is free, the above argument can be carried through in the

category of free complexes.

iii) (Relative Splitting) If ~he splitting of the boundary is

already given, then we can modify the construction so that the

result has the given splitting of the boundary.

2.3. Splitting lemma for geometric quadratic Poincare complexes

and pairs

Let (C,l)J) be a strictly n-dimensional geometric quadratic

Poincare complex on X with radius less than cS, and let Y be a sub-

set of X. We would ]ike to use 2.2.4 to split (C,ljJ) into two

parts, one lying over Y and the other mostly over X - Y, whose

common boundary lies along the border between Y and X - Y:

C:

X:

40

C'-C"

/

1-

I I \

---~

'. --~·-----.. '------... ______ /

y x - y

! /

/

As C', we can use the following subcomplex of C:

c ~ = c .1 y-i8. l l

Then C11 = C.C/C') n-"~ 11· es X Y-n8 2 2 4 · · 1 over - ; . . gives an equ1va ence

with radius 0 from (C,~) to a union such that everything has radius

less than 28. The only defect is that the common boundary B of 2.2.4

may lie all over X. It is easy to cut off the portion of B lying

over Y-n8 : Bis chain equivalent to s- 1C(pc(l+T)w0Pe>· Next notice

that Pc(l+T)~0 p~ is a chain equivalence over X - Y8 . This produces 8 I

a "chain contraction" over X - Y for some 8 1 > 0. We wiJl use

this, trying to eliminate the portion lying over 8 I x - y

We begin with defining "chain contraction" mentioned above.

Let (C,d) be a geometric R-module chain complex on X, and Y be a

subset of X.

2.3.l. Definition

41

A collection {s.: C. 1 1

Ci+l} of R-module

homomorphisms is called a chain contraction of C over Y if

d.+1s. + s. 1d. = 1 1 1 1- 1

on Y.

A chain contraction {si} over Y has radius less than E if each si

has radius less than i=;.

Roughly speaking. if there is a chain contraction of C over

Y~ we can do the "localized folding argument" over Y to eliminate

most of the portion of C over Y. See. [Co §14] for the standard

folding argument.

2.3,2. Lemm~ If a geometric R-module chain complex (_C,d)_ on X with

radius less than E has a chain contraction over Y with radius less

than ~ and satisfies the following:

for Si, < k,

then C is 2E chain equivalent to a geometric R-module chain complex

(C',d'l on X with radius less than E such that

0 for Si, < k,

ii) C' = C on X - Y, and

iii) -2' there exists a chain contraction of C' over Y -

with radius less th~n 3E.

Proof: Let {s.} denote the chain contraction of Cover Y, and let 1

42

i, j, p, q denote the following canonical inclusion maps and

projection maps:

i q

CkjX-Y-E > -s <:- ck CkjY .

p j

Notice the identities dk+lskj j and dkj = 0. Define (C',d')

I -s as in Diagram l. Tis the trivial complex with Tk+2 = Tk+l =Ck Y ,

T Q, = 0 otherwise, and dT = l: Tk+2 -7 Tk+ 1; and T' is the de-

suspension of T. There are obvious chain equivalences f: C ----7>

C a:i T and h : C'@ T' ---'?" C'. Define a chain map g

c I EB TI by

g = l 9,

if 9, i- k+]

Since gk+l can be decomposed as

g is an t:-isomorphism. Composing f, g, and h, we obtain a 2t:

chain equivalence between C and C'. A desired chain contraction

{s~} of C1 over Y-s is defined by s~

t (sk+l

9, ~ k+3.

0 for 9, ~ k, sk+l

S for Q,

dk+3 dk+2 dk+l dk dk-1

C: · · · --) ck+3 ck+2 .--~ ck+l ck ck-1 .,

f l '{ -1 f

dk+3 dk+2 dk+l dk dk-1 C fli T: · · ~ ck+3 ck+2 ck+l

/,

ck -----7' ck-1

C' $ T'

C'

... --/ ck+3

0

dk+3 dk+2 dki

------? ck+2 /-ck+l

$ If) . / SkJ

pdk+l -E -------:;.. Ck-1 ?' ckjx-Y ~

'11 0

/ c I -E k y ckjy-E __ ~ ckjY-E

I -1 1

dk+3 hl dk+21 h pdk+l dki

· · · --> ck+3 --------? ck+2 -E

~ ck+l -----7 ckjx-Y ----T ck-1

0 (})

c I -E k y

skj

Diagram 1.

~ .!'-

45

Using this lemma repeatedly~ we get the following:

2.3.3. Corollary Fix a positive integer n. For any E > 0, there

exists 6 > 0 such that for any strictly a-dimensional geometric

R-module chain complex C on X with radius less than c5 which has a

chain contraction over Y with radius less than 6. there exists a

strictly n-dimensional geometric R-module chain complex C' on X

with radius less than E satisfying

= C' [Y-E n-2 0,

ii) C = C' on X ~ Y,

iii) there exists an R-module homomorphism s

with radius less than E such that

d s = 1 and sd = 1 n n

iv) C is E chain equivalent to C'.

-E on Y

C' -- C' n-1 n

It is in general impossible to finish this elimination, but

if we "stabilize" everything, we can avoid the difficulty as follows.

(We will use a special case of this which corresponds to taking

the product with JR.) Let (E,dE) be a strictly 1-dimensional

geometric Zl-module chain complex on a metric space Z such that

E0 E1 = Zl[P], where P is a set of points of Z. Then C' 0 E

is a strictly (n+l)-dimensional geometric R-modu1e chain complex

with radius less than t on X x Z (we measure the radius after

projecting points tQ Xl:

c I 13 E n 0

46

C' •'il E n-1 0

0-----'? C' ~ E n 1

$ -------------7 IP

C' n-1 ~ El

n 0 ( (-) I 0 dE) ( dn l

d 0) 1 0 n .

and (C' 8 E). [Y-E x Z l

0 if i ~ n-2. The maps

(s'. 0 1) $ (s'. 1 0 1): (C'. ~Ea) ~ (C'. 1 ®El) l l- l l-

where s' = s s'. = 0 if i 1 n-1, define a chain contraction of n-1 ' i

C' ~ E over Y~E x Z with radius less than E. Use 2.3.2 to get a

strictly (n+l)-dimensional geometric R-module chain complex C on

X x Z with radius less than E such that

C. [Y- 2E x Z = 0 if i ~ n-1 l

the boundary map d:Cn+l ------7" Cn is given by the following matrix:

n

d = \ (-) 1 © dE

d ~ 1 n

: (C' @ ?l[P]) ~ (C' JY- 2E 0?Z[P] ) ~cc·~ ?l[P]) ~ (C' l~ ?l[PJ), n n-1 n n-

where j is the inclusion map of C~_ 1 [Y- 2 E into C~-l Now consider

47

the following E-isomorphism of Cn+l to itself:

h =C -sj

-2s where q is the project ion map of C~-l onto C~_ 1 J Y -. Then on

y-3E x Z, h = d. If we replace the boundary map d: Cn+l -7 Cn

-1 by dh , we get a new geometric R-module chain complex C', which

-1 is E-isomorphic to C and the boundary map dh is the identity on

y-3E x z. Now we can delete c~+1ly-4 E x z and c~1y-4 E x z from

C'. Thus we obtain the following:

2.3.4. LeJTUUa Fix n and E. For any s > 0, there exists c5 > 0 such

that for any strictly n-dimensional geometric R-module chain complex

C on X with radius less than c5 which has a chain contraction over Y

with radius less than 6, there exists a strictly (n+l)-dimensional

geometric R-module chain complex C on X x Z with radius less than

E (measured in X) satisfying

i) C lies over (X - y-E) x Z,

ii) C ® E = C on (X - Y) x Z,

iii) C 3 E is E chain equivalent to C,

In our application of this lemma for splitting, we will use

the following complex as E. Consider a geometric ?l-module ?l[?ZJ

48

on m1 which is generated by all the integers, and a ~-module

isomorphism t: Zl [ZZ] -~ Zl (~j defined by

t ( [n) ) = [n+ l] .

Define a strictly 1-dimensional geometric ZZ-module chain complex

E by

E = E 0 1 zz [zz]

d = t-1 : El

Although E is not finitely generated, we define a symmetric

structure ¢ on E as shown below .

. l ... -" E :

0

E

For s ~ 2, ¢ s

0

l 0

0 -1

0 =t

-1 o; d~~ = t 1. Obviously ¢0 is an isomorphism,

thus (E,¢) is a I-dimensional "symmetric" Poincare complex.

Now we state the stable splitting lemma for geometric

quadratic Poincare complexes and pairs.

2.3.5. Lemma Fix n. For any E > 0, there exists 8 > 0 such that

for any strictly n-dimensional geometric quadratic Poincare

complex (C,~) on X with radius less than 6, and a subset Y of X,

there exist two strictly n-dimensional quadratic Poincare pairs

49

on X with radii less than s:

( B. ') c I ' ( () \jJ I , -ti;) )

CB --· ~ c". c ol/J 11, zµ))

such that

i) -E C' (resp. C") lies over Y (resp. X - Y ),

ii) C is homotopy equivalent to their union along B,

iii) (B,l/J) 0 (E.¢) is s homotopy equivalent to a strictly

n-dimensional geometric quadratic Poincare complex on X x IB.

which lies over (YE - y-E) x IB..

