Rigidity and crystallographic groups, I

Invent. math. 99, 2548 (1990) Inventiones mathematicae © Springer-Verlag 1990

Rigidity and crystallographic groups, I

Frank Connolly 1 and Tadeusz Ko~niewski2 1 Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA 2 Department of Mathematics, University of Warsaw, PKiN IXp., 00901 Warszawa, Poland

1. Introduction and statement of main results

This is the first in a series of articles about rigidity theorems for crystallographic groups. The main theorem is (1.1). It is an "h-cobordism rigidity" statement. It implies (see (1.6)) that any crystallographic manifold, with odd order holonomy, which is h-cobordant and simply homotopically equivalent to the standard one is actually homeomorphic to it. In the second part of this work [9], we will exploit this result to prove that one can remove the word "h-cobordant" from the previous sentence.

The main theorem concerns an involution on the group Wh~P'°(Mr). This is the group of G h-cobordisms of the flat torus Mr, of a crystallographic group F with holonomy group G. (Definitions of these terms appear in (1.4).) We prove a vanishing result of the Tate cohomology:

(1.1) Theorem. Let F be a crystallographic group with odd order holonomy group G. Assume no f ixed sets of Mr have dimension two, three, or four and no gap between nested f ixed set components has codimension two. Then /~*(Z/2Z; top, p Wh G (mr)) = O.

(1.2) Some perspective. Notions of rigidity in topology have a long history. One can say that the results of Bieberbach [-3] provided an important early paradigm for these. He proved that two isomorphic crystallographic groups, F and F' of rank n, must actually be conjugate in Aff(n), the group of affine transformations of n-space.

Much later a more topological result in this spirit was proved by Hsiang- Shaneson, [22], and independently by Wall [35]: If F and F' are isomorphic free abelian groups acting properly, with compact quotient on euclidean spaces R" and R"', then n=n' and F and U are conjugate in Homeo(R").

At almost the same time, A. Borel made the now well known conjecture, in an informal conversation, to the effect that any two isomorphic, torsion free, properly discontinuous, co-compact subgroups of Homeo (R") must be conjugate. This has since been verified in the case when the group, F, is a torsion free virtually nilpotent group (see [-16, 17]), and in the case when the group is a

26 F. Connolly and T. Ko~niewski

torsion free virtually poly-Z group (see [19]). A recent announcement [21] promises the same conclusion when the group is "negatively curved" - that is to say it acts uniformly on R" preserving some metric of negative curvature.

It is worth noting that each of the results of the last paragraph relied on a previously proved "h-cobordism rigidity" result, stating that Wh(F)= 0, where Wh(F) denotes the Whitehead group of F (see [-2, 18, 35, 20]). But in all of the above results the group F is torsion free, and the action on R" is free. When F has subgroups of finite order, one cannot expect Wh(F) to vanish since F often contains finite groups which are retracts of it, and the Whitehead group of a finite group seldom vanishes. Moreover, as our results show, it is entirely unrealistic to expect that I/IZ/~t°p'P/AAr"'~G v'-rJ ~ will vanish, because of the nonvan- ishing of Nil. Our main theorem, however, provides a new kind of analogue of these "h-cobordism rigidity" results.

(1.3) A somewhat vaguely stated conjecture exists, generalizing Borel's conjecture to the case when F has torsion. It is due to F. Quinn (see [26]). As a motivation to the reader we wish to state this rigidity conjecture in somewhat sharpened form.

One must necessarily make some hypotheses in order to have any hope of a rigidity theorem when F has elements of finite order. The conclusion of such a theorem would be that a certain F-manifold )~ should be equivariantly homeomorphic to some standard F-manifold, M r. But the hypotheses must at least insure that they are properly F-homotopy equivalent. It is our point of view that therefore the first appropriate hypothesis is:

(,) F acts properly on M with compact quotient. Moreover the fixed set of each finite subgroup of F is a contractible manifold, flatly embedded in each bigger fixed set.

When F is virtually torsion free, and acts on manifolds ~r and Mr so that (*) holds, there is a F-homotopy equivalence J: /~ -~ /~r , unique up to F-homotopy. See ([8], Lemma 2.1). Let G=F/Fo where Fo denotes a normal torsion free subgroup of finite index in F. Write M and Mr for ~l/Fo and ifIr/F o. Then ] induces a G-homotopy equivalence J: M ~ Mr. A rigidity theorem would assert that J is G-homotopic to a homeomorphism, but this can only be true if the Whitehead torsion of J is zero and J can be chosen to preserve strata. This torsion is measured in the equivariant Whitehead group, umtop,,,,,G p ~,,r~, ~ A~t ~ introduced under various names by Quinn and by Steinberger-West (see Chap. 2). We write this torsion z(M), viewing it as an absolute invariant of (M, F). So our second hypothesis is:

(**) z (M)= 0 and J preserves strata. But (*) and (**) are insufficient for the desired rigidity conclusion. Indeed,

even for a crystallographic group of the form Z" x ~ Z/2 Z, S. Weinberger has given a quite simple example [36] of a failure of rigidity for manifolds in which (.) and (**) hold. It seems to us reasonable that the failure of rigidity in such examples can be traced to elements of the group F which preserve one geodesic or a parallel pencil of geodesics in a flat 2-plane of Mr while reversing the sense. Algebraically, this translates into the existence of two-torsion in the holonomy of a rank two crystallographic group. That is why we propose a third hypothesis whose aim is to evade the baneful effects of Nil and UNil:

Rigidity and crystallographic groups, I 27

(***) Assume that F contains no subgroup isomorphic to D~ x Z or to H x~K where K is the fundamental group of the Klein bottle and H is a finite group whose order is divisible by the second power of some prime.

Here is our sharpening of the Borel-Quinn conjecture.

Conjecture. Let F be a discrete group of isometries of a Hadamard manifold Mr. Let M be a F manifold of the same dimension as A4r. I f the hypotheses (*), (**), (***) hold, and no fixed sets have dimension three or four, and no gap between fixed sets has dimension two, then M and ~C4r are equivariantly homeomorphic.

In this paper we are going to prove this conjecture in the special case when F is an odd-holonomy crystallographic group and M and Mr are equivariantly h-cobordant. See Corollary (1.7).

(1.4) Some basic definitions. A crystallographic group F, of rank n, is a discrete subgroup of the group of isometries of R" (with euclidean metric) such that R"/F is compact. The set of translations in F forms a normal free abelian subgroup of rank n (a result of Bieberbach). This is the "translation subgroup", which we write Ar. It is equal to its own centralizer in F. F/Ar is called the holonomy group of F, and is written Gr. It is a finite group. The torus R"/Ar is written Mr and is called the flat torus of F. The group Gr acts by isometries o n M r .

The "topological equivariant restricted Whitehead group" of Mr is therefore defined (see Chap. 2 for a formal definition). It is an abelian group denoted

top,p WhG (Mr) where G=Gr. This group parametrizes h-cobordisms on Mr in the following sense. Let W be a locally linear Gr-manifold whose boundary consists of Mr, and a manifold M, such that the inclusions Mr-* W~-M are Gr-homotopy equivalences. This is an h-cobordism on Mr, and it defines an element z(W, Mr) in l/l//at°p'P/)~Ar ~ " " G ~"-r1" z(W, Mr) is zero if (W, M r ) ~ ( M r x l , Mr). According to a theorem of Quinn [-27] and, independently, Steinberger [31], if no fixed set of Mr has dimension two, three, or four, and no gap between nested fixed set compounts has dimension one or two, then the rule

(W, Mr) -* z(W, Mr)

provides a bijection from the set of Gr-homeomorphism classes of h-cobordisms on Mr and Vv'h~P'P(Mr).

The Gr-homotopy equivalence M i ,Mr above induces an isomorphism

of Whitehead groups: function,

by the rule:

Wh~p,p(M ) i. ,Wh~p,p(Mr). We can then define a

. : l, Vh~P,p(Mr)__ ~ T~top, o t~ / t , , , , 6 ~l~,r~ ~

* (W, Mr) = i, (W,, M).

This functfon * is actually a group homomorphism and **=identi ty. (As a proof of this, note that in Connolly-Lfick [10] it is shown that in WhPL'p(Mr) one has that • is a homomorphism and **=ident i ty ; but in 2.14 below, we

28 F. Connolly and T. Ko2niewski

show that the map Wh~L"(Mr)~ umtop,,t~,~,,,~G v,lrj~ is an epimorphism. The result follows).

This involution , , allows us to def ine/4*(Z/2Z; Wh~P'°(Mr)), the Tate cohomology, whose vanishing is our main result.

