G-CW-surgery andK0(ℤG)

Math. Z. 179, 11-42 (1982) Mathematische Zeitschrift

�9 Springer-Verlag 1982

G-C W-Surgery and Ko(2E G)

Robert Oliver 1,, and Ted Petrie 2,**

1 Matematisk Institut, Ny Munkegade, DK-8000 Aarhus C, Danmark 2 Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903, USA

Let G be a finite group. All maps between G-spaces (spaces with G-action) will be assumed equivariant. For any G-spaces X and Y,, a map f : X ~ Y is called a pseudo-equivalence (p.e.) if it is (non-equivariantly) a homotopy equivalence; f is called a quasi-equivalence (q.e.) if it induces an isomorphism on ~1 and integral homology. The basic problem addressed here is

(0.1) Given f: X~Y.. When does there exist a G-space X'~_X and a pseudo- equivalence (or quasi-equivalence) f ' : X ' ~ Y extending f?

The procedures for constructing such extensions (X', f ' ) will be referred to here in general as "G-surgery".

The obstructions to answering (0.1) depend, of course, on the category in which one is working. The obstruction which seems to be common to all categories, an invariant 7(f)e/~0(7ZG ) depending on the mapping cone Mz of f , was defined separately by both authors ([-11] and [14]). Formally, many of the ideas there are equivalent, but the motivations and applications are quite different. Oliver used 7(f) to solve (0.1) in the category of G-complexes when Y is a point; while in Petrie's work, this is just one part of the obstruction theory [-14-17] arising when considering (0.1) in the category of smooth G-manifolds.

Our goal here is to isolate y(f), by studying (0.1) in the category of finite G- complexes (where no other algebraic obstructions occur). But a principal motivation is still surgery on closed G-manifolds, and so many of the technical results proven here are stated with greater emphasis on generality than is needed for the applications in this paper.

Note that without some side conditions, (0.1) is trivially solved (for G- complexes) by taking X' to be the mapping cylinder of f (in particular, "surgery" in this sense on CW-complexes is meaningless in the non-equivariant case). The side conditions imposed are restrictions on the isotropy subgroups allowed in X ' - X . Such conditions arise naturally: e.g., one might demand that

* Partly supported by a National Science Foundation summer grant, partly by a Sloan fellow- ship ** Partly supported by National Science Foundation contract no. 76-06652

0025-5874/82/0179/0011/$06.40

12 R. Oliver and T. Petrie

(X')U=X ~ for certain H_~G, or (as in the smooth category) that X' have the same isotropy subgroups as X.

The treatment of (0.1) will now be sketched. By a "family" of subgroups of G we mean any set of subgroups closed under conjugation; in particular Iso(X) denotes the family of isotropy subgroups of a G-space X. Let N (or N(G)) denote the family of subgroups of G of prime power order. A family Y is called connected if for any PeN, there is a unique minimal subgroup /Se~- containing P; when this holds we write ~ = {/~l P ~ N } _ ~ .

For simplicity, we consider (0.1) here only in the case where YP is connected for all P e N (the general case, although often more interesting, requires the language of "G-posets" developed in Sect. 1). The main result (Theorem 3.2) then takes the following form:

Theorem. Let f: X ~ Y be any map between finite G-complexes, such that ye is connected for all PeN. Let ,,~ be any connected family containing Iso(Xu Y), and let y ' c ~ be any subfamily containing ~ . Assume also that Xa +O or ~,~ ~_N. Then there is a finite G-complex X ' ~ X such that I s o ( X ' - X ) ~ ~' , and a q.e. f ' : X'--* Y extending f, if and only if

[Y]-[X]EA(G,~)+~2(G,~ ' ) in (2(a).

Here, ~2(G) denotes the Burnside ring of G (under tom Dieck's definition [6]). f2(G,~ is simply the subgroup generated by all orbits [G/H] for H 6 ~ ' ; while A(G, ~ ) is characterized (Proposition 1.6) by

A (G, ~ ) = {IX] - lef2(G) [ (X, x) a finite contractible based

G-complex, Iso (X - x)_~ ~} .

The condition "XG:#~ '' can be removed if certain other conditions on Y are met.

The proof of the theorem splits into two main steps. The first (Sect. 2) is to extend f to a map fz: X I ~ Y such that (1) I s o ( X 1 - X ) _ ~ , (2) [Y] -[X1]~A(G,~') , (3) ~1(fl) is an isomorphism, and (4) for some n, /~,(Mj-1) =H,(MI1) is a projective 2~G-module. The projective obstruction is then defined:

7(L) = ( - 1)" [-H,(MI~)] ~/s (77 G).

Condition (2) insures that 7(fl) lies in a certain subgroup Bo(G,~)~Ko(ZG), designed to be the group of "removable" projective obstructions for G-CW- surgery with isotropy subgroups in Y.