A strictly (n+l)-dimensional geometric quadratic Poincare pair

on X with radius less than c5 has a similar splitting into two

geometric triads.

Proof: If (f:C------7D. (c5\jJ,1jJ)) is the given pair, define sub-

complexes C', D' of C, Das follows:

C' c.1Y-(i+l)c5 i l

Then f(C'.) SD'.. The conditions 2.2. land 2.2.2 are satisfied l l

because (l+T)\jJ0 (c' 0 ) s_ c0 and (l+T)olj!0 (D'n+l) s.. n0. It is easy I o I 0 I

to construct chain contractions of B and c· over Y-o \./ (X-Y ) I

for some c5', and we can apply Lemma 2.3.4 to B and c·. The proof

for complexes is similar.

so

We will call (B,ljJ) ® (~,\jJ) the external suspension of (B, \jJ)

and denote it by E(B, \jJ).

2.4. Glueing ~nd splitting over~ triangulation ~f ~manifold

The notion of pairs(= 2-ads) and triads(= 3-ads) naturally

extends to "(k+2)-ads." ([R4J) These are defined inductively as

follows:

(l) a (k+2)-ad x is a collection of

i) k+l (k+l)-ads 30x, ... , (lkx satisfying

d. Cl .x J l

if 0 ~ i ~ j ~ k,

ii) a pair Cl .x ~ II x ll (Ii x It is the underlying l

chain complex of x), and

(2) suppose there are k+2 (k+2)-ads a0y, ... , 3k+ly which

satisfy (>'<), then a0y \J ••• v 3iy are defined inductively

(on i) using the glueing method of §2.1.

Let x be a (k+2)-ad. When i 1 < ••• <in, we define Cl{. . }x 1v l1·····l,Q,

by

An a-dimensional quadratic Poincare (k+2)-ad x is a (k+2)-ad

x together with structure maps

(\jJ) : llCl X\ln-1 al-r-s --~ Ii Cl xjj, a s Cl a ac{O,l, ... ,k}

such that

51

(1) 3.x is an (n-1)-dimensional quadratic Poincare (k+l)-ad l

for each i, and

(2) ( va.x ~11x11, (tj; ,vtj;.)) is an n-dimensional quadratic l ¢ l

Poincare pair.

Here [al denotes the size of the subset c;i,. and l/tjJ. is defined using l

the glueing operation of §2.1 repeatedly,

Let X be a metric space. If we use geometric chain complexes

on X, we can define geometric quadr~ic Poincare n-ads _9E_ X.

Such a thing has radius E(resp. support CS X) if all the involved

chain complexes, chain maps. chain homotopies, and the possible

compositions of maps have radii E(resp. support C). An n-ad x

is called special if 3{ }x = 0. Recall that an "n-ad" 0,1, ... ,n-2

in the usual sense is a topological space together with n-1 subsets.

We ca 11 this a -~-opo logical n-ad to distinguish it from n-ads of

chain complexes. Let (X, c\,~X) be a topological n-ad of metric space.

A geometr:_ic quadratic Poincare n-ad x on (X, c\~X) is a geometric

quadratic Poincare n-ad on X such that 3.x lies over 3.X. Let l l

(6, 3,"6) be A topological (n+2)-ad induced by an a-simplex. An

(n+2)-ad on (6, 3,"6) is automatically special.

As an applic2tion of previous sections, let us consider the

following problem. Let M be an a-dimensional compact manifold

with a triangulation K, and fix a metric on M. Suppose each

a-simplex 6 of K is given a geometric m-dimensional quadratic

Poincare (n+2)-ad x on (6, 3,"6) such that

52

n-1 2.4.l. (compatibility) if, for an (n-1)-simplex (J of K, (J.(J . 1.

We would like to glue all these Cn+2)-ads to get a geometric

quadratic pair on (M~ (JM). We can also consider a problem of the

inverse direction. Notice that there is a small difficulty. For

example~ consider an annulus (Fig. 1 (i)) with 2n 2-simplices

which are. given geometric quadratic Poincare special 4-ad

structures. Let us try to glue these together inductively in the

order as shown in Fig.l(ii). This turns out to be a wrong order.

If we try to glue the (2n-l)-st 2-simplex, then we have to do so

along the union of a l~simplex and a a-simplex, which is not

codimension one, But if we glue (1) and (2), (3). and (4), ..• ,

(2n-l) and (2n) respectively, then we. have n 2-cells such that

any two 2-cells are disjoint or meet along a codimension one cell.

Nqw we can glue these in any order, This is the basic idea.. First

we glue locally so that the local blocks behave nicely, and then

we glue the blocks (in any order). When we split something~ we

first split it into several blocks so that each block is a union

of simp1ices in some nice way, and we split each block into pieces.

Let M b.e an n-dimensional compact manifold with a triangulation

K. Assume that K is the first barycentric subdivision of another

triangulation L. For each vertex v of L, its star S(v) in K, or

a dual cone, is the local block mentioned above. Two such dual

53

(i)

------- ------(2.) -=? . '

.. . ---~.:;;-

( _)__ Y\ - I ) --ci-42. - simplex (ii)

Figure 1. Glueing on an annulus.

54

cones are either disjoint or meet along codimension 1 cell(s).

The glueing and splitting problem over S(v) can be solved by

looking at the link L(v) of v in K. Note that L(v) is an (n-1)-

dimensional sphere and the triangulation is the first barycentric

subdivision of another. Thus we can keep on reducing the dimension

until the link becomes a circle as above, and in this case there

is a natural order of 2-simplices and glueing and splitting can be

done. Thus we have

2.4.2. Theorem (Glueing over a manifold) Let K be the first

barycentric subdivision of a triangulation of a compact n-dimen~

sional manifold with a metric, and suppose each n-simplex 6 is

given an m-dimensional geometric Poincare special (n+2)-ad on

(6, 3'1<6) which are compatible on common faces (in the sense of

2.4.1). Then one can glue them together to get a geometric

(I ) I ....

55

quadratic Poincare pair on K_ such that it$ boundary lies over

3K.

2.4.3. Theorem (Stable splitting over a manifold) Let us fix an

integer m, and let K be the first barycentric subdivision of a

triangulation of a compact manifold M with a metric. Let E be any

positive number. Then there exists o > 0 such that any geometric

rn-dimensional quadratic Poincare pair with radius < o on (M, 3M)

can be stably split into pieces each of which lies in an E-neighbor-

hood of the corresponding simplex of K.

2.4.4. Remarks 1) Until now we have considered chain complexes

with constant coefficients. The result can be extended to the

case arising from stratified systems of fibrations (1.2.2 and

1.2.3), if the filtration is compatible with the triangulation and

each simplex is sufficiently small.

2) If a splitting of the boundary is already given, then the

result has the given splitting of the boundary.

3. Surgery space and as.sembly

3.1. Surgery spaces

In 3.1 and 3.2, we will immitate what Quinn did in[Q3, §SJ.

Fix a stratified system of fibrations p: E ---7 X, an integer n,

and a covering space E of E. A primitive k-simplex x (of degree

n) is a strictly (n+k+Q,)-dimensional geometric _quadratic Poincare

special (k+2)-ad on pr: E x JRQ, ----7 X, where r is the obvious

projection E x JRQ, -7 E. Here we are using E x JR.9' as the fixed

covering space of E xJR.R,. We always assume that x has bounded radius

when projected to 1R.R, and that it is locally finitely generated on

X xJRQ,. If x has radius E and support C, then its faces 30x, ... ,

<\x are primitive (k-1)-simplices with radius E and support C.

For a primitive k-simplex of degree n with radius C an·:l

radius E, we have the following operations.

(1) Reduction Suppose C' <;,; C is compact, and E' ?- E. Then x

can be regarded as a primitive k-simplex of degree n with

support C' and radius E 1 • This is called a reduction of x.

( 2) S . Th 1 . . ._,.R, . uspension e externa suspension operation w =

L: x ... x L: defined in §2.3 can be naturally extended to an

operation on primitive simplices. Q, The result L: x has the

same support and radius as x.

We define the space of quadratic Poincare ads.

56

57

3.1, J. Deftnition JP n (X; p; E) is the 6-set with simplices (which

will be cal.led ela,borate simplices) defined inductively: an

elaborate 0-simplex is a primitive 0.,-simplex of degree n, i.e.,

a strictly Cn+,Q,)-dimensional geometric quadratic Poincare special

,Q, . 2-ad (= complex) on pr: E x IR ~ X for some ,Q,( with unrestricted

compact support and radius). An elaborate k-simplex o consists of an

underlying primitive k-simplex Joi of degree n, together with k+l

elaborate (k-1)-siwplices a0o, ... , ako. We require these to satisfy

the usual a.a. identities, and in addition require that the l J

external suspension of a reduction of the underlying primitive

(k-1 )-simplex I 3 .o J of Cl .o be different from the i-th face Cl. j o J . l l l

of the underlying primitive k-simplex loJ only in the structure

mcips, And that the structure maps be homcilogous by a chain whose

radius is less than that of jo[. We define the support and radius

of an elaborate simplex o to be those of Joi.