The group Wh~v'P(Mr) is shown in Chap. 2 to be a direct sum of groups of the form Wh(A)/Wh(A)e. Here A is crystallographic and Wh(A)c denotes a controlled Whitehead group which we prove to be isomorphic to a subgroup of Wh(A). The classically defined involution on the Whitehead group then allows us to cons ide r / t* (Z /2Z; Wh(A)/Wh(A)c). The involution • is not always a direct sum of such classically defined involutions. In case G has odd order, however, • is a direct sum of classically defined involutions (see Chap. 2). So (1.1) reduces to:

(1.5) Theorem. Let F be a crystallographic group. Then the controlled Whitehead group Wh(F)c injects into Wh(F) and the algebraically defined involution on Wh(F) (see 1-24]) preserves Wh(F)¢. With respect to this involution, if bGrb is odd, then /~* (Z/2 Z; Wh (F)/Wh (F)~) = O.

We want to explain now some of the geometric consequences of (1.1). Let G be a finite group and let X be a compact G-ANR. By Sh(X) we

understand the set of equivalence classes of pairs (M, f ) where M is a closed locally linear G-manifold and f : M ~ X is a G-homotopy equivalence. (M, f ) is equivalent to (M', f ' ) if there is a G-h-cobord ism, W, as defined above, connecting M and M' and a G-map, F: W ~ X , extending f and f ' . Analogously one defines S(X) by requiring f, f ' and F above to have zero torsion in WhiP(X). Of course these "structure sets" S(X) and sh(x), are empty if X is not sufficiently nice. Rigidity theorems amount to the statement that S(X) consists of a single element when X has the form M/F0 and F acts on M satisfying (,), (**), and (***). On the other hand, the map s h ( x ) ~ s ( x X S ~) obtained by sending (M, f ) to (M x S a, f x 1) is a monomorphism, so if rigidity holds for F and for F x Z, then sn(x) must also consist of a single element. This helps to explain the significance of:

(1.6) Corollary. Let F be a crystallographic group with odd order holonomy and no fixed sets of dimension two, three, or four, and no gaps between nested fixed sets of codimension two. Then the natural map S(Mr)~Sh(Mr) is an isomorphism.

As a special case, we have:

(1.7) Corollary. Let F satisfy the hypotheses of (1.1). Let ()f/l, F) satisfy (*) and (**), and assume that M=l f l /Ar is Gr-h-cobordant to Mr. Then M is Gr-homeomorphic to Mr.

Proof of (1.7). (]fI, F) determines an element (M, f ) in S(Mr) which goes to zero in Sh(Mr) and therefore by (1.6), (M, f ) is equivalent to (Mr, id.). By the s-cobordism theorem mentioned in (1.4), it follows that M is Gr-homeomorphic to Mr.

(1.8) Here is an outline of the rest of the present article. In the rest of Chap. 1, we collect various generalities about crystallographic groups. The plan is to


split those having odd order holonomy groups into the following three classes and then prove Theorem (1.5) in each case:

Case A: When Hi(F; Z)+0. Case B: When Hi(F; Z)=0 , but Gr is not a hyperelementary group. Case C: When H 1 (F; Z )= 0, and Gr is a hyperelementary group. In Chap. 2 we show that for each crystallographic group with no fixed sets

of dimension two, three or four, and no gaps of dimension one or two, the exact sequence of Steinberger-West (see [32]) reduces to a short exact sequence:

0 ~ Wh~L'P(Mr)~--* Wh~L' P(Mr)-~ Wh~P'P (Mr)--~ O.

This is of fundamental importance in our proof. In Chap. 3, we prove the Theorem (1.5) in Case A, on the assumption that

(1.5) holds for crystallographic groups of smaller rank than F. In Chap. 4, we prove the Theorem (1.5) in Case B, on the assumption that

(1.5) holds for crystallographic groups of finite index in F, whose holonomy groups are hyperelementary.

In Chap. 5, we prove Theorem (1.5) in Case C, on the assumption that (1.5) holds for finite index subgroups of F whose holonomy group has order strictly less than that of [Gr[.

Taken together then, Chaps. 3, 4, and 5, prove (1.5) and hence (1.1). In Chap. 6 we prove the Corollary (1.6) by proving the existence of a Rothen-

berg exact sequence of structure sets. We now collect some foundational results about crystallographic groups.

They were defined in (1.4). Rather than approaching these as isometry groups of R", however, it is advantageous to take the following abstract approach.

(1.9) Definition. A group F is called an abstract crystallographic group of rank n, if F contains a normal, rank n, free abelian, finite index subgroup A, which is its own centralizer in F.

Such a subgroup A must be unique (for A is the centralizer of any of its finite index subgroups; but if A' is a second such subgroup, A c~ A' has finite index in A and A'). This makes A a characteristic subgroup of F. So each abstract crystallographic group F has a canonical exact sequence,

(1.10) 1 ~ A --* F - ~ G ~ 1,

where G is a finite group. G is called the holonomy group of F. It acts by conjugation on A. A is called the translation subgroup of F.

(1.11) Construction. We show here that each abstract crystallographic group F of rank n is isomorphic to a cocompact discrete subgroup of the group E(n), of rigid motions of euclidean n-space.

Let #: G ~ A u t ( A ) be the G module structure on A defined by (1.10). The exact sequence (1.10) thereby defines a cohomology class )~ in H2(G; A). Let

30 F. Connolly and T. KoSniewski

i: A ~ A ® R denote inclusion (@ means ®z). Since G is finite, H2(G; A ® R ) = 0 . Hence i, (Z)= 0. This means one has a commutative diagram:

1 , A , F ~ G , 1

1 , A ® R ,(A®R) x~G ,G ,1

where G acts on A ® R via the mapG " >Aut(A) i, ,GL(A®R). Since

H1(G; A ® R ) = 0 , the map t is well defined up to a conjugation by an element of the subgroup A®R. The group of affine transformations of the vector space A ® R is:

Aff(A®R) = (A®R) x ~GL(A®R),

and the map, (ida x , i , #): (A®R)x ~ G ~ ( A ® R ) x rGL(A®R), is an embedding since A is its own centralizer. This yields an embedding,

e=(idA x~i, #)z: F--+Aff(A®R),

so that the diagram below commutes:

1 ~ A , F j, le 1 , A @ R ,Aff(A®R)-

, G ,1

l i. kt

, G L ( A ® R ) ,1

e is well defined up to conjugation by a translation. We can choose an isomorph-

ism A ® R P ~R" relative to which the G action defined by i , p passes to an

orthogonal action of G on R". Then, the group of rigid motions E(n) contains fl e (F)fl-a. This exhibits F as a discrete cocompact group of E(n) and the embedding is unique up to conjugacy by an element of Aft(n).

Henceforward we will write 2Q r for A ® R with this F action by isometries. The holonomy group G acts by isometries on the flat torus M r = A ® R / Z , with the metric induced from Mr- We conclude:

(1.12) Proposition. Each abstract crystallographic group F, of rank n, has a canonical action as a cocompact discrete group of isometries on some n-dimensional euclidean space Mr.

A crystallographic manifold is a pair (M, F) where F is an abstract crystallographic group and M is a F manifold satisfying (*).

The next proposition is a compendium of elementary results about crystallographic groups and their crystallographic manifolds.

(1.13) Proposition. Suppose F is a crystallographic group with translation and holonomy groups A and G respectively. Let (All, F) be a crystallographic manifold and M = M/A.


a) There is a unique F-homotopy class of F-maps J: M --* Mr. b) The map J" M-* Mr, induced by J, is a Gr-homotopy equivalence. Let j: F ~ Gr be the natural homomorphism and let H be a subgroup of Gr. c) MH4=O iff the cohomology class z(F), defined by (1.10), is in the kernel

of the homomorphism rest: HZ(G; A)~H2(H; A). d) I f M n 4=0, then the set rco(Mn), of path components, is in bicorrespondence

with the group Hi(H; A). The fundamental group of each path component of M n is isomorphic to H°(H; A)~-A H and its dimension is equal to the rank of A n"

e) Let H be an isotropy subgroup of F acting on ]f/I. Then p(iVl n) is a single component of M s(m, denoted M[H]. Every component of every fixed set of every isotropy group of M has this form. Moreover (l~ n, Nr(H)/H) is a crystallographic manifold; the translation subgroup of Nr(H)/H is A n" H/H ~ An.