The second step is to extend f to a q.e. by eliminating H,(MI,). The procedure is based on three technical results, Theorems 4.3, 4.5, and 4.12. The first says that Bo(G,J~)=Bo(G,~), insuring that the extension to f ' : X ' ~ Y can be made with Iso(X'-X~)_~ ~_~ i f ' (thus meeting the original side condition). Theorems 4.5 and 4.12 involve decompositions of Bo(G, ~).

One more observation is worth making here. Suppose that X and Y are smooth G-manifolds. The connectedness condition on Y easily implies that Iso(Y) is a connected family: for any PeN, yv= Up yG,. This is a finite union.

G-CW-Surgery and Ko(~.G ) 13

ye is connected and contains each Yay as a properly imbedded submanifold. Thus ye_yG, for some y~ye, and P=Gr~Iso(Y). (This, of course, is the original motivation for the idea of a "connected" family.) If, furthermore, f is a degree one map, then Iso(Xu Y) is connected, and Iso(X)_ Iso(X u Y) ~ [i6]. Thus, the restrictions in Theorem 3.2 fit very naturally into smooth G-surgery; e.g., setting ~ = Iso(X w Y) and g*-'= Iso(X).

Applications of Theorem 3.2 are given in Sect. 3. These take the form of describing the structure of Iso(X) and X G, as X varies over G-complexes q.e. to a fixed G-complex Y The main technical but important Theorems 4.3 and 4.12 are deferred until Sect. 4 so as not to interrupt the course of the geometrical argmnents leading to Theorem 3.2 and its applications.

1. Preliminaries

Let Y(G) denote the set of subgroups of G with G acting by conjugation, Let ~___ ~9"(G) be the set of subgroups of G of prime power order.

For any G-space Y, define

it(r)= H H~G

For ~e~0(Y u) set p(~)=HeY(G), and let I~l c y be the underlying space of the component e. Order/ /(Y) by setting ~<fl whenever ]c~] ~lfll and p(e)~-p(B). If Y is connected, then H(Y) has a unique maximal element m: with p(m)= 1 and

Note that p: tl(Y)--,9"(G) is a map of partially ordered G-sets. In this paper, the term "G-poset" will refer to a partially ordered G-set H = H ( Y ) for some connected G-space Y, with p : / / ~ 5 : ( G ) understood as part of its structure. The following properties of G-posets are immediate; they can, in fact, be used to axiomitize the definition of a G-poset, if desired.

Lemma 1.1. Let H be a G-poset. For any ~ H and H c_p(cQ there is a unique ~ E H with c~>=c~ and p(~u)=fL If ~, fl, 7EH are such that p(cQ~_p(fl)~p(7) and ~>=y, fi>_)', then ~>[3. F1

The term "G-complex" is used here to mean a CI~complex with G-action which permutes the cells in such a way that if gEG sends a cell to itself, it does so via the identity map. A //-complex Z, for any G-poset //, is a finite G- complex with base point q and sub-complexes Z c_Z (with qsZ~) for all aE//; such that Zg~--gZ~ for any geG, Z ~ Z p for ~<fl, and

ZU= V Z~ for Hc_G. #(~)=H

The chief examples of H-complexes come from maps f: X--,Y between finite G-complexes, when H=II(Y). Let X + and Y+ denote the spaces X and Y with disjoint basepoint added (fixed by G), and let M l, denote the mapping cone o f f Write

Y~ = ]c~l, X~ = X p(~) c~f - t (y~),


and let f~: X~-+Y~ be the restriction o f f Then X +, Y+, and Mr can be given the strucure of H-complexes by setting (X+)~=(X~) +, (Y+)~=(Y~)+, and (Mfl~ = M s . (Note that X and Y themselves are not H-complexes.)

Alternatively, let Z be an arbitrary H-complex (H any G-poset). Then, for any G-invariant subcomplex A c Z containing the basepoint, A and Z/A have natural induced structures as H-complexes:

A~=Ac~Z~ and (Z/A)~=ZjA~ for e~H.

For any G-poset H, set ~(H)={e~fI[p(c~)~N}. By a family , ,~_H is meant any G-invariant subset. A family Y~_H is called connected if m e ~ , and for each ee~(II), the set {fl_<eJfleff} either is empty or has a unique maximal element d. For connected i f , we set

= {41 ~ e ~ ( u ) } _~ y .

The term "G-family" will often be used to mean a family in some G-poset, without specifying the G-poset. Many of the constructions are independent of the G-poset (though whether or not a family o~__cH is connected does unfor- tunately depend on H).