Suppose C S X is compact and E > 0, then JP (X,C,p,c;E) is n .

the subset of JP (X;p;E) made up of all the simplices with support n

containing C and radius not exceeding E. If E is the universal

cover of E. then we omit E in these notations and write JP (X;p) n

and JP (X,C.p,E) respectively. We introduce a control of radii n ..

and supports in the following definition,

3.1.2. Definition Suppose Xis 1oc;::i1ly compact metric. p: E ·---?

58

X is. a stratified system of fibrations, and E is a covering space

of E. Then a k-simplex of 1L (X;p; E) is defined to be a simplicial n k map 6 x [O.,co) ~ JP n (X; p; E) which satisfies the fo.llow:ing

condition: there are a sequence of compact sets C. ~ X with C. l l

X, and a sequence E. of numbers monotone decreasing l

to 0, such that the image of /'ik x(i,co) lies in JP (X,C.,p,s.;E). n i · l

If E is the universal cover of E. we write 1L (X;p) instead of , n

JL (X;p;E). n

Since we are interested in JP (X;p) and JL (X;p), we only deal n n

with these in the rest of this section. The general E will appear

in the definition of a homology spectrum in §3.2.

First of all, JP(X;p) (we omit the subscript n) satisfies

the Kan condition. Consider for example two elaborate 1-simplices

o0 • o1 of 1P(X:p) with a0o 0 = a0o 1 . The Kan condition asserts

that there is an elaborate 2-simplex T with 30T = o0 , a11 = o 1 .

We construct such T as fol lows. Suspend I Oo I ' I 0 i l , and I aooo I if necessary so that we can make the union [o0 ! U [o 1 ! along

Ja 0o0 !. Consider the trivial cobordism x between Jo0 ! v [o 1 I and itself. Then T is given by (x;o0 .o 1,o0 \./ o1), where o0 \J o1

denotes the elaborate I-simplex ( [o0 [ V [o 1 i: a1a 0 , 31o1) ·

The same argument works for any simplicial map /\ ~ 1P (X;p) . . n .

Thus JP(X:p) satisfies the Kan condition.

59

3.1.3. Remark The same technique will be used quite often in this

chapter. Stippose we have a S~ad (x,a0x, ,,. ,8 3x);

Then considering the trivial cobordism t of 30x t: 31x to itself,

we can triangulate this as follows:

dl x ---1,,-/_aox t ~"----- ---

-----~ Cl 2x - x

·--·----

This will be called the "triangulation argull'._~~s_."

The next result describes the spectrum structure.

3.2.4. Theorem Let p be as in 3.1.2. Then there is a natural

homotopy equivalence T: 0, ILn(X;p) ---> ILn+l (X;p) ·

Proof: A k-simplex of 0, 1L (X;p) is a 6-map o: 6k x (0, 00) x I -.;> n

1P (X;p). n

We define TO: L'ik x [0, 00 ) --~ 1Pn+l(X;p) as follows.

Let T be an m-simplex of 6k x [0, 00). O(T x I) consists of several

simplices of 1P (X;p). Since o(T x 0) = o(T x 1) = 0, if we glue n

60

these together After some necessary suspensions, we can regard

O(T XL) to b.e an ro-simplex of JPn+l(:X;p), and this defines a

k-simp lex To of JLn+l (X; P) ·

We will show that T is a homqtopy equivalence. First of all,

each 0-simplex of JP n+l (X;p) can be 'naturally regarded as a

1-simplex of JP (X;p) with two 0-faces. A 0-simplex o: 1-0,00) --7 n - ~

JP n+l of JLn+l (X;p) can be expressed as· in the following picture.

0 0 (j)_ ---------------- 0 -- .... -------------- 0 -----

o(O) o( [O, lJ) o(.l) o([l, 2]) 0(2)

0 ---------------- 0. ---------------- 0 -----0 0

By inserting trivial cobordisms we can "triangulate" this (3.1.3):

0 0 ---------------- 0 0 ---------------- 0

o(O) ,,//,/_,,-··

/ o(O)

,/

,,./' _,//

a (1 ) /,;.-ci( 1 ) a ( 2)

o(l)

o(O)

o( 0,1 ) /// o( 1,2 ) / 0 ---------------- 0 ---------------- 0 -----

0 0

This defines a 0-simplex L0, 00) x I ------? IP (X;p) of ~ 1L (X;p). n n

If we apply T to this, then the result is different from the

original by trivial cobordisms; therefore these two can be connected

by a 1-simplex of JLn+l(X;p); i.e., T maps into every component.

Next consider an element of the relative homotopy TI.(T). It J

61

represented by a map o: 6j x [0, 00) ------7 JPn+l(X;p) such that . . . -1

ol3.6J = 0 for i < j, and ol3.6J = Tp for some p: 6J x [O,oo) x I 1 J

--7' JP (X;p) n We need a deformation of a rel Cl.CT to a map in the J

ima.ge of T. An extension p': 6j x[O,oo) x I-- JP (X;p) of p n

can be defined by first letting p'(T) = 0 for Tin ( LJ 3.6j) x .. 1 1<]

[Q,cll) x IV 6j x [0,oo) X {0,1} and then using the triangulation

argument. The natural cobordism will give a simplex connecting

a and Tp'.

3.2. Homology theory and assembly map

Let (F, Cl~F) = (F, 30F, ... , Cl.F) be a topological (j+2)-ad. " J

We consider only special ones; i.e., we assume,"\ Cl.F = ¢. Fix 1

a covering space F of F.

3SF' for S ~ {0,1, ... j}.

ClSF denotes the restriction of F over

We construct a 6-set IL (F,Cl~F;F) n "

modifying the construction of JP . n

If j = 0, TI.., (F;F) n

ILn(F,¢;F) is defined to be JPn(i<,7<,F-7 ~~,O;F), where~'< denotes

a single point. If j > O,a primitive k-simplex x for IL (F,Cl~F;F) n "

is a strictly (n+k+j+£.)-dimensional geometric quadratic Poincare

(k+j+3)-ad on F x JR£, --? >'< such that

(1) it has the same structure as 6k x 6j; i.e., the faces

have two indices: 3 Qx, a~ {0,1, ... ,k}, B ~ {0,1, ... ,j}, and a,µ

3 Qx = 0 if a = { 0, 1 , ... , k} or S = { 0, 1, ... , j } , a,µ

c2) a Qx is actually a Cn+k+j+£.-!al-IS!)-dimensiona1 geometric a,µ £,

quadratic Poincare (k+j-jaj-jsj+3)-ad on ClSF x JR ----7 * and

62

(3) everything has radius 0 and support *·

The i-th face of a primitive k-simplex x is defined by using the

first index, for i= 0, 1, ... , k. The elaborate simplices of

IL (F,o*F;F) are defined by allowing stabilization and reduction n ,

in the identification of faces as before.

For each i = 0,1, ... ,j, we have a natural map o.: n.. (F,oJF;F) 1 n •(

,...__ -?- n.. (o.F,a*a.F;o.F) by taking the i-th face with respect to the

n 1 • 1 1

second index.

The next step is to do the above construction to each simplex

of a simplicial complex and fit them together using the second

index to form a "bundle of spectra." Let us consider a stratified

system of fibrations p: E ----7-X = IKI, where K is a simplicial

complex. For each j-simplex 6 of K, we have a topological (j+2)-

-l -1 ad (p (6), p (3*6)). Let E be the universal cover of E and let

p-1(6)- be the restriction of E to p-1(6). Now apply n.. ( , ; ) n

to this for each 6, then it defines a 62-set (= 6-(6-set)) n.. (p): n

IL (p)(j,k) n

-1 -1 -1 -{IL (p (6) ,p (3 .... 6) ;p (6) )(k) j 6 is a j-simplex of K}. n "

denote the geometric

realizations. We define the geometric realization j n..n(p) j

I n.. (p) I = LL n

63

where the equivalence relation is generated by: (a,t) <= I ]L (p-1(6), n

- 1 -l - I I I I -l p ('d*6);p (6) ) x 6 is equivalent to ( 'di a,t) E- ]Ln(p ('di6),

-1 . -1 - I p ( a ... 'd. 6) ; p ( 'd. 6) ) x 'd. 6, provided t eo- 'd. 6 < 6. We have a ~ 1 1 1 1

projection map p*: IILn(p)I ~ !Kl defined by p*(a,t) = t. p*

has a natural zero-section i: IK I ----7- I ]Ln (p) I. I ]Ln (p) I has a

triangulation obtained by assembling the standard product

triangulation of 6k x 6j.

-1 -1 Notice that we can regard a k-simplex of ]L (p (6),p ('d.6); -n ~

-1 - -1 -1 -1 -p (6) ) as a (k+l)-simplex of ]L-n-l(p (6),p ('d*6);p (6) ),

by adding a trivial face as follows.

b ---·-· -·----] ---- c. x

---------------- d

\

0

- ()------- c I> ~()

- l) I I

This correspondence can be realized as a map from the reduced

I -1 -1 -1 - I suspension of ]L_n(p (6),p ('d*6);p (6) ) x 6/i(6) to

I -1 -1 -1 - I ]L-n-l(p (6),p ('d*6);p (6) ) x 6/i(6):

·'f.

>

(Two *'s are identified in each picture.)