Proof a), b), c), d) and the first two assertions of e) are all special cases of the results of [8]. See 2.2 of [8]. The final sentence of e) amounts to the claim that Nr(H)/H is a crystallographic group whose translation subgroup is A n. H/H. Now it is clear that A u" H/H ~-AU= A c~ Nr(H) is a normal free abelian group of finite index in Nr(H)/H. So we must prove An'H/H is self centralizing in Nr(H)/H. We take any bigger subgroup in Nr(H)/H, say A/H. Since An'H ,t= A, we conclude that j(H) is a proper subgroup ofj(A) in G. Since j H is an isotropy group of G, it follows that each component of M j(m has dimension less than dim MJ(n). This says that rank A ~<rank A n, by part d). In particular then, A does not centralize A n ~- A n. H/H. []

In Chap. 3, we will make crucial use of the following result.

(1.14) Lemma. Let A be an abstract crystallographic group. Let ~: A-*A be an automorphism such that the semi direct product F=A x , Z is also abstract crystallographic. Then some positive power of ~ is an inner automorphism of A by an element of A~. I f the holonomy group, Gr, has odd order, then this positive power can be taken to be odd.

Proof Suppose [Gr[ = n. Then ~": A ~ A is the restriction of an inner automorphism of F by an element of At, and so it induces a map of the exact sequence of A"

0 ~Aj ,A ,G A '0

0 ,A~ ~A ,G~ ~0.

Therefore it determines an element [z], in the cohomology group HI(Ga; A~). (Specifically, for each x~A, c~"(x)x-'= z(x), where z is a crossed homomorphism from Gn to Aa). H'(G~; Aa) has exponent n, since IGal=n. Hence n [ z ] = 0 , and nz is a principal crossed homomorphism. It follows that the map (7")": A ~ A is given by inner automorphism by an element of A~. []

32 F. Connolly and T. Ko~,niewski

2. A short exact sequence of Whitehead groups

In this chapter, we will review equivariant Whitehead theory, and prove a number of results which we will need about equivariant K-groups, culminating in the fact that the forget-control map is a monomorphism of K-groups in the case of a crystallographic manifold.

(2.1) Let G be a finite group of isometries of the connected compact Rieman- nian manifold M. Assume the set aM of singular points of the action has codimension greater than 2. Let F be the group ~I((M x GEG)). We can identify F with ~ ( ( M - c r M)/G), as well as with the group of isometries of the universal cover M which cover isometries of G. We then have a natural exact sequence:

I ~ F o ~ F J , G ~ I

where Fo = tel(M). We are interested in the controlled Whitehead group, Wh(F)c, with M as control manifold.

This group, Wh(F)~, is defined in various places. Quinn, [25], defines it, under the name of Hi(M/G; S(G.)). Chapman, [6], defines it under the name of Wh(X,p), where X=MxG(EG) k and (EG) k is a skeleton of EG chosen so as to preserve local compactness and 2-connectedness, and p: X ~ M / G is the natural map. Later Quinn, [-28], defines it under the name of LimWh(X, p, e)

e ~ O

where X = M x G EG and p X--* M/G is the natural map. Later still, Steinberger and West, [32] also give a definition of the group we want. They would call it Wh~L(M, rel aM)c where aM denotes the union of the nonfree orbits of M. In Quinn, [25], our group is given the name Ha(M/G; Wh(r)), where r can be taken to be M xGEG~M/G.

(2.2) For the reader's convenience, and for the sake of definiteness, we outline a definition of Wh(F)~, following the spirit of Steinberger and West [32] because of the transparency of that approach.

Let X be a metric space and a finite G - C W complex, on which G acts by isometries. We consider pairs (Y, f ) where f : Y ~ X is a G-strong deformation retraction and (Y, X) is a relative finite G-CW-complex. Following Cohen [7], one defines the sum of two of these:

(Yl , f0+(Yz , f2) means (YlWxYz,f lwf2).

Given (Y~, f~), i = 0, 1, and CEPL G-maps gi: Z ~ Y~ so that fo go, f l gl : Z ~ X are equivariantly e-homotopic, (when measured in X), we say (Y, fo )~(Y, fl). Let Xo be a G-subspace of X.

The set of equivalence classes, under the equivalence relation generated by ~~, of pairs (Y, f ) such that each member of the homotopy, f , has diameter <e and f - l ( X o ) = X o, generates a group, Wh~e(X, relXo)~. One writes


Wh~L'p(X, relXo)~ for the subgroup of Wh~L(X, relXo)~ represented by pairs (Y, f ) in which f and n n Y~/ = X~ vo Y~> n whenever x ~n x ~ > n Define:

Wh~t' P (X, rel Xo)c = Lim Wh~ t ' °(X, rel Xo) ~. e~0

If in the above discussion, Y is only required to be a compact G-ANR (instead of a finite complex) and the maps gl are only required to the CE-maps, one gets a group i/lf/~top, pgy .... G w~,relXo)~- (A G-map f: A ~ B is a G - C E map if, for each b~B, f-~(b) is Gb contractible to a point in any A-neighborhood of f-'(b)).

We define Wh(F)~ as Wh~L'p(M, rel aM)~ where M and G are as in the first |Jf/at°p'P(A'4 rel aM). paragraph of (2.1). Similarly define Wht°P(F) as .... G t~,,,

(2.3) We now make the further assumption on M that for each subgroup H of G, the subset an(M)={xeMn: G~+H} has codimension greater than two in M n, at each point ofan(M ). Let M~ denote a component of M n not contained

g M~ = M~ }. W(H, ~) acts by isome- in an(M); let W(H, ~) denote {gENn(H)/H: n n tries on m~. Let 17V(H,~) denote n,(M~× w(n.~)EW(H,~)). We get an exact sequence:

1 -+nl(Mff)-* W(H, ~)~ W(H, ~)~ 1.

With M~ serving as control manifold, we get a group Wh(ITV(H,~))~ as in (2.2), and a map:

Wh ( ~TV(H, ~))c ~ WhPL' p (M)~

obtained by sending a pair (Y, f ) to the pair ((G ×N(n,,)Y)uMvM, f') where f ' is the G map which agrees with f on 1 × Y

According to Steinberger-West, [32], these maps coalesce to produce isomorphisms:

Wh(~V(H,a))c -- 'Wh~L'p(M)c (H, a)

where the sum is over a set of conjugacy class representatives (H, ~) of all subgroups H of G and all components ~ such that M~ is not contained in all(M).

There is a similar isomorphism of uncontrolled Whitehead groups. WhPL'p(M) can be defined as WhPL'p(M)e, where e = ~ . According to Illman [23] or Rothenberg [29] there is an analogous isomorphism:

Wh(W(H, ~)) ~ ,Wh~L'p(M). (H,a)

For similar decompositions of Wh~L(M) and w~PL(M)c see [23], [31]', the sum ranges over a larger index set in these cases.


(2.4) We now further assume that the F action on the universal cover, M, of M, satisfies (*) of (1.3). According to Connolly-Ko~niewski ([8] Lemma 2.2) the isomorphisms in (2.3) reduce to:

Wh ( N r ( H)/ H)~ ~" , WhPG L" p ( M)c H

Wh(Nr(H)/H) ~ , Wh~ L' "(M) H

~. Wht°V(Nr(H)/H) ~ ' I/IZ/~t°P'PgA/g~''"G ~*'~J

H

But H now ranges over a complete set of conjugacy class representatives of the isotropy subgroups of F; the control manifold for Wh(Nr(H)/H)c is the component p((l~)n) of M ju.

(2.5) The lower K-groups, K_i(F), were first defined by Bass (see [1], p. 664). It is well known that /~-i(F) can be identified with the subgroup of elements of Wh(F × Z i+~) which are invariant under all transfer maps f* , where f : Z ~+1

Z ~+a is any homomorphism whose matrix is diagonal relative to the standard basis. So one defines /£_~(F)¢ as the subgroup of elements of Wh(ffxZi+l)c which are invariant under all transfer maps f * : Wh(F x Z ~ ÷ ~)c-* Wh(F × Z i + ~)~ where f is as above. The control manifold is M × T i ÷~ ; but since these elements are invariant under transfers given by covers T i + ~ T ~+~ which are expanding diffeomorphisms, we may just as well take M as the control manifold. Analo- gously, one defines /~PL tM~ -i,G~ j as the elements of Wh~L(M × T i+1) which are invariant under the above transfers, with similar definitions for /~PL tM ~ and - i , G ~ ]c

gto_~, G(M)" According to Steinberger-West [32] there is an exact sequence:

R _ ~ ( r ) ~ ~ k _ , ( r ) -~ R'_°~ ( r ) ~ k _ ~_1 ( r ) ~ - ~ R _ ~ _ , ( r )

for all i= > - 1 , and a corresponding exact sequence of equivariant K-groups:

~PL ~PL ~top ~ K_i,o(M)~K_i,G(M)~_~P_L._I G(M)c_..+ ~PL tlt/r~ K-i ,o(M)c , -i-l,a,,, , ,~.