As a special and illustrative example, take H=H(pt). It is isomorphic to ,Y(G) via p. A family f f_c~(G) is connected if for each Pe~ a either {H

P I H~J ~} is empty or it contains a unique minimal subgroup P. Let i f _ H be any G-family. An ~-complex is a H-complex Z such that, for

any eeH, z~ = u {Z~ : fie,~-, fi < e}

The idea is that Z should be built up by attaching "a-celts" G/p(e)x D" (to be defined more precisely later) for e e Y only. When H = Y(G), the condition just reduces to requiring Iso(Z-q)__ff .

The following will be crucial to the constructions in Sect. 2, and is the reason for defining connected families:

Proposition 1.2. If o~ ~_H is a connected G-family and Z any JZ-comptex, then Z~ =Z~ for any o~e~(H).

Proof. Z~= U zp=z~, since d is the maximal element of {fieff[fl<e}. [] e f

Consider again the case of a map f: X-~ Y between finite G-complexes, with H=H(Y). The following families in H will be used in later sections:

g ( Y ) = { e e u l p(~)~IsoO~)}

~ I = {c~eHIp(e)eIso(X~ w Y~)}.

These can be thought of as analogs in H(Y) of Iso(Y) and Iso(My); note that Y+ is an -~(Y)-comptex and M s an J~r-complex.

For any family if___ H, now let YJ(G, ~-) denote the group of equivalence classes of ~'-complexes under the relation

Z ~ W if z(Z~)=)~(~) for all eeH:

where )~ denotes Euler characteristic.

G-CW-Surgery and Ko(TgG ) 15

Addition is given by wedge product, and zero is the equivalence class of a point. Note that ~(G,~) can be regarded as a subgroup of f2(G,//). Denote the equivalence class of an ~-complex Z by [Z]~Q(G,~).

The following reiations are clear (just compare Euler characteristics):

Proposition 1.3. (a) Let A ~_ Z be any inclusion of ~-complexes ( ~ any G-family). Then

[ z ] = [A] + [Z/A] in ~(C, g).

(b) Let f: X ~ Y be any map of finite G-complexes, and//--H(Y), 7hen

[M),]=[Y+]-[X +] in f2(G,H).

(c) Let f be as in (b), and f ': X' ~ Y any extension o f f (X'~X), Then

[Mf,] = [ M f ] - [X'/X] in f2(G,H). []]

The following technical resntt wilt be needed in Sect. 2.

I, emma 1.4, If Z is an 2Y-complex (~7c_ H any family) and x(Z~)= t for all c~m~, then

[Z]=0 m f2(G,~).

Proofi Let ~ - ~ be a minimai element, and consider 2--{..) Z~. This is an gag

invariant subcomplex of Z; and (by minimality of c~) 72r for an), fl~// is a wedge product of the Z~ for certain g~G. Since )~(Zg~)= t (g~e.Y), this implies [ Z ] = 0 in f2(G, ~), Thus Z/Z is an (~'-a~)-complex with [Z/Z]=[Z], and induction on Igl shows that [Z/Z]=O. []

Let H be a G-poset. The basic elements of f2(G,H) come from H-complexes (~) for ~ / / : (e) is the disjoint union of G/H (//=p(~)) with a basepoint q, and has H-structure defined by

(m)~={q}u(Q) gH) for /~ / / .

Write [c~] = [(~)]eg2(G,/7). Note that (~) is an ~--complex for any family ~-_~H containing cc

The space (c~) is a special case of the notion of attaching an "~-ceti" or "i- celt of type c~" to a //-complex Z. Such an operation is defined by an attachment map

~o: G/I-I• (H= p(~))

such that e(gHxS~-~)c_Zg~. The new space Z ' = Z U ( G / H x O ) has an ob- q~

vious H-complex structure:

(Z')p=Zcw(g~r ~) for ricH.

Note that the attaching map ~0 is determined, up to homotopy, by an element of ~,_ 1 (Z~).


Clearly, any H-complex is built up from its basepoint by attaching a-cells for c~eH; it is an ~-complex if and only if it is built up by attaching a-cells for c~Y. Note that (c~) is just the result of attaching a zero-cell of type ~ to q. In fact, if Z' is obtained from Z by attaching an/-cell of type ~, then

[Z ' ]=[Z]+(-1)~[c~] in f2(a,H).

Proposition 1.5. Y2(G, ~ ) is the free abelian group generated by [e] for c ~ / G . (Compare [6].)

Proof Since any Y-complex is built up by adding ~-cells for ~ f f , and since adding an a-cell has the effect of adding _+ [~] to the class in Y2(G, ~-), this group is clearly generated by the [c~ l for e ~ . It remains to show that these elements are linearly independent.

Suppose y' n~[fi]=0 for n~28. Then ~n~2((fi)~)=0 for all ~ (setting Ile~lG

2(X)=z(X)-1 for any space X). Let c ~ / G be a minimal element with G#=0. From the definition of (fi), 2((fi)~)=0 unless gfl__<~ for some g; and (~ being minimal) this occurs for at most one fi with n,4=0, and gf l=e. So we may assume fl = ~, and

0--n. 2((~)0 = n,. IG/p(~)l.