64

This assembles to a map from the reduced suspension of \IL (p)l/i(X) -n

to I JL._n_ 1(p) \/i(X), and it giv~s a well-defined map I JL_n(p) \/i(X)

-----;;"" SI (I IL-n-l (p) I /i(X)). Taking nn-j of thi's h "" we ave a map

IL_,~(p) is Quinn's "ex-spectrum" LQ3, §8].

3.2.1. Definition The homology spectrum JH(X; JL.(p)) is the

SI-spectrum defined by JH(X; JL(p)) = lim Sin(\ lI.. _n (p) j /i(X)). n700

3.2.2. Proposition The functor JL.( , ; ) which was used to

construct JH is homotopy invariant; i.e., 1) a homotopy equivalence

homotopy equivalence E ~F induces a homotopy equivalence

JL(E, 3*E;E) -=> JL(F, o*F;F), and 2) 3i: JL(F -F x !;,n) -----7 lI..(F x oi6n,F x o*oi6n;F x oi6n) is a homotopy equiva-

lence.

Proof: (1) By assumption, they use the same coefficients;

therefore the result is obvious.

(2) The coefficients are constant.

is defined as shown below:

The homotopy inverse of o. l

65

Aj k Use the trivial cobordism to fill in u x 6 .

also give the necessary homotopy.

The same trick will

According to Quinn[Q3, §8], this implies that IB( ;1L(p))

is a homology theory on the category of polyhedra with stratified

systems of fibrations with polyhedral fibrations.

Let us think about triangulation of ! 1L_n(p) J for a moment.

Each of its building blocks 6k x 6j is given a structure of

quadratic Poincare (k+j+3)-ad of dimension (n+k+j+9.,) on

P-l ( ") x IB.9., ____,,. -,',·. h . 1 . 3 1 3 u · , By t e triangu ation argument . . , we may k .

assume that each m-simplex c in a triangulation of 6 x 6J is

given a structure of quadratic Poincare special (m+2)-ad of

dimension (n+m+£) -1 on p ( p,._o) (1his is possible

because p is a fibration over the interior of each simplex of K.)

Replacing this map by -1 9., p (p,._o) x IB. ----7 P,.p, we can regard it to

be a structure on p x IB.9., = pr : E x IB.9., --7 X with radius :;::

diameter of p,._o.

66

Using this, let us define A.: J

JH. (X; JL(p)) ---7 1L . (X;p). J -J

A k-simplex of rln-j (I 1L_n (p) I /i(X)) n-j k is a map p: S x 6 ---7-

\ 1L _n (p) I /i(X). By modifying p a little if necessary, we may

assume that there exist a compact codimension 0 submanifold V

n-j k I of S x 6 and a map p': V --7' \ 1L_n(p) such that p sends the

complement of int V to the base point [i(X)] and olV factors

through p'. For each (n-j+k)-simplex 6 of V, p'(6) is given a

structure of (-j+k+£)-dimensional quadratic Poincare special

(n-j+k+2)-ad on p x JR.£ with radius= the diameter of p*p'(6).

Glueing all these , after a barycentric subdivision if necessary,

we obtain a (-j+k+£)-dimensional quadratic Poincare special

(k+2)-ad on p x JR.£ with radius

which is a simplex of JP . (X; p). -J

max (diameter of p_,_p' (6)), 6'=V ~

If we use finer triangulations

(e.g. barycentric subdivisions) of V and in-, (p) \, the radius of -n

the result becomes smaller, and it differs from the original

result by several inserted trivial cobordisms and a small change of

the positions of some of the generators of modules. Therefore

there is a homotopy from the simplex in 1P corresponding to the

original triangulation to the one corresponding to the subdivision.

Consider this as 6k x [t,t+lJ ---7 JP .(X;p). Repeated application -J

k of this generates a map 6 x [0, 00) --7 JP . (X;p). Since the -J

radius goes to 0 as t ~ oo, this defines a simplex of 1L . (X;p). -J

This is a very elaborate version of the "assembly map" defined

by Quinn in his thesis [QlJ.

67

The following is the main theorem of this chapter.

3.2.2. Theorem (Characterization theorem) Let p: E ~X be a

stratified system of fibrations, where X is a finite polyhedron.

Then the assembly map A.: JH. (X; Il..(p)) --7" Il.. . (X;p) is a homotopy J J -J

equivalence.

Proof: First fix some notations. Since X is compact, we will

assume that everything has support X in the following proof.

Embed X in Sn-j for a sufficiently large n, and let W and r denote

a regular neighborhood of X in Sn-j and the retraction: W ~ X.

Let p-: E ~W be the stratified system of fibrations obtained

as the pull-back of p by r. p- has the advantage that the base

space is a manifold.

We will show that A. J

maps into every component. Let o be a

0-simp lex of Il.. . (X; p) ; o -J

is a map from [O,oo) to

a sequence E. monotone decreasing to 0 such that l.

IP .(X;p) -J

o( [i,oo))

with

IP .(X,p,E.) = IP .(X,X,p,E.). o(O) can be regarded as an (£-j)--J l. -J l.

dimensional quadratic complex on p- x m£ with radius Eo· Try to

split o(O) into pieces lying over the simplices of W(l), the

first barycentric subdivision of W. Choose E which is sufficiently

small compared with the size of any simplex of 1/l). The stable

splitting theorem (§2.4) gives o >0 such that if o(O) has radius

less than o, then each piece lies over an E-neighborhood of the

68

d .. 1 fW(l) correspon ing simp ex o . We may assume that this is the

case. If E0 ~ o, then choose i such that Ei< o and use another

0-simplex o' defined by o'(t) o(t+i), which is in the same

component as o. Construct a map of p to itself with radius E which

sends p--l(E neighborhood of 6) to p--l(6) for each simplex 6( W(l).

This map induces an E isomorphism which makes each piece lie exactly

over the corresponding simplex of W. There are only trivial ads

(i.e., ads made up of zero complexes) over the simplices in 3W;

therefore this gives a 0-simplex p of ~n-jCI IL (p) l/i(X) after -n a suitable "subdivision." We will show that A. sends p into the

J

same component as o. First of all, if we glue all the ads on

simplices of (a subdivision of) W(l) together, then we get a

quadratic Poincare complex (A.p)(O) which is cobordant to o(O). J

o(O)

r L

(A.P)(O) J

This is the first step of the inductive construction of a 1-

simplex T connecting a and A.p. Assume we have filled in J

2-simplices of W .(X;p) up to£ as follows: -J

a

A.p J

T(O)

a (O)

(A.p)(O) J

69

a ( £)

T ( 9,)

(A. p) (9;) J

0(9,+l)

(A.p)(Q,+l) _J

A.p J

The next step is obtained by applying the "barycentric subdivision"

shrinking argument to the union off£,£+1]) v T(£) 1.. A.p([£,£+1J) J

and then applying the triangulation argument to the resulted

cobordism.

o[£,£+1J o(£+1) / ----------·- ·-- - . ./-

/ ---------- --/ ----- ------/ - -

-,,'--- --

----1) T(Q,) -_ -~;~~--~:~~~1 ~_:_-:::-:-_::__~.:..=-=·~d

T ( 9,) shrunk

A.p[Q,,9,+lJ J

Thus A. maps into every component. J

Next we will show that the relative homotopy group Tik(Aj)

vanishes fork > 1. The idea is basically same as the first part.

k Its element is represented by a map o: 6 x O,co) -- IP .(X;p) -J

A.p for some p: J

sn-j x 6k-l --7 In.. (p) [/i(X) -n

such that p(Sn-j x Cl6k-l) = [i(X)j.

70

Let P': V ----;;- I lL (p) I be the map defined on a codimension 0 -n

submanifold of Sn-j x 6k-l associated with P. (See the definition

of A .• ) J

n-j We may assume X and W(c S ) are subcomplexes of V

(c.sn-j x 6k-l) via sn-j = sn-j x (center of 6k-l) c sn-j X Ak-1 Ll '

by using subdivisions if necessary. When we defined A., we J

!l associated a (n-j+k-1)-ad on p x JR to each (n-j+k-1)-simplex 6

of V. But notice that it can be also regarded to be over the

simplex 6, choosing the centers of simplices as the basis

elements. Similarly "subdivided" small thing,p-, can be also

realized on V. The map p*p': V ----7'X puts them back into the

original things over X. Now we replace p by PA for which

everything lies on X (and hence on W), via homotopy. The following

diagram shows the construction.

p p PA (not so small) (very small) (very small)

v

1 2

n-j k-1 t On S x 6 x O,lJ we apply the triangulation argument to

the cobordism between the original and the subdivided things.

71

On Sn-j x t,k-l x (1,2], we immerse the mapping cylinder of p~~p'

whose cylindrical part is a composition of very thin cobordisms.

P and this cylinder should be chosen very small so that we can

do splitting here.

Thus, from the beginning, we may assume that p is lying over

X. Now apply the relative version of splitting lemma to the

pair induced by o over W, then we can obtain a k-simplex of

~n-j<I 1L_n(p) j/i(X)), p0 , as follows:

Do the same argument as in the first part to obtain a homotopy

which pushes o down to Ajp0 . This completes the proof.

The proof actually gives the following:

3.2.3. Corollary (Shrinking lemma) Let p be as in 3.2.2. Then,

for any positive integer n, there exists a positive number E such

72

that for any 0 < 6 < s, there is a function("shrinking function")

s: (n) 1P. (X,p,6) ---r IL. (X;p) such that the following composition is J . J

homotopic to the inclusion map:

. (n) 1P.(X,p,6)

J

S restriction _ __,,> IL. (X;p)

J 1P.(X,p,6),

J

( f.)(n) 6 where 1P. X, p. u denotes the subset of JP. (X, p, ) which J . J

is made up of ads of dimension less than or equal to n.