(2.6) Consider the coefficient system (in the sense of Bredon [4], p. 1-9 ff.) which assigns to each subgroup H of G the group Kq(H), and to each inclusion of subgroups H ~ K, the induction homomorphism ind , : Kq(K)c ~ Kq(H)c. On the G-manifold M this defines an equivariant homology theory, H~(M; KSq(Gx)).

2 - -H~(M; Rq(Gx)) if (2.7) Lemma. There is a spectral sequence such that Ev, q-- q< 1, and Ev~q abuts to K.p+q(F)~ if p + q < 1.

Proof. This is a special case of the spectral sequence introduced by Quinn, [25], Chap. 8; especially 8.7. []

(2.8) Corollary. I f M and F are as in (2.1),/(,(F)c is a finitely generated abelian group for all n < 1.


Proof This relies on the fact tha t Kq(H) is a finitely generated abelian group whenever H is a finite group and q < 1. When q is negative this is due to Carter [5]. For q = 0 and 1 the result can be found in Bass [1]. From this and (2.7), the result is immediate. []

(2.9) Lemma. Assume M and F are as in (2.1). Then there is a number e.>0, such that the natural map K.(F)c ~ K,(F)~ is injective for all n < 1.

Proof By the spectral sequence and Carter's results [5], /(,(F)~=0 if n< --2. So it is enough to prove the result for each fixed n< 1. Give each group/(,(F)~ the discrete metric and discrete topology. Then /~,(F)c is a closed subgroup of the direct product, I](K,(F)~/0 and is therefore a complete metric space.

According to the Baire category theorem, since /~,(F)~ is countable by (2.8), at least one of its points must be an open set, and so by translation arguments, 0 must be open in K,(F)¢. But the product topology then implies that there are finitely many integers: i~ > i 2 > . . . > i t , and finitely many open sets U~ in Kn(I')l/ij such that /£,(F)~c~(U, x g2... X Ur)={0 }. If the Uj are chosen so that the relax-control maps send Uj into Uj+,, this implies that the map /~,(F)~

/£, (F)~ sends only 0 into U1 if e-1 = i~. Hence/£, (F)~--,/~, (F)~, is injective. []

We now specialize to the case of crystallographic groups. If F is a crystallographic group, an s-expansive endomorphism is a G-map,

g: Mr ~ Mr such that a) the induced map, denoted g , , on Hi(Mr; Z)=A, satisfies: g, (a)=sa for

all aeA. b) the induced map on the universal covers, ~: Mr ~ ~tr expands all distances

by a factor of s. (That is to say, d(~(x), ~(y))= s d(x, y) for all x, y in Mr.) If g: M r ~ M r is s-expansive, it is clear that gk is sk-expansive. Such a G-map g induces a map of Borel constructions, g x 1 : Mr x ~EG

~ M r x a E G , and the induced monomorphism on ~Zl(Mrxc.EG) ( ~ l ( M r x r E F ) ~ F ) , is denoted f = ( g x l ) , : F--*F. It is clear that f l A = g , and that

the resulting map of quotients, F / A ~ F / A is the identity. The map f : F ~ F is only well defined up to a conjugation by an element of A, because the map g does not preserve the base point of Mr; we call f the expansive monomorphism induced by g. The map ~: Mr---,Mr is also well defined only up to composition with an element of A. ~: M r ~ M r is f-equivariant, because ~ x 1' M r x EG

IVI r x EG is f-equivariant. According to Epstein and Shub [12], for any s - 1 mod IGrl, there is an

s-expansive endomorphism g: Mr ~ Mr.

(2.10) Lemma. Suppose F is a crystallographic group and s is an integer prime to IGJ. For any s-expansive endomorphism g: Mr-*Mr, and any subgroup H of Gr, there is an integer k such that map gk: M r ~ M r sends each component of mHr to itself.

Proof We may assume that Mr n is non-empty. According to Connolly-Ko~niewski ([8], Lemma 2.2a) there is a free transitive action of Hi(H; A) on the set ~o(mr n) defined as follows. Each component [s] in ~o(mr n) has the form p((Mr) r)


where p: lfflr~M r is the universal cover, and K is a subgroup of F of the form s(H), where s: H-~F is a homomorphism such that i s=inclus ion: H~G. Each [z] e l la (H; A) is a class of crossed homomorphisms [z: H~A]. The action,

Hi(H; A)× 7ro(Mrn) ~ Zto(Mr H)

sends ([z], Is]) to I-t], where t: H ~ F is given by the formula: t(h)=z(h)s(h). With this description, it is evident that the map g , ' 7z0(Mrn)~Tzo(Mr ~) is

equivariant with respect to the map Hi(H; A) (flA), ~HI(H; A}, where f : F ~ F

is the expanding monomorphism defined by g. Now ( f lA) , is an isomorphism since f IA =( × s), and s is prime to IHI. Hence g,: Z~o(Mrn)~rco(Mr u) is a bijection of a finite set; therefore (g , )k=l for suitable k > 0 (e.g. if k is chosen so that s k= 1 mod IHI). []

(2.11) Lemma. Let F be a crystallographic group, and let g: M r ~ M r be an s-expansive endomorphism. Assume that, for each finite subgroup H, g sends each component of M~ to itself Let B be any Z [1/s] module. Then the transfer map g!: H , ( M r u, anMr; B)~H,(M~, anMr; B) is an isomorphism.

Proof Each component M~ Is], of each fixed set M~, is again a crystallographic torus, and g: Mnr[s]~M~[s] is s-expansive (its crystallographic group is Nr(s(H))/s(H), for any isotropy group H; here s is a monomorphism H-~F).

We first show that g~: H k ( M r ; B) --* H k (Mr; B) is multiplication by s"- k, where n = d i m Mr. Now g,: H,(Mr; B)-~ H,(Mr; B) satisfies:

g, g~=( × s"): H,(Mr; B)--,H,(Mr; B),

and g,: H 1 (Mr; Z )~ H1 (Mr; Z) is multiplication by s. Since Mr is a torus,

g , : Hk(Mr; Z)~ Hk(Mr; Z)

is multiplication by s k for all k. By the universal coefficient theorem it follows that g~: Hk(Mr; B)~ Hk (Mr; B) is multiplication by s"-R. Therefore

g': Hk(Mnr ; B)~ Hk(Mnr ; B)

is an isomorphism for all k and all H. A Meyer-Vietoris argument shows

g~: H,(aHMr; B)~ H,(aHMr; B)

is also an isomorphism for all k and all H. By the five lemma,

g': H , ( M r n, trnMr; B)~ H,(Mp, anMr; B)

is also an isomorphism. This proves (2.11). []

Rigidity and crystallographic groups, l 37

(2.12) Lemma. Let F be a crystallographic group, and let g: M r ~ M r be an s-expansive endomorphism. Assume that s is prime to IGr]. Then for all i< l, the localized transfer map of controlled K-groups (with Mr as control space),

[1/s] KT(r) ®Z [1/s]

is an isomorphism,/f i = 1, 0, - 1, - 2.

Proof We use Lemma (2.7). The filtration on /~i(F)c has length 2+i. So it is enough to see that g~: E2@Z[1/s]-~E2®Z[1/s] is an isomorphism. But for each subgroup H of Gr, the map

Hk(M~, anMr; B) *' ,Hk(M H, aHMr; B)

is multiplication by s r k where r = d i m Mr H, B = K ~ ( H ) (constant coefficients). Since g is N(H)/H equivariant, it follows from the spectral sequence for homology with local coefficients that the map:

HN(rl)/HtA/tH aHMr; Kq(H))®Z[1/s] g' uN(H)/HtA~H anMr; Kq(H))®Z[1/s] k ~2*1F, ) l a k ~lvaF

(local coefficients), is an isomorphism. Hence by an exact sequence argument it follows that the map:

H~(Mr; Kq(Gx))®Z[1/s] g~ ,Hap(Mr; K,q(Gx))®Z[1/s] =E2,q®Z[1/s]

is an isomorphism. []

(2.13) Proposition. Let F be a crystallographic group, with holonomy group G and let g: M r ~ M r be an s-expansive endomorphism. Assume that s is prime to ]GI and to the torsion in Ki((F))c for all i<l . Then g~: KI((F))c~Ri(F)~ is injective for all i <= 1.

Proof This is clear from (2.12) because K I ( F ) ~ i ( F ) ~ ® Z [ 1 / s ] is injective. []

We now arrive at the main result of the chapter.