Now let us consider a special case when Xis a single point *·

In this case, IL. (1~; E -7 7~) is homotopy equivalent to 1P. (*, E ~ *, O) J J

= IL.(E). J

3.2.4. Proposition There is a natural isomorphism

e -00 1 (TI) n

where B is the classifying space of the group TI. TI

1 -··1 is the limit lim 1 (-j) of Ranicki' s lower 1-groups n n

j? [Rll. 1(-j)(TI) is defined to be the kernel of the product of

n

projection maps

(1) j 1n+j+l (TI x T ) ,

where T denotes the infinite cyclic group. The map 1(-j)---? n

L(-j-l) is induced by the map n

(1) ( x Tj+l) Ln+j+l TI

73

which corresponds to taking product with s1 .

Notice that the external suspension operation introduced in

§ 2. 3 actually, changes a geometric ?Zn-module chain complex C on X

into a geometric 7l[TI x T]-module chain complex C ~ E. Here we

are regarding E to be corresponding to s1 So if C represents

an element of L(l)(TI), then C ® E represents an element of n

Let us project this to (1) Ln+l (TI). Then the result is

C © E(E), where E(E) is the 72-module synnnetric Poincare complex

1 of S . Obviously E(E) is cobordant to 0, and hence so is

This implies that C ~ E actually represents an element of

c.L~!{<nxT).

C .~ E(E).

L (O) (TI) n

Proof of 3.2.4: First let us define 8. An element of 1(-j)(TI) n

can be represented by a free (n+j+l)-dimensional quadratic

Poincare complex over 7l[TI x Tj+l]. Consider this as a geometric

quadratic Poincare complex on B TI ·+1

x JR] --7 *. Obviously it has

a bounded radius in the JR.j+l coordinate. So we can regard this

as an n-simplex of JL0 (BTI) with empty boundary, and hence as an

1 element of TI n..0 (B ) . (x S ) on the left side corresponds to n TI

(xJR.1) on the right, so the diagram

74

L (-j) (TI) n

-----'>

L (-j-l) (TI) n

commutes, and it defines a homomorphism e of the direct limit.

Next we show that 8 is onto. Take an element of Tin JL0 (BTI)

and represent it by a (?Zn-module) quadratic Poincare complex

on B XJR.j ---7 *for some j. Call the complex C. Split C along TI

BTI x lR.j-l x O; we obtain a decomposition C ~ C+ UBC_, but B may

lie all over B x lR.j. Recall that B ~Eis homotopy equivalent TI

to another complex, say D, which lies in a small neighborhood

of BTI x lR.j-l x 0. We should note that not only B ~ E but also

D can be regarded to be a ?l[n x T]-module quadratic Poincare

complex. Consider the external suspension ID, which is a

?l[TI x r 2J-module quadratic Poincare complex on BTI x lR.j-l. We

claim that IC is cobordant to ID. Note that both IC and ID has

a splitting along D: IC ~ (IC+) V D(IC_) and ID = (l::+D) VD(I_D).

IC $ (-ID) is cobordant to

Notice that (IC+) vD(-I+D) (resp. (IC_) v'D(-I_D) ) lies over

B xlR.j x LO,oo) (resp. B xJR.j x (-oo,O]), then the following lenuna TI TI

implies that these are cobordant to 0, and hence IC is cobordant

75

to ID. 2·

Repeat this process until one gets a 7l [nx T .1]-module

quadratic Poincare complex. By the construction this represents

an element of L(l-2j)(TT) and hence an element of L-00 (TI). n n

3.2.5. Lemma A d . p. ~ 1 B XJR.j ny qua ratic oincare comp ex on TI

which lies over BTI x JR.j-l x [O,oo) is cobordant to zero.

Proof: Let F be such a complex, and let t denote a parallel

·-1 translation of BTI x JRJ x JR defined by t(x,y,z) = (x,y,z+l).

Then

gives the desired cobordism.

Let us go back to the proof of 3.2.4. We will prove the

injectivity of 8. Pick an element x in the kernel of 8. We may

assume that its image is cobordant to O. Apply the same argument

to this cobordism as in the onto part. This will show that x = 0.

This justifies the following notation:

-00 3.2.6. Notation TI ~O(B) = L (B ). n TI n TI

4. Crystallographic groups

4.1. Preliminaries on crystallographic groups

We begin this section by reviewing some work on

crystallographic groups by Farrell and Hsiang in [FH2]. (See

also [Ch], CFJ, and [Wo].) A group r is crystallographic if it

is a discrete co-compact subgroup of E(n), the group of rigid

motions of Euclidean n-space. Identify JR.n with the group of

n n translations of JR, then E(n) =JR ~ O(n). The intersection of

r and ]Rn is the maximal abelian subgroup of r with finite index,

which is denoted by A and is called the translation subgroup of

f. The finite factor group f/A, called the holonomy group of f,

is denoted by G. The rank of r is the rank of A. For any positive

integer s, r = f/sA and A = A/sA. T and T denote the infinite s s n

cyclic group and the finite cyclic group of order n respectively.

4.1.1. Examples (1) D denote the co-dihedral group; i.e., D 00 00

E(l) is the subgroup generated by x i--7 x+l and xt---7 -x (where

X E JR),

(2) See [FH2, ~4] for the definition of 2-dimensional crystallo-

graphic groups of type 1, 2, and 3.

The following is a structure theorem of crystallographic

groups.

76

77

4.1.2. Theorem Let r be a crystallographic group of rank i(~ 2).

Then, either

Cl) r f' ~ T where f' is a crystallographic subgroup of rank

i-1; or

(2) r maps onto some crystallographic group rA of rank m(~ 1)

with holonomy group GA, and there are a crystallographic group

r- of rank n(; 1) and an infinite set of positive integers s each

of which is relatively prime to /GA/ such that if H is a maximal

hyperelementary subgroup of fA and /GA/ divides /HI, then there s

exists a crystallographic group TI together with a group monomor-

phism e: TI ~ rA and a group surj ection n: IT ----7' r- satisfying

-1 e (TI) = q (H')' where H' c. rA is conjugate to H and q: rA --7 rA -- s s

is the canonical projection. Furthermore there exist a 8-equi-

variant bijection g: JR.ID~ JR.ID and an n-equivariant affine

surj ec tion h: JR.ID __ ,,,. JR.n such that

for each tangent vector X of JR.ID.

Proof: This is implicitly proved by Farrell and Hsiang in [FH2].

Their Theorem 1.1 has three cases. Their case (i) is our (1).

In case (ii), there is an epiIDorphisID from f to a non-trivial

crystallographic group IA with holonomy group GA and there is an

infinite sequence of positive integers s - 1 mod /GA/ such that

78

any hyperelementary subgroup of r; which projects .onto GA projects

isomorphically onto GA. The proof of Theorem 4.4 in [FH2] gives

an s-expansive endomorphism e:rA -----7 rA and a 8-equivariant

diffeomorphism g: JRm ~ JRm such that 6(f'') = q- 1(H') where H'

is a subgroup of rA conjugate to H. Therefore let r- = IT = rA, s

n = 1, and h = 1. Then

(l/s) Jxl ~ (2//S) Jx[ ·

In case (iii), f maps onto a 2-dimensional crystallographic group

rA of type 1, 2, or 3, and Theorem 4.2 of [FH2J gives exactly what

we want. D is used as r-. co

We will also need the following result of Farrell and Hsiang.

4. l. 3. Lemma ( [FH2, Lemma 1. 2 J ) Let ¢: f ----;> f A be an epimorphism

between crystallographic groups r c E(£) and rA c: E(m). Then

there exists a ¢-equivariant affine surjection F: JR'-' ----7" JRm.

Let us consider the action on JR£ of a crystallographic

group r of rank £, with holonomy group G. The action may not be

free, since r may have torsion, but its translation subgroup A

acts on JR£ freely and the orbit space is the flat torus T£

£ The finite group G acts on T as a group of isometries such that

JR£/f = T£/G, where T£ is given the natural induced metric.

79

Therefore we can use the argument for a finite group action on a

smooth manifold. If (H) is the conjugacy class of the subgroup H

of G, then Y(H) denotes the subset of T9'/G consisting of the points

x such that the isotropy subgroup of a point in Ti lying in the

orbit xis in (H). {Y(H)} gives a stratification. Enumerate the

conjugacy classes of subgroups of G: (H0), (H 1), ... , (HN), such

that

(1) HO G,

(2) ~ 1,

(3) if (H.) S (H.), i.e., H. is a subgroup of some member l J l

of (H.), then i ~ j. J

Define closed sets X. of Ti/G by l

X. l L) y(H)'

(H) 2 (H.) l

then Ti /G = ~ 2 ~-l 2 ... ?. x0 is a closed filtration of T9' /G

x. l x. 1 l-

Xi-1 = y(H.)' l

has a neighborhood U. in l

with the cone bundle structure:

U. ~---? X. - X. l' l l l-

Furthermore, for each i,

i T /G - X. 1 together l-

such that for j > i, U. n (X. - X. 1) are sub-cone-bundles and 1. J J-

these sub-cone-bundles together form the cone bundle.

Let Wf denote a free contractible f-space. f acts freely

on Ri x Wf diagonally.

80

4.1.4. Proposition The projection p: ( JRQ, x Wr)/1---? JRQ,/f

is a stratified system of fibrations.

Proof: For each orbit x & JRQ,/f, define f to be the isotropy sub-x

group of a point of JRZ which is in the orbit x. r is well-defined x

up to conjugacy. The map p has point -1 inverses p (x) = Wr/f , ' x

which are classifying spaces for f . This proposition will be x

proved if one can show that (fx) is constant on each stratum Y(H)"

Let x be a point Q,

in Y(H)' and pick up a pointy e JR in the pre-

image of x. Let y' (- TQ, denote the orbit by A to \vhich y belongs,

and ass11me H is the isotropy subgroup of y' . TI denotes the

quotient map: f ---3> G. Obviously f ~ TI-l(H), and the index x

[n- 1 (H) : A] is finite. -1 Let TI (H) = A 1 1_1 A2 l' •.. •.J Am be the

coset decomposition by A. Each coset A. contains a unique l

nlr : f ----7 His x x

an isomorphism. Let r denote the projection JRQ, ----7- JRQ,/r, then

-1 -1 r acts uniformly on r (Y(H)), and r (Y(H))-----? Y(H) is a cover-

ing space. For any x' in the same component of Y(H) as x, we can

actually show that r x Q,

YI E:° JR Of XI •

r ,, by carefully choosing the lifting x

Next let us see what good thing will happen if f satisfies

(1) or (2) of 4.1.2, respectively. Supposer= f' ~ T.

81

Then by 4.1.3, the epimorphism ¢:f ----7 T induces a ¢-equivariant

ff . . . F JRQ, = d h . . F a ine surJection : ----"' ill., an ence a proJection :

~ JR./T = s1 . It is easily seen that F is a fibre bundle with

fibre JR.£-l /f'. Suppose f satisfies (2), and let ¢ be the epi-

morphism: r ------7 rA. Lets and H be as in (2), and we assume

H = H' for simplicity. Define a crystallographic group C(c., f) of

-1 -1 rank 9, by C = ¢ (q (H)). The epimorphism ¢induces a cp-equi-

· ff" · · f JR,Q, JR.m variant a ine surJection : ---7 -1 The maps f, g , and

h together induce a surjection:

,Q, m -1 m n -a : JR /C ---,-.,. JR /q (H) ---7 JR /IT ----7 JR /r .

-1 9, n we will denote the composition hg f: JR -----=;>JR by a-. For an

orbit x E JR.n/r-, define a subgroup 6x of C by 6x = (ne- 1 ¢JC)- 1 (r~);

i.e., -1 6 is the set of elements of C which leaves a- (y) x

invariant for some y t:· JR.n on the orbit x.

up to conjugacy. Then the point inverse

6 is well-defined x a- 1(x) is JR.9,-n/6.

x

If we consider the projection pH: (JR.£ x Wr)/C-? JR.Q,/C, then

the composition apH: ( JRQ, x Wr) /C---? JR.n/r- has point-inverses

-1 9,-n (apH) (x) = (JR x Wr) /6x. Notice that apH is a stratified

system of fibrations, and that we can use the same filtration

and the neighborhoods of strata as those for the projection

JR.n-----;.. JR.n/f-, since 6 depends on r-. The good thing with a is x x

that, if we transfer things on JR.Q,/r to things on JR.Q,/C without

changing the radiu£ a lot, then by choosing s sufficiently large,

we can make the transfer image very small on JR.n/r-.

82

In this section we review Dress' s· induction theory[D],

which will be necessary in the proof of the main theorem.

[tD,§6J of tom Dieck is also a good reference. Let G be a finite

group and let G-set be the category of finite G-sets and G-maps.

Let Ab denote the category of abelian groups. A bi-functor

is a pair of functors, where M* is contravariant and M* is

covariant, We require these to coincide on objects, and we

write M(S) = M"<(S) = M~_(S) for a finite G-set S. If f: S --~ T

is a morphism, the notation f*

often used.

4.2.1. Definition A bi-functor M (M>'<, M~,<): G-set --· Ab is

a Mackey fun<~tO£ if it satisfies

(1) for any pullback diagram in G-set

the diagram

F u -------»S

T------~V

f

M(U) ----~M(S)

H* r r h''

M(T) -------'> M (V)

83

commutes, and

(2) the two emb.eddings S ~ S lL T ~ T into the disjoint

union induce an isomorphism

M*(S .U. T) ---> M*(S) ta M*(T).

Let Mand N be bi-functors, A natural transformation of

bi-functors X: M -'J' N consists of a family of maps X(S): M(S)

~ N(S), indexed by the objects of G-set, such that this family

is a natural transformation M* -7 N* and M,'( ---?- N,~.

Let M be a Mackey functor and S a G-set. We can define a

new Mackey functor MS by

M i<(f) s . M*(idS x T)

and the projection map p: S x T ~ T define a natural transforma-

S tion of bi-functors 8 : M ~MS b.y

s 8 (T) = p* M(S x T).

4.2.2. Definition A Mackey functor M is S-injective if 8S is

split injective as a natural transformation of bi-functors.

Let S be a G-set. We define s0 to be a point and Sk =

84

S x ••• x S (k copies), p.; ·· i

sk+l --7 sk denotes the projection

which omits the i-th. factor, 0 :.;:: i ~ k. The. important property

of an S-injective Mackey functor is the following.

4.2.3. Proposition [D, Proposition 1.1'] If a Mackey functor M

is S-injective, then the following two sequences are exact:

do d1 d2 1. 0 -~ M(SO) ~ M(S1 ) ------7' M(S 2) -? ...

2. o~ M(SO) ~ M(S 1) ~ M(S 2) 4--

k do dl d2

dk i k i where L: (-1) p;~, dk L: (-1) pi*" i=O i i=O

We introduce another notation. If H is a subgroup of G, let

G/H denote the set of (right) cosets of H. G/H is naturally a

transitive G-set. Actually any finite. G-set is G-isomorphic to

a disjoint union of these. We often write M(H) instead of M(G/H).

If HS KS G and f: G/H--;. G/K is the canonical map, we will call

the map

f><: M(K) = M(G/K) -7- M(G/H) = M(H)

the restriction from K to H and denote it by res~, and call the

map

f,,: M(H) = M(G/H) -..:r M(G/K) = M(K)

the induction from H to Kand denote it by ind~. If K = G,

we write and · d · de in H = in H" We will rewrite the

two exact sequences in 4.2.3 using this notation. Let S =

85

L.LHi=FG/H, where F is a farnil y of sub.groups of G. 0 Then M(S ) =

M(G) .;i.nd M(S) = e H-"'FM(H)' 0 The projection p: S ~ S is made up

of the canonical maps fH: G/H---? G/G; therefore

'07H c..: F M (H) ---T M ( G) .

These maps are also called (the product of) the restricti0n ma~(.~_)

and (the su~ ~) the induction map (~)respectively. Finally M(S 2 )

can be written as -1

©11 K H(H !'\ (gKg ) ) , where for a fixed pair ' , g

(H, K) E F x F. g runs over the double coset representatives of

H, K. ln our application we are only interested in these, and we

can restate 4.2.3 as follows:

il G/H. If a Mackey functor M is H <i:- F

S-injective, then we have the following exact sequences:

-1 0 ------?- M(G) ---7 ~H..; F M(H) ----7> G M(H "(gKg ) ) H,K,g

-1 0 ~ M(G) ~- 6JH E:- F M(H) <:;..--- ~ K 0 M(H () (gKg ) ) .

' ' l:1

Next we consider some sufficient condition for a Mackey

functor to be S-injective. Let M, N, and L be bi-functors

G-set ---7 Ab. A pairi12_g M x N ----? L is a family of bilinear

maps

86

M(S) x N(S) ----,:;> L(S); (x,y) ~ X•Y, S f:.. Ob(G-set)

such that for any morphism f: S ~ T, one has:

L*f(x•y) (W~fx) • (N*fy) • x ..; M(T), y "' N(T)

x •(Ni/y) L7/ ( (W'fx)• y), x E- M(T), y E N(S)

(M,./x) ~y L,~f(x •(N>':fy}), x <;,. M(S), y (- N(T).

A Green functor U: G-set -- Ab is a Mackey functor U

together with a pairing U x U ~ U. If lJ is a Green functor, then

a left U-module is a Mackey functor together with a pairing

U x M ~ M such that via this pairing M(S) becomes a left

U(S J-module,

The following proposition is very important in the theory

of S-injectivity.

4.2.4. Proposition [D, Proposition 1.2 J Let U; G-set----'> Ab

be a Green functor; and let S he a G-set. Then the following

assertions are equivalent.

(l) The sum of the induction maps U(S) --7 U(SO) is surjectiye.

(2) U ia S-injective.

(3) All U-modules are S-injective.