(2.14) Theorem. Let F be a crystallographic group, with holonomy group G. Assume the control space below is Mr/G. Then:

a) For each i< 1, the forget control map Ki(F)c ~ Ki(F) is injective. b) The exact sequence of Steinberger-West, mentioned in (2.5) reduces to:

K~,~(Mr)~K~,6(Mr)~O fora l l i< l .

Proof b) is an immediate consequence of the decompositions of (2.4) along with a). So we prove a), in case i= 1. Let x be any element in Wh~e(Mr)c which goes to 0 in Wh~L(Mr). x is represented by a G-strong deformation retraction (Y,f) where f : Y ~ M r and the homotopy ft: Y-~ Y has diameter 6, say. Since


x goes to 0, there are CEPL maps: Y, h Z k 'Mr so that fh, ,~krelMr where

k is also a G-strong deformation retraction. The diameter of this homotopy is D, say, (measured in Mr/G ). Now apply the transfer g!: Wh~L(mr)c--* Wh~L(Mr)~ for an s-expansive endomorphism, for any s. The element g~(x) is represented by (Y',f ') , the covering of (Y,f) for which the projection map (Y',f')--*(Y,,f) restricts to g: M r--, Mr. But (Y', f ' ) has diameter 6Is and the homotopy f ' h',-~ k' has diameter D/s. This means that for every s sufficiently large, g~(x) goes to 0 in Wh~L(mr)~ where e is as in (2.9). But this means, by (2.9), that g~(x)=0 for all s sufficiently large and all s-expansive endomorphisms g. By (2.13) this implies that x = 0. []

Note. If we now set i= 1 and restrict our attention to summands corresponding to the (H, e) described in (2.3), we are left with:

0 -~ WhgL' ° (Mr)~ ~ Wh~t'p(Mr)~ Wh~P'°(Mr) ~ --*0

3. Proof of the main theorem in case A

In this chapter, F denotes a crystallographic group with odd order holonomy group, having the form:

F=A x~Z.

It follows that A is also crystallographic, r a n k ( A ) = r a n k ( F ) - 1 , A~ =Arc~ A, and that Gr = GA x ~ Z/n Z for some odd integer n. We are going to prove that:

(3.1) If "forget-control" induces H*(Z/2Z;Ki(A)c)-L-,I~*(Z/2Z;Ki(A)) for each i < 1, then it also induces an isomorphism,

/4" (Z/2 Z;/~i (F)c)--~n* (Z/2 Z ; / ( , (F)) for each i __< 1.

Let Z/n Z---, SO(2) be the standard inclusion. This gives an action of Z/n Z on S 1, and therefore a Gr action on S 1 whose kernel is Gn.

(3.2) Lemma. There is a Gr-equivariant map ~t: M r ~ S 1 which is a fiber bundle with typical fiber M~; the structure group is the group of GA-equivariant homeomorphisms of M ~.

Proof. We will write RA for R®zA, t: F ~ R A r x ~ G r for the embedding of (1.11). F acts on RAt via an embedding RAr x ~Gr~Af f (RAr) . The epimorphism r: F ~ Z extends to an epimorphism of groups, RAr x pGr--*R; this gives a homomorphism of quotient groups (RAr)/Ar x ~Gr~R/n Z and if we restrict this homomorphism to the normal subgroup (RAr)/Ar it gives a Gr-equivariant map re: M r ~ R / n Z = S L n is a bundle and its fiber is (RAa)/An=MA as required. []

Now the involutions on the equivariant Whitehead groups do not usually respect the splittings into non-equivariant Whitehead groups introduced in (2.4). But in the case of Mr this splittings is preserved because each component of


each fixed set has trivial N(H) /H normal bundle, and because Gr has odd order. This is a consequence of Connolly-Lfick ([10], Theorem 4.1).

We are going to use the "twisted Bass-Heller-Swan formula" which was proved by Farrell and Hsiang in [15]. This is an exact sequence:

(3.3) W h ( A ) 1-~ , ;, W h ( A ) i ;, W h ( F ) / C , /~o(A) l - e , , /~o(A)

where C denotes C(ZA, c0@C(ZA, c~ -~) and C(ZA, #) denotes the twisted nil- group introduced by Farrell in [13]. C is a direct summand of Wh(F). Since the group W h ( F x Z ~+1) splits up into summands specified by their behavior under the various transfers of Z ~+~ and these summands are preserved under maps of form f x 1 : F x Z i + 1 ~ F' x Z i + 1, the exact sequence above yields an exact sequence for each i =< 1 :

(3.4) k z ( A ) ' - ' * ,K,(A) ~ ,Ki(F)/Cg ,_R,_,(A) 1-~, ,K~_,(A)

where Ci denotes the summand of C(Z(A x zi+l) , 0~ x I )®C(Z(A x zi+I), ~-1 x 1) which is invariant under the transfers given by the monomorphisms (x s): Z ~+~ ....> Z i +1.

Now we prove an analogous formula for controlled K-groups:

(3.5) Lemma. There is an exact sequence:

£,(A)c 1-~, ,g,(A)c ' ,g,(r)~ ,g,_,(A)c '-~" ,R,-,(A)c

Proof This will use the Meyer-Vietoris exact sequence of controlled K-groups proved in Steinberger [31] (see (8.1)). Let R/n Z - * R / Z = S 1 be the obvious map.

Let p denote the composite map Mr ~ , R / n Z - - * R / Z = S 1. Write S 1 as a union

of two hemispheres: S a = D u D '. Since Gr acts trivially on R/Z, p- l (D) and p- l (D ' ) are Gr-submanifolds of Mr and this gives a Meyer-Vietoris exact sequence:

g~.~(p-'(?b))~-~ gPL , , , - , ,D . @geL, - , .D,- -~ k~.~(Mr)¢ i,G~" ~ J J,: i,G[P [ JJc --~ /~PL " - 1 __~ ~PL t =- 1 , D , , G/~PL " - 1 , D , , -

But p - 1 (D)..~ M~ x D x a~ Gr, so if we restrict ourselves to the summand corresponding to the principal orbit types, we get an exact sequence:

~ i ( A ) c @ ~ i ( A ) c a , ~ i ( a ) c @ ~ i ( A ) c .i~ K i ( r ) c I.). K i - 1 (A )c@Ki - l (Z~ )c

4O

where a denotes the matrix of maps:

F. Connolly and T. Ko~niewski

I] By cancelling out redundant summands of this sequence we are reduced to:

~i(A)~ ,-~ , g i ( A ) ~ i ( F ) ~ i _ l ( A ) ~ 1-~ ~,~i_l(A)c" []

The forget-control map gives a map from the exact sequence of (3.4) to the exact sequence (3.5). This can be reduced to the following map of short exact sequences:

(3.6) 1 1 o ~ (~,(A))~ , ~ , ( r ) / q (g~_l(A))" -,o,

where, for any au tomorphism of a group, e: K ~ K , we write K" and K, for the kernel and cokernel respectively of 1 - e .

We can now complete the proof of (3.1). By hypothesis we have that:

I]*(Z/2Z;I~i(A)~ ) ~ , /4*(Z/2Z;/£i(A)). Also some odd power of ~: A ~ A is an inner automorphism by (1.14). Hence the induced map in K-theory, (also denoted e) is an au tomorphism of odd order, say n. Let ~ = 1 + e + . . . + e"- l . Let K=/~i(A)¢ and K'=Ki(A ). We will first prove that K ' ~ K '~ and K ~ K ' ~ are Tate homology equivalences.

Now, Z K n ( 1 - e ) K and K/(XK+(1-~z)K) are both n-torsion groups, so their Tate cohomology vanishes. Since 0 ~ Z K n (1 - ~) K ~ 2;K (1 - ~) K ~ X K + (1 - e) K ~ 0 is exact, this shows that the sum of two inclusions gives Tate homology equivalences:

Z K O ( 1 - ~ ) K ~ K and ZK'®(1--~)K'-~K'.

Now K ~ K ' is a Tate homology equivalence, so X K ~ X K ' as well as ( 1 - - ~ ) K ~ ( 1 - ~ ) K ' are too. The exactness of the sequences 0 ~ ( 1 - - ~ ) K ~ K ~ K ~ 0 and 0 ~ K ~ K ~ ( 1 - ~ ) K ~ 0 (and the K' analogues) yields, via the

t five lemma, that the maps K ~ ~ K ' ' and K, K , are Tate homology equivalences. The five lemma applied to (3.6) therefore implies that the map:/£i(F)c ~ ff~i(l')/Ci is also a Tate cohomology equivalence. But since the involution interchanges the two summands of Ci, it follows that this is an extended Z / 2 Z module, and so is Tate eohomologically trivial. Hence the map Ki(F)c~Ri(F) is also a Tate cohomology equivalence. This completes the proof of (3.1). []

R i g i d i t y a n d c r y s t a l l o g r a p h i c g r o u p s , I 41

4. Induction techniques: proof of the main theorem in case B

In this chapter we will prove case B of Theorem (1.3). For any group F and any group homomorphism j: F ~ G to a finite group

G, and any subgroup H of G, we denote the group j-I(H) by Fu. Typically in this chapter, F will be a crystallographic, G will be its holonomy group and j will be the natural epimorphism.