4.2.5. Example [D, Theorm 2 J Let GW(G, 2Z) be the equivariant

Witt ring of Dress [D, p.293 J. GW(-, 2Z) induces a Green functor

87

and

$He:- F GW(H, ?Z) ®A -----7- GW(G, ?Z) 9-J A

is surjective in any of the following cases:

1. F family of cyclic subgroups of G, A= ~(the rationals),

2. F =family of p-elementary subgroups of G, p odd, A =?Z[~],

3. F family of 2-hyperelementary subgroups of G,

A =?Z( 2) =?Z[l/3,1/5, ... J, 4. F union of the families in 2 and 3 above, A =?l.

As was mentioned before, any finite G-sets S is G-isomorphic

to a disjoint union G/H. of homogeneous G-sets G/H .. l l

If M is

a Mackey functor, M(S) ; ~ M(G/H.) = $ M(H.). In fact we can l l

redefine a Mackey functor M to be a bi-functor from _Q_ to Ab,

where G is the category of subgroups of G whose morphisms H -----:;..

-1 K are triples (H,g,K) such that g E G and gHg is a subgroup of

K, satisfying:

l. for any isomorphism f: H ---7 K, f*f,., is the identity M(H)

---,'> M(H),

2. for any inner conjugation f = (H,h,H), h E H, f* and f*

are identity : H(H) -~ M(H), and

3. the double coset formula holds: Let L and L' be subgroups

of the subgroup H of G. Suppose H has a double cadet decomposi-n

tion H = LJ i=l

Lg . L I ' g . E H . l l

Then

(L,e,H)*(L',e,H)* n L:

i=l

88

( ( L t -1) ) (L ( t -1) -1 t )..._ L g. g. ,e,L ~. g.L g. ,g. ,L ". 1 1 " 1 1 1

Green functors and modules over a Green functor are redefined in the

obvious way. Given a Mackey functor in this sense, we can

construct a Mackey functor in the original sense.

4.3. Induction theorems

In this section we construct two Mackey functors which are

modules over Dress's equivariant Witt ring (4.2.5), and, by

applying the facts in ~4.2, we obtain two exact sequences (4.3.5).

Let f be a crystallographic group of rank £ satisfying

4.l.2 (2), and consider an exact sequence 1 -7 K -7 r---;. G ~ 1,

where G = f~. For a subgroup H of G, C = CH denotes the subgroup

¢-l(q-l(H)) of f and pH denotes the projection (IR£ x Wf)/C -7

£ IR /C as before.

4.3.1. Definition £ £ Let Hj (IR /C; 1L(pH)) = Tij 1H( IR /C; lL(pH)),

where JH( ; ) is the homology spectrum defined in 3.2.1.

Let us define a Mackey functor M: _Q_-----?> Ab as follows. For Q,

a subgroup H of G, define M(H) to be Hj ( IR /CH; 11( pH)). Suppose

f = (H,g,K) is a morphism from H to K, and let '( be any element

of r such_ that q¢(y) = g. Then f induces a map f#: pH -----7 PK

between stratified systems of fibrations; i.e., we have a

89

commutative diagram

f It ( JR9, x Wr) /CH ______ __,,-. ( JR9, x Wr) /CK

PH l 9,

where f# and f# are maps induced by the action of y on JR x wr

and JR9, respectively. f# is a finite covering. We have the

following two operations corresponding to f#.

defined as follows. If M is a geometric LZCH-module on pH

generated by {xa} ~ (JR£ x Wr)/CH' then f*M is the geometric

LZCK~module on pK generated by {f#(xa)}. As the fixed lifting 9,

of f It (xa) in JR x Wr, yxa will be used. f,'<M is a direct sum of

copies of M, when viewed as aLZCH-module. Abstractly it is just

a tensor product. A LZC8-module homomorphism h: M --7' M'

naturally induces a LZCK-module homomorphism f*h: f*M-----;> f,._M'.

(If g = 1 and '( = 1, then f*h = 1 8l h : f H = LZC © M ---7' ·k K LZC

H f,./W<) LZCK ~LZC M' = f,.<M'.) By the natural identification

H f*(M*), f* operates on quadratic Poincar~ complexes ([RJ,§2.2]),

90

{x } a follows. If M is a geometric ?ZCK-module on pK generated by

S ( JR.9, x Wr) /CK, then f>'<M is a geometric ?ZCH-module on pH

-1 generated by f# {xa}; i.e., f*M is a free abelian group generated

-1 -1 by the points in y (q ({x })) , where q is the projection a JR.9, x Wf ---'> ( JR.9, x Wf) /CK' and the ?ZCH-module structure is

obtained by forgetting the action by elements in

For a ?ZCK-homomorphism h, f*h is defined by f*h

induce the desired f*.

-1 y c y -K -1 = y hy. These

Obviously, f,.< does not increase the radius. On the other

hand, f* may increase the radius; but the result has radius at

most 2jGKj times as large as the original radius, where GK is

* the holonomy of CK. Therefore f* and f induce maps

---7" TI..( JR.9, /CK; pK) and TI..( JR.9, /CK; pK) --7-Il...( JR.9, /CH; pH) respectively.

By the characterization theorem, these induce the desired maps

f~: M(H) ----7 M(K) ,, and f*: M(K) ---7 M(H).

Obviously M is a bifunctor, and satisfies the first two conditions

of Mackey functor.

4.3.2. Proposition H satisfies the double coset formula, and

hence is a Mackey functor.

91

-------! I

\ i

.,.,_~~---~~~·__;

I \

fhQ.;·,. ·-Q. ! ,-

1 '''Li'-. =- \ /CTI<..

~\ I • / , • I .• /-\ I \ oV J·)

\ 0 I ~ ~ ------.../

Figure 2. The change of radius by restriction.(g = 1)

92

Proof: Let L and L' be subgroups of a subgroup H of G, and suppose

H has a double coset decomposition H n

n

l) i=l

Lg . L I ' g . ~ H • l l

Let

CH= l_) CLg.CL, be a corresponding i=l l

double coset decomposition - -

of CH' where g. c.· CH such that q<jl(g.) l - l g.. Let P be the pullback:

l

i.e. ,

p ________ ,_,,, ( JR..Q, x Wf)/c1 ,

l (L' ,e,H)I!

( L, e, H) ff

p {([x], [yJ) E ( IR.Q, x Wr)/CL x ( IR.Q, x Wr)/CL,

.Q, [xJc = [yJc (-: (IR x wr) /cH}.

H H

Then it is easily verified that the following map

[xJc -1 Lng.L'g.

l l

t------.,,. ([xJ, [g.- 1xJ) l

.Q, is a CH-isomorphism, where x E, IR x wr and I ] is the corresponding

orbit. Therefore we obtain a pullback diagram:

Un i=l

93

-1 -1 (Ln(g.L'g. ),g. ,L'),1 n l l l 1r x.,

----------~(JR x Wf)/CL'

I I

(L',e,H)I/ l ( JR 9, x W f) I CH

(L,e,H)ll

and the double coset formula for modules and chain complexes

are easily derived from this. Since direct products of Poincare

complexes correspond to glueing along empty (i.e. O) boundary,

this establishes the desired double coset formula for M.

GW(H, Zl) acts on M(H) by tensor product. Recall that

GW(H, Zl) is constructed using H-spaces. An H-space is a Zl-free

(left) ZlH-module N together with a symmetric H-invariant

non-singular form f: N x N ~ 7l. Let Ni< = Hom(M, 7l), then Ni<

is also a (left) ZlH-module. An element h ~ H acts on N* by

-1 h-a(y) = a(h • y) for a E N*, y 0 N. By letting (CN)O = N*

and (CN)i = 0 for i # 0, we have a Zl-module chain complex CN.

We define a 0-dimensional symmetric Poincare structure ¢f: N

~Ni< by ¢f(x) = f(x,-). By assumption ¢f is an isomorphism and

H-invariant. If {el' ... ,en} is a basis of N~~ and H is a free

ZlCH-module with a basis {o 1 , ... ,om}, then Ni< ~ M is a free 7ZCH-

94

module with a basis {e. ©a.}. Here we use the diagonal action l J

of ZlCH. CH acts on N>'< via H. ,Q,

In ( 1R x Wr) /CH, the generator

""' d h · d i· n 1R,Q, W ei ~ aj correspon s tote same point as aj' an , x f'

the lifting of e. 0 a. is chosen to be the same point as the l J

.lifting of oj. Tensor products with (CN'¢f) described in (R3, §1.9)

induce the desired action of GW(H, Zl) on M(H).

4.3.3. Theorem Mis a GW(-, Zl)-module.

Proof: Same as the proof of (FHl, Theorem 2.3J.

Another Mackey functor M' : _Q_ --7 Ab can be defined by

Notice that ( JR'<(, x W ) /C f H

is a c1 assifying space of CH. f,., and f* are defined in the same

way as before. This time we do not have to worry about the radius.

We can prove the following in the same way.

4.3.4. Theorem M' is a GW(-, 2Z)-module.

As an immediate consequence of 4.3.3 and 4.3.4, we have the

following theorem.

4.3.5. Theorem Let F denote the family of conjugacy classes of

maximal hyperelementary subgroups of rA. Then the following s

95

sequences are exact.

$ Q,

H. (JR /CH" Ka -l; Il...(p -1))' H,K,g J g 0 H~gKg

0--~ L ~co (( JRQ, (resH)

L~(( JRQ, x Wr) /r) ei x Wf)/CH) J HE F J

$ 1~00(( JRQ, x Wf)/C _ 1) H,K,g J Hf)gKg

There are similar exact sequences for induction maps.

4.4. Calculation of surgery groups

The following is the main result.

4.4.1. Theorem Let f be a crystallographic group of rank Q,

with no 2-torsion, and p the projection ( JR,Q, x Wr)/f --7

9, JR /f, where Wf is a contractible free f-space. Then there

is a natural isomorphism

a Q,

H. (JR /r; Il...(p)) J

Proof: The map is induced by the following composition:

9, lli . (JR /f; JL(p))

-J

96

A . F -J 9,

---·JL.(JR /f;p) J

where A . is the assembly map and F is the restriction map to O; -J

i.e., if pis a k-simplex of JH .( JR9,/f; JL(p)), then FA .(p) -J -J

. (A .p)(O). We will prove the isomorphism inductively on the -J

size (rank(f), [G[) off using the lexicographic order.

If 9, = 0, then f = 1 and ( JR9, x Wf)/f is a single point.

Since IB .U:; JL(p)) "'JL.(*;p) ( =E.(*)) by 3.2.2, the theorem -J J J

is obvious in this case. If 9, = 1, then f T and hence the

argument below for crystallographic groups satisfying 4.1.2 (1)

with rank > 1 can be applied.

Now assume that 9, ·~ 2. First suppose that r satisfies (1)

of 4.1.2. See Diagram 2. The first row is the exact sequence for 9, 9,-1 - .. 9, 9,-1

the pair ( 1R /r, 1R /f'); notice that (JR /r)/( JR /f') =

9,-1 + Z( JR /f') , where Z is the reduced suspension. p' denotes the

· · f 1R9,-l Ir' · ' · h · · restriction o p over ; i.e., p is t e proJection

( 1R9,-l x Wr) /r' ------7- JR9,-l /r'. The second row is the well-known

exact sequence

due to Wall, Shaneson, Farrell and Hsiang. By 5-lemma, a is

proved to be an isomorphism.

Next suppose r satisfies (2) of 4.1.2. We first show that

_(, I '{'

r ::i:

: :....

... I

w.

8 -

-;;3

-

~

~

I ~

-I

-t------·------

-J

x :<::

J t=t

._, -

Ill

-J

"O

._,

._,

._,

I I

r

* '\;

r ::i:

: :,.

..,.

~ (.

..;.

--

;;::!

-~

~

---

----

?<::>

x t:1

:<::

c=

t-'·

'""""j

Ill

Ill

._

, -

00

-"O

t1

J ._

, Ill

._

, ._

, 9

I

~ N

~ r

::i::

w.

~ w

. I

I

--

-.,-

-...

-~

~

?<::>

?<::>

I I

,_.

-< -

x J

:<::

J ._

, c=

-J Ill

-"O

._,

._,

._,

I ~

~ t:-'

::i::

w.

I w

. I

8 I

--

--

-.....,

~

;:'O ~

~

I I

--

-x

J :<::

--:

! Ill

._

, r=;

---:l - "O

._,

._,

._,

1 ~

L6

98

the map being considered is injective. Suppose y is an element

of the kernel. £ r Represent it by a 0-cell P of 1H . (JR /-; JL(p)). -J

A .P(O) represents the image a(y) by a. Kan condition implies -J

that there exists a 1-simplex o of JLj(( JR.£ x Wr)/f) which

connects A .P(O) and 0. Let the radius of P be 6 measured in -J

£ £ JR ;r. It is automatically finite, because JR /f is compact.

Choose a positive number s in (2) sufficiently small so that

4lcl6K//5 < e:

where K is the Lipschitz constant of the affine surjection

f: JR.£ ~ JR.m induced by the epimorphism (jJ: I --7 f"' and E is

the positive number posited in 3.2.3, when we consider JR.n/r-.

( E depends only on 1Rn/r-, its filtration, the neighborhood

system, and the dimension of the thing being considered.) Now

we have a commutative diagram:

H. ( 1R£,lf; JL(p)) ( resH)

t I o~ $ H j ( JR C ; JL (PH) ) J H

! I t I a aH Jt (resH)

L -:-°" (( JR,Q, _oo £ 0 ----7 x Wf) /f) e L. (( JR x Wr)/CH)

J H J

where each exact row comes from the restriction maps corresponding

to the maximal hyperelementary subgroups Hof f"'. For H such s

that lcAI does not divide IHI, the size of C = ¢- 1(q- 1(H)) is

smaller than the size of f. So by induction hypothesis, aH is

99

an isomorphism, and hence resH(y) = 0. For H such that !GA! does

divide IHI , we have a shrinking map a: 1R.£/C ------7 1R.n/r-, and

n/ -resH(o) has radius less than E on 1R. f . Therefore the image

which is represented by resH(A .P(O)), is 0. We claim that the -J

above map is an isomorphism. For the convenience of the proof,

let us replace

where v6 is a carefully chosen vertex of 6. See [Q3, Proof of

8.6]. Notice that IL((apH)- 1 (v6 )) = IL((1R.£-n x Wr)/6), where

x = v6 , and that the size of 6x is strictly smaller than the size

of r.

Here p is the projection ( JR.£-n x W ) I 6 ~ 1R.£-n I 6 . So, x r x x

JH( 1Rn/r-; IL(apH))

n/ - I £-n; I "" JH( 1R. f ;J._L JH( R 6; IL(p )) x 6,) x x

lim ;lcUClim ~"tk! IL k .(p) !IC m. 9'-n/6)))/(1R.n/f-) - -l x x

i-7~ k-;-,,.,

lim ltj CJ..0 IL_ . ( p) I ) I ( 1R. £I CH) j "°"7"" J

. £ lim QJ I IL_. (pH) I I ( 1R /C8 ) j~'JC J

£ 1H ( 1R. /CH; IL (pH) ) .

100

Now this implies resH(y) = 0. The restriction map

(resH)H is injective; therefore, y is O; i.e., a is injective.

The onto part is similar. _co 9,

Pick any element in Lj ( JR x Wr) /f),

represent it by a quadratic Poincare complex with radius 6, and do

the diagram chase as in the proof of 5-lemma. We need to use

the next column this time, but it is already known to be isomorphic

or at least injective. This completes the proof.

4.4.2. Remark In 4.4.1, we assumed that f has no 2-torsion.

This is because 4.1.2 may not be true for T ~ T2 and the

induction step does not proceed. So the theorem is true even

for r with 2-torsion if there appear no crystallographic groups

of the form T ~ T2 in the induction steps.

(Ch]

[Coj

[CH]

[tDJ

LD]

[FJ

(FHlJ

[FH2]

[KS]

[Ql J

L"Q2J

[Q3J

l}n J

[R2]

[R3J

Bibliography

L. S. Charlap, Compact flat Riemannian manifolds I, Annals of Math., 81 (1965), 15-30.

M. Cohen, !:_ Course in Simple-homotopy Theory, Springer Graduate Texts in Math., 10 (1973).

E. H. Connell and J. Hollingsworth, Geometric groups and Whitehead torsion, Trans. AMS, 140 (1969), 161-181.

T. tom Dieck, Transformation Groups and Representation Theory, Springer Lecture Notes in Math., 766 (1979).

A. W. M. Dress, Induction and structure theorems for orthogonal representations of finite groups, Annals of Math., 102 (1975), 291- 325.

D. Farkas, Crystallographic groups and their mathematics, Rocky Mountain J., 11 (1981).

F. T. Farrell and W. C. Hsiang, Rational L-groups of Bieberbach groups, Comment. Math. Helvetici, 52 (1977), 89-109.

~~~~~-' Topological characterization of flat and almost flat Riemannian manifolds Mn (nf3,4), preprint.

R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Annals of Math. Studies, 88 (1977), Princeton Univ. Press.

F. S. Quinn, A geometric formulation of surgery, thesis, Princeton Univ., 1969.

Ends of maps I, Annals of Math., 110 (1979), 275-331.

~~--~-' Ends of maps TI , to appear in Inventiones Math.

A. A. Ranicki, Algebraic L-theory, TI : Laurent extensions, Proc. London Math. Soc., (3) 27 (1973), 126-158.

----- , The algebraic theory of Surgery I. Foundations, Proc. London Math. Soc., (3) 40 (1980), 87-192.

_____ , Exa_s:! Sequences in the Algebraic Theory of Surgery, Princeton Univ. Press, 1981.

101

102

[R4J A. A. Ranicki, Classifying spaces in the algebraic theory of surgery, to appear.

[SJ E. H. Spanier, Algebraic Topology, McGraw-Hill, 1966.

[wJ C. T. C. Wall, Surgery on Compact Manifolds, Academic Press, 1970.

lWoJ J. Wolf, Spaces of Constant Curvature, McGraw-Hill, 1967.

The vita has been removed from the scanned document

SURGERY GROUPS OF CRYSTALLOGRAPHIC GROUPS

by

Hasayuki Yamasaki

(ABSTRACT)

Let f be a crystallographic group acting on the n-dimensional

Euclidean space. In this dissertation, the surgery obstruction

groups of r are computed in terms of certain sheaf homology

groups defined by F. Quinn, when f has no 2-torsion. The main

theorem is :

Theorem : If a crystallographic group !' has no 2-torsion,

there is a natural isomorphism

Il -CO. a H,._( JR /f; TI...(p)) -7 L,., (f).