We are going to prove:

(4.1) Suppose F is a crystallographic group. Suppose for each hyperelementary

subgroup H of Gr, tha t /4*(Z/2Z; I(i(Fn)c ) ~" , /4*(Z/2Z; Ri(Fu)) for each i< 1.

Then/4*(Z/2Z;/~i(F)c) -- ,/4*(Z/2Z;/£~(F)) for each i< 1. This is going to be an immediate consequence of the following induction

theorem. Let G be a finite group of isometrics of the compact Riemannian manifold

M. We shall write E for M x GEG. Let F denote ~I(E). Write Ki(ff)c for the controlled K-group on E, with controls measured in M/G. It has a description by means of geometric groups, as explained in [28] (also, see below). Both /(~(F)c and /£~(F) come with an involution obtained by reversing the direction of loops in M, allowing us to consider their Tate cohomology. Let Jt ~ denote the class of hyperelementary subgroups of G and cg the class of all subgroups of G. We will prove:

(4.2) Lemma. The functors assigning, to each H in cg, the groups /~*(Z/2Z; K;i(FH) ) and /4*(Z/2Z;/~i(Fu)c) respectively are Jf-computible in the sense of Dress [11]. (Recall a functor F is Jf-computible if F(G)= Lira F(H)).

HE~-~ o

Why (4.2) implies (4.1): If F ( - ) and Fc(-) denote /4*(Z/2Z;/~i(F~_)) and /4*(Z/2Z;/(~(Ft_))~) respectively, then F(G)=LimF(H)), Fc(G)=LimF~(H))

H ~ ¢ HE~'~ '°

and the inverse limit of the isomorphisms in the hypothesis of (4.1) yields the isomorphism in the conclusion.

The rest of this chapter is devoted to a proof of (4.2). We will show that each of these functors/4*(Z/2Z;/~i(ff//)) and tQ*(Z/2Z; Ki(l'n)c) is, in the sense of Dress [11], a Green module over an H-compatible Green functor. (4.2) is then an immediate consequence of the main results of [11].

Recall Swan's ring Go(Z G) is the Grothendieck group of Z G modules which are finitely generated free abelian groups. It acts on Wh(F) as explained in Farrell-Hsiang [14] (Briefly, if (P, a) denotes a projective F module and an automorphism a: P~P, and if N is a ZG module which is a finitely generated free abelian group, then a®l" P ® z N ~ P ® z N represents [P, a] . [N]). Since this induces an action on Wh(F x Z i+1) which sends transfer-invariant elements to themselves, each /£~(F) is a module over Go(ZG). It is shown in [14] that this makes the functor H~K~(Fu) a Green module over the Green functor H -~Go(ZG).

Now in a similar way, Wh(F)~ is a module over G0(ZG) as we now explain.


Let X denote a function from a finite set IXI, to E. (Write x for an element of IXl and Ix] for its image in E, so that X={(x , [x]): x~lSl}). The geometric group on E, denoted Z[X], is, by definition, the free abelian group on X. We consider two such geometric groups Z[X] and Z[Y] on E to be equivalent

if there is a bijection IXI--,IYI such that the composition [XI--'IYI r.-.,E is g . A morphism 0" Z [ X ] ~ Z [ Y ] of geometric groups on E is an element of the free abelian group Mor (Z[X] , Z[Y]) with basis consisting of the set of all (x, z, y) where r is a curve in E from Ix] to [y] and x~[X[, y~[Y[. (Any path f: I~E and any reparametrization of it, f(b: I~E, where ~b(l, 0,1)~(I, 0, 1), are understood to give the same curve). Two morphisms 0: Z[X]~Z[Y] and

0': Z [ X ' ] ~ Z [ Y ' ] are equivalent if there are equivalences IX[ * ,IX'[ and

IY[ *' ,[Y'[ of geometric groups so that the induced group isomorphism Mor(Z[X],Z[Y])~Mor(Z[X'],Z[Y']) sends 0 to 0'. One can compose any

two morphisms, Z [ X ] 0 , Z [ Y ] ~' , Z [ Z ] : (lY'l,a, Izl)o(IxI, T, lYl) is defined

as (Ixl,~ ~r, Izl)" 6~, (Kronecker delta). Let IXl denote the pullback of the universal cover /~ under the map IXI--,E; it is a free F-set. Once complete sets of orbit representative of [)~l and I~'l are chosen, say {xl, 22. . . 2,} and {Yl, Y2... )Tin}, each morphism 0: Z [-X]--*Z I-Y] specifies a homomorphism of two based right ZF modules, 101" Zl)?l--,Zl~'l; I(x, ~, Y)I means the ZF map sending ff to ~g where ~g is the terminus of the lifting of r starting at 2; I(x, r, Y)I sends all other elements of IXI to 0. This yields a homomorphism, Mor(Z[X],Z[Y])

H o m z r ( Z [A'[, Z Irl), denoted 0 ~ [01. We can define the diameter of 0 as the smallest e > 0 such that for each

(x, ~, y) appearing with non-zero coefficient in c3, the set z(I) lies in the closed z-ball around x, where the distance function on E is pulled back from that on M/G. An e-homotopy of 0 consists of a homotopy with endpoints fixed, for each (x, %, y) appearing with non-zero coefficient in 0, from ro to a path z~, such that the e-ball around x contains the track of the homotopy. We say that 0 is an e-isomorphism if there is an e-morphism 0': Z[Y]--*Z [X] so that 00' and 0'0 are e-homotopic to the identity. (In particular, 101-1 = 10'1.) We say 0 is a geometric isomorphism of diameter <5, if there is a bicorrespondence

[XI j 'IYI such that 0 has the form ~ (x, z~,j(x))ex, where each e~ is a unit x ~ X

of Z and Zx lies in an e-ball around x and around jx . Any two e-morphisms over E, say 0i: Z [ X i ] ~ Z [ Y / ] , i=1 ,2 , can be added, since the sum 01+02 is naturally an element of:

Mor(Z [X 1 w X2], Z [Y1 w Y2]) (disjoint unions).

We write this sum 01@02. Any morphism of the form 0Gg, where g is a geometric e-isomorphism is called an z-stabilization of 0. A finite composition of automorphisms of Z [X], say A = A1 A2 ... A,: Z [X] ~ Z [X] is an e-deformation if each IAil has a triangular matrix for some ordering of the basis X, (depend- ing on i) with + l's and - l ' s on the diagonal; and if, in addition, e exceeds the diameter of each product path z, ... z 2 zl for which some term of the form


(ui, zi, ui+l) appears in Ai with non zero coefficient. If 0, (?': Z [ X ] - , Z [ Y ] , we say 0' is an e-deformation of c~ if we can write c3'=At?B where A and B are e-deformations. This symmetric, reflexive, relation is not transitive but if t? and t?' are equivalent under the transitive relation generated by e-deformation, then a stabilization of t?' is a 9e-deformation of a stabilization of & (See [6] Th. 3.5 or [25], p. 384 or [-28] p. 16). Moreover if 0: Z rx]-- , z [-Y] is an p-isomorphism, with e-homotopy inverse ~?', then the sum ~?0)~': Z [ X w Y]- - - ,Z[Xw Y] is a 3e deformation of the identity. (See [28], p. 15, or Milnor, [24], § 1, Eq. (2)).

The above discussion makes it clear that the set of all e-isomorphisms of all geometric modules on E, modulo the equivalence relation generated by e- stabilization, e-homotopy, and e-deformation, forms an abelian group. This is Wh(F)~. One defines Wh(F)c as ,Lim Wh(F)~.

The "forget control" map Wh(F)~--, Wh(F) sends the class of 0: Z [X] --* Z [Y] to the class of the matrix of the lSI relative to the matrix given by the bases )~ and

We now define the two operators resg and indg in the controlled context. For any geometric group Z [X] on E = M x GEG, one defines a geometric module Z[res~X] on E H = M x a E G , by specifying that IresGuX[ is the pull back of

the covering E n p ,E by the map IXI--*E given by X. Then res~nX= {((x, 8), 8): x~lXI, pS=[x]} is a function from IresgX] to Ea. On the other hand if Z [X]

is a geometric group on E u then the composition [XI-*E n p ,E is denoted indu~X which allows us to form the geometric group Z[-indgX] on E. If ~: Z[-X]-~Z[Y] is a morphism of geometric groups on E (resp. Eu), it is clear we can form morphisms of geometric groups: resGu0: Z [resu G X]-~Z[res~ Y] (resp. indg 8: Z [ind~ X] -~ Z [ind~ Y]). For example, rest(x, ~, y) = ~ ((x,O), ~ ~, (y, j~), where ~ ~ ranges over all liftings of z to Eu and 8, ]' are

T~

its endpoints. Now if ~: Z [ - X ] - , Z [ Y ] is a morphism of geometric groups on E, and

N is a Z G module which is a finitely generated abelian group, we define a morphism 0-N as follows. Pick a base B, over Z, for N. Form the geometric groups Z IX x B] and Z [-Yx B] where X x B-- {((x, b), [x]): beB, x~lX]}. If 0=(x, z, y), and if the lifting of z starting at ~, ends at yg, then define 0- N in Mor(Z [X x B], Z[-Yx B]) by the rule:

~?. N = ~ ((x, c), ~, (y,c)) n~,o where for each b in B, b g - 1 = ~ nb,c c. (b,c)eBx B ceB

This insures that if N' is a second ZG module which is a finitely generated abelian group, then (0. N ) . N ' is equivalent to ~ - (N®zN' ) (if the chosen basis of N Q z N ' is {b®b' lb~B, b'eB}). It also insures that the "forget-control" map Wh (F)c ~ Wh (F) is a map of modules over Go (Z G).

If we change the base B = {b~, b2, ..., b,} of N to another base, say B '= {bl + b2, b2, . . . , bn} then the morphism 0. N changes to a new morphism, say (~?. N)', but it is easy to see that (~-N)'=A(c~.N)A' where A and A' are deformations

44 F. C o n n o l l y a n d T. Ko~niewsk i

of diameter 0 (the paths involved in A and A' are constant paths and the matrices IAI, ]A'] are triangular).

Moreover if H is a subgroup of G and • is a morphism of geometric groups over En and M is a Z G module which is a finitely generated abelian group, then ind,(0, res t (m))= ind~(c~)- M. Also, if 0 is a morphism of geometric groups over E, and M is a Z G module which is a finitely generated abelian group, then rest(0 - M) = res~(O), rest(M).

The above discussion makes it clear that Wh(F)c is a module over Go(ZG), and that the rule H~-*Wh(Fu)c is a Green module over the Green functor H~-+Go(ZH).

Therefore for each integer n, /~n(Z/2Z; Wh(F)c) is a module over I4°(Z/2Z;Go(ZG)). The rules H~--~/~"(Z/2Z; Wh(FH)c) and H~-*/4n(Z/2Z; Wh(FH)) are Green modules over the Green functor H~--*/4°(Z/2Z; Go(ZH)).

According to Dress [-11], the proof of (4.2) is then immediate from the following lemma"

(4.3) Lemma. The Green Jhnctor H~-~H°(Z/2Z; Go(ZH)), on the class of subgroups of G, is Jt~-computible.

Proof We must show that the map ~ H°(Z/2Z; Go(ZH)) H ~,~cg

zi,d~ ,HO(Z/2Z; Go(ZG)) has 1 in its image. To do this we employ Dress's

ring GW(G,Z) of pairs (M,b) where M is a ZG module which is a finitely generated abelian group and b: M × M ~ Z is a nonsingular, G-invariant, symmetric bilinear form. According to Dress, H--*GW(H,Z) is H-computible. So 1 is in the image of

GW(H, Z) YindGH :' GW(G, Z).

Since Ad(b)" M~Hom(M,Z)=M* is an isomorphism, the forgetful map (M, b ) ~ M, yields a ring homomorphism:

GW(G, Z)-* H°(Z/2Z; Go(Z G))= Ker {(1 - * ) : Go(Z G)--, Go(Z G)}.

Since the following diagram commutes and the verticals are ring homomorphisms,

Y, /-/°(Z/ZZ; Go(Z/-/)) ~i°d~ ,HO(Z/ZZ; Go(ZG) ) H e ~ '

GW(H, Z) ZindGH ' GW(G, Z), H~.,~

it follows that 1 is in the image of the bottom map. This proves (4.3), and therefore also (4.2) and (4.1).

Rigid i ty a n d c r y s t a l l o g r a p h i c g roups , I 45

5. Proof of the main theorem in case C

In this case we use the basic structure theorem of Farrell and Hsiang on crystallographic groups (see [18], Theorem 3.1) plus the G-version of the e-approximation that was proved by Steinberger and West (see [33], Theorem 1).

For any integer s - 1 mod [Gr[, the group sAr is a normal subgroup of F. Set As=A/sAr, F~=F/sAr and let q: F~F~ be the natural epimorphism. In this section we will let ~ denote the class of hyperelementary subgroups of ~. Then as seen in Chap. 4,

Ki(F)/FIi(F)~ ~ lim /(~(F~)//~(Fn)~

H e ~,~g

( 5 . 1 ) /4~(Z/2Z; Rg(F)/~ii(F)c)~lim /4~(Z/2Z; ~Ig(FH)/Ki(F~rL) Heo~ o

because all of these are Green modules over the Green functors Go(ZG) or H ° (Z/2 Z; Go (Z G)). We will show:

(5.2) Assume that F is a crystallographic group with odd order holonomy, with no fixed sets of dimension two, three, or four and no gaps of dimension two. Assume CaseC holds for F and assume that, for all i, /4*(Z/2Z; ff~,i(ff')/Ki(ff')c)=O for any subgroup F' of finite index in F for which [Gr,[ is a proper divisor of [Grl. Then, for all i , /~*(Z/2Z; Ki(F)/K,i(r)c)=O.

Proof of (5.2). Since ]Gr[ is prime to [As[, we see HJ(Gr; As)=0 for j = 1,2. Hence the sequence

1 -~As--*F~-~Gr-~ 1

splits, F~=Asx~Gr, and there is just one conjugacy class of subgroups of F~ which are sent isomorphically onto Gr by the natural epimorphism F~Gr. But as shown by [18], Theorem 3.1 and [16], Theorem 1.1, for infinitely many s = 1 mod [Grl, this is the unique conjugacy class of hyperelementary subgroups of F~ whose order equals that of Gr. We now choose some subgroup K in this conjugacy class. If H is any group in ~'~ not conjugate to K, then the holonomy group of Fu is isomorphic to a proper subgroup of Gr (see (1.13@ while G~, the holonomy group of FK is just f~(F) where f~: F-~ F is an s-expansive monomorphism. So by hypothesis one has that / t*(Z/2Z; Ki(Fl~)/gi(Fu)c)=O if H is not conjugate to K. Hence (5.1) gives a monomorphism:

(5.3) L*: /4*(Z/2Z; g,(r)/g,(r)c) , I~*(z/2z;

Each element z of/( i(F) is representable by a G-equivariant strong deformation retraction on Mr x ($1) i+1 as in [33], with diameter d, say, when measured in Mr/G. The diameter of fs*(Z) is then dis. Once s is sufficiently large, by the main result of [33], f~(z) is in Ki(F)c. Hence for any [z] in HJ(z/2z; Ki(F)/Ki(F)c), there is an integer s > 0 such that fs~[z]=0. By (5.3), this shows that HJ(Z/2Z; Ki(F)/K~(F)~)=O. This proves (5.2). []


6. A Rothenberg type exact sequence

The main result of this section is (6.3) which is an exact sequence relating the set of homotopy structures of a G-manifold, Sh(M), and the set S(M) of simple homotopy structures of M with the equivariant Whitehead group. It is therefore stems from a large tree of results, the roots of which are located in [30].

(6.1) Assume X is a locally linear G-manifold, and none of its strata have dimension two, three, or four. Assume also that no fixed set component has codimension one or two in any bigger fixed set component. The main result of [31] allows us to identify WhT°p'°(X) as the set of equivariant locally linear h-cobordisms on X. This permits one to define an involution on WhT°p'P(X) by reversing the direction of an h-cobordism, as explained in Chap. 1.

The involution * allows us to define a Z /2Z action relative to which we can discuss/ t*(Z/2Z; WhT°p'°(X)).

(6.2) Lemma. Let F be a crystallographic group. Let (M,f)eSh(Mr). Assume no fixed sets have dimension two, three or four, and no gaps between these have dimension one or two. Then z(f) represents an element of I~ 1 (Z/2Z; WhT°p'P(Ma)). Moreover, if (W,F) is an h-cobordism from (m , f ) to (m',f ') then z ( f )=z( f ' ) if and only if z(W; M, M')~ Kernel ( id- *).

Proof Proposition 3.2 of [10] says that z(f)+*z(f)=--~¢(zG(Mr)) where x~(Mr) is the equivariant Euler characteristic (defined in [10], 1.5). But each fixed set component (Mr~)~ of Mr is a torus (by Proposition (1.5)). Hence the (non-equivariant) Euler characteristic of the pair ((M~),, (M~n)~) is zero. It follows from its definition that zG(Mr) must also be zero. Hence z ( f )+ * z ( f )= O, proving the first claim.

The proof of the second claim is routine:

~(f')=~(F)+ ~(i')=z(F)+ z(W; M', M)

=T(f)--T(W, M, M') + z(W; M', M)

= z ( f ) - z ( W ; M, M') + *T( W; M, M ' ) = z ( f )

if and only if ~(W; M, M')~Kernel( id-*) . []

Lemma (6.2) allows us to define an action o f /4° (Z /2Z ; Wh~°P'P(Mr)) on the set S(mr): if (M,f)~S(mr) and [x]~/4°(Z/2Z; WhT°p'P(Mr)) there is an h- cobordism (W;M,M')~WhT°p'P(M), in the kernel of ( id-*), such that f , r(W; M, M')=x, and a G-homotopy equivalence F: W ~ M r extending f. Set [x]. (M, f ) = (M', f ' ) where f ' = F IM'. Then (M', f ' ) ~ S (Mr) by Lemma (6.2).

If (W; M, m') represents zero in n o (Z/2 Z; Wh~ °p'p (Mr)) then (m, f ) = (M', f ' ) because (W; M, M') can be factored into a cobordism of the form:

(V; M, N)+u(V; N, M).

Thus we have defined an action of H°(Z/2Z; WhT°p'P(Mr)) on S(Mr). Now suppose (Mi,f/), i=0, 1, are in S(Mr). (a(Mo,fo)=c~(Ml,fl) in Sh(Mr)if and only if there is an h-cobordism (W; Mo, M0 connecting (Mo, fo) and (M1, fl).


By L e m m a (6.2), the t o r s i o n o f ( W ; M o , M 1 ) r ep resen t s an e l e m e n t of / 4 ° ( Z / 2 Z ; Wh T°p" P(mr)). L e m m a (6.2) a l so shows tha t Image(~b) cons is t s o f t hose

(M, f ) in Sh(Mr) w h o s e t o r s i o n is in I m a g e ( i d - * ) . T h e r e f o r e we have p r o v e d the f o l l o w i n g p r o p o s i t i o n :

(6.3) Propos i t ion . Let F be a crystallographic group with no f ixed sets of dimension two, three, or four, and no gaps of dimension one or two. Then the action defined above provides an exact sequence:

/ 4 ° ( Z / 2 Z ; Wh~°P'P(Mr)) ,S(Mr) ~' ,Sh(Mr) ~ , / - ) ~ ( Z / 2 Z ; Wh~,°P'°(Mr)).

I t is exac t in the s t r o n g sense tha t qS(x)=q~(x') if and o n l y if x a n d x ' a re in the s a m e o rb i t o f n ° ( Z / 2 Z ; Wh~ °p' p (mr)). []

Proof of Corollary 1.5. This is i m m e d i a t e f r o m (6.3) a n d (1.3). [ ]

Acknowledgements. Both authors wish to thank the University of G6ttingen and its S.F.B. (Geometric and Analysis), for their hospitality during the summer of 1988, when this paper was written.

Bibliography

1. Bass, H.: Algebraic K-Theory. New York: W.A. Benjamin Inc., 1968 2. Bass, H., Heller, A., Swan, R.: The Whitehead group of a polynomial extension. Publ. I.H.E.S.

22, 64-79 (1964) 3. Bieberbach, L.: Ober die Bewegungsgruppen der Euklidischen Rfiume. I: Math. Ann. 70, 297 336

(1911); II: Math. Ann. 72, 400M12 (1912) 4. Bredon, G.: Equivariant Cohomology Theories. (Lect. Notes Math., vol. 34). New York Berlin

Heidelberg: Springer 1967 5. Carter, D.: Lower K-theory of finite groups. Comm. Algebra 8, 1927-1937 (1980) 6. Chapman, T.: Controlled simple homotopy theory and applications. (Lect. Notes Math., vol. 1009).

New York Berlin Heidelberg: Springer 1983 7. Cohen, M.: A course in simple homotopy theory. New York Berlin Heidelberg: Springer 1973 8. Connolly, F., Ko~niewski, T.: Finiteness properties of classifying spaces of proper 1' actions.

J. Pure Appl. Algebra 41, 17-36 (1986) 9. Connolly, F., Ko~niewski, T.: Rigidity and Crystallographic Groups II. (in preparation)

10. Connolly, F., Lfick, W.: The involution on the Equivariant Whitehead Group. J. K-Theory (to appear)

11. Dress, A.: Induction and structure theorems for orthogonal representations of finite groups. Ann. Math. 102, 291-335 (1975)

12. Epstien, D., Shub, M.: Expanding endomorphisms of flat manifolds, Topology 7, 139-141 (1968) 13. Farrell, T.: The obstruction to fibering a manifold over a circle. Indiana Univ. Math. J. 21,

315-346 (1971) 14. Farrell, T., Hsiang, W.C.: Rational L-groups of Bieberbach groups. Comm.Math. Helv. 52, 89-109

(1977) 15. Farrell, T., Hsiang, W.C.: A formula for K1(R,[T]). Proc. Syrup. Pure Math, vol. 17, Applications

of Categorial Algebra. American Mathematical Society: Providence 1970 16. Farrell, T., Hsiang, W.C.: The topological-euclidean space form problem. Invent. Math. 45, 181-

192 (1978) 17. Farrell, T., Hsiang, W.C.: Topological Characterization of flat and almost flat manifolds, M",

n4:3,4. Am. J. Math. 105, 641-672 (1983) 18. Farrell, T., Hsiang, W.C.: The Whitehead group of poly-(finite or cyclic) groups. J. London Math.

Soc. 24, 308-324 (1981)


19. Farrell, T., Jones, L.: The surgery L-groups of poly-(finite or cyclic) groups. Invent. Math. 91, 559-586 (1988)

20. Farrell, T., Jones, L.: K-theory and dynamics I. Ann. Math. 124, 531-569 (1986) 21. Farrell, T, Jones, L.: Compact negatively curved manifolds are rigid (announcement) 22. Hsiang, W.C., Shaneson, J.: Fake Tori. In: Cantrell, J.C., Edwards, C.H. (eds.) Topology of Mani-

folds. Chicago: Markham 1970, pp. 18-51 23. Illman, S.: Whitehead torsion and group actions. Ann. Acad. Sci. Fenn., Set. A I. Mathematica

vol. 588, 1974 24. Milnor, J.: Whitehead torsion. Bull. Am. Math. Soc. 72, 358 426 (1966) 25. Quinn, F.: Ends of Maps II. Invent. Math. 68, 353~424 (1982) 26. Quinn, F.: Applications of topology with control. In: Proceedings of the 1986 International Con-

gress of Mathematics, vol. 1, American Mathematical Society, Providence, 1987, pp. 588-602 27. Quinn, F.: Homotopically Stratified Sets. J. Am. Math. Soc. 1, 441~499 (1988) 28. Quinn, F.: Geometric algebra and ends of maps. Lecture notes of CBMS conference lectures

at the University of Notre Dame, August 1984 (unpublished) 29. Rothenberg, M.: Torsion Invariants and finite transformation groups, Proc. of Symp. in Pure

Math. vol. XXXII, part 2: Algebraic and Geometric Topology, pp. 267 311 30. Shaneson, J. : Wall's surgery obstruction groups for G × Z. Ann. Math. 90, 296-334 (1969) 3l. Steinberger, M.: The equivariant topological s-cobordism theorem. Invent. Math. 91, 61-104

(1988) 32. Steinberger, M., West, J.: Equivariant h-cobordisms and finiteness obstructions. Bull. Am. Math.

Soc. 12, 217-220 (1985) 33. Steinberger, M., West, J.: Approximation by equivariant homeomorphisms. Trans. Am. Math.

Soc. 302, 297-317 (1987) 34. Steinberger, M., West, J.: Controlled finiteness is the obstruction to equivariant handle decomposi-

tion (to appear) 35. Wall, C.T.C.: Surgery on Compact Manifolds. New York: Academic Press 1970 36. Weinberger, S.: Personal Communication

Oblatum 5-III-1989

Rigidity and crystallographic groups, I

Documents

Transcript of Rigidity and crystallographic groups, I