Then G = 0 , a contradiction; and so all coefficients must be zero. []

In order to handle the aspects of G-surgery relating to the projective class group, we introduce the following notation. A G-resolution is a finite G- complex X which is "k-dimensional and (k-1)-connected for some k, and for which Hk(X) is 28G-projective. An ~-complex which is a G-resolution will be called an ~-resolution (or occasionally, when needed for emphasis, a (G, Y)- resolution). If X is a k-dimensional ~--resolution, set

?G(X) = ( -- l)k [Hk(X) ] ~/~ o (~ a);

where /~o(2gG) is regarded as the group of finitely generated projective 7/G- modules modulo free modules.

If X is an Y-resolution, then X is contained in an .~-complex X* obtained from X by adding free cells G x D', and having z(X*) = 1. To see this, it suffices to remark that (with k = dim(X)), the rank of Hk(X ) is a multiple of the order of G by ([22], Lemma 8.1). Even though X* is not uniquely defined as an ~-- complex, its class in f2(G, ~ ) is: X~ = X* for any c(:t: m. Now set

{X}=[X*]~O(G,J~), and

Bo(G, ~ ) = {76(X)1X is an ~--resolution, {X} =0}.

Note that Bo(G,,~ ) is a subgroup of/(o(TZG), since

?a(ZX)=-yG(X ) and ?;~(X v Y)=76(X)+?~(Y)

for ~-resolutions X and Y of equal dimension with {X} = {Y} =0. (If one wants to add 7G(X) and ?6(Y) for X and Y of different dimension, m-cells G x D ~ can be added to one of them to equalize dimensions without changing 76.)

G-CW-Surgery and Ko(7I G ) 17

Set @(G, Y) = {{X} IX is an ~-resolution}.

The above remarks show this to be a subgroup of (2(G, @). Likewise,

{({X}, 7G(X))I X an ~-resolution}

is a subgroup of ~?(G,Y)X/(o(ZG), and so there is a well-defined "obstruction" homomorphism

r~: ~(G, g)--,~o(TZ G)/Bo(G, g), (r~({X})=~,~(X)+Uo(G,g)).

Define A (G, Y) = Ker(F~). The group r has a simple combinatorial description in terms of

Euler characteristics (see Proposition 2.6). It is important as an approximation to A(G, @), which is characterized by the following geometrical property (compare [12]).

Proposition 1.6. A(G, ~ ) is precisely the set of elements in f2(G, Y ) respresent- able by contractible ~-complexes.

Proof. If X is a finite contractible Y-complex, then [X]~b(G,~-) and Ff([X]) =0 by the definitions. Conversely, if x~A(G,Y)=Ker(F~), then, from the way F~ was defined, there is an ~-resolution X with {X}=x and yG(X)=0. In other words, Hk(X) is stably free (k = dim(X)), and so free orbits of cells G x D i (i=k and k + l ) can be attached to X to produce a contractible ~-complex X'. By definition, IX'] = {X} = x. []

2. G Surgery and the Projective Class Group

The basic constructions for "G-CW-surgery" will now be described, except for those involving calculations in/(o(ZG). First, the simpler case of constructing ~-resolutions is considered, giving a combinatorial description of q~(G,~). Afterwards, the procedure for extending a map f: X--*Y (meeting certain conditions) to a map f ' with M s. an ~-resolution is described - the techniques are basically the same, but it is here where it becomes clear why the ideas of G- posets and families are necessary.

The idea of all of these constructions is to construct R-acyclic complexes or R-homology equivalences ( R = ~ , ) or ~) in two steps. First, cells are attached to push all homology into a single dimension. One then checks whether the remaining homology group is free (over an appropriate group ring), and if so attaches more cells to remove it. The conditions for the modules being projective or free are similar to those needed for constructing contractible complexes in [11], and are covered in the following five lemmas. Compare also [14].

Lemma 2.1. Let R ~_II~ be a subring, and X a finite k-dimensional G-complex such that/t ,(X; R ) = 0 for i+k. Then Hk(X; R) is R[G]-projective if and only if, for any prime p not invertible in R and any p-subgroup I + P ~ G , X P is 7Zp- acyclic.

18 R. Ohver and T. Petrie

Proof By Lemma 2 and Proposition 4 in [10], Hk(X; R) is R[Gl-projective if and only if Hk(X; 2g~p)) is 7/~v)[G;]-projective for all pr Gv being any p- Sylow subgroup of G. It thus suffices to prove the lemma when R = 7@) and G = G v is a p-group.

Assume first that X e is 7/;-acyclic for all p-subgroups (thus all subgroups) I+P~_G. Then {Xe: I+P~_G} is a set of gp-acyclic subspaces closed under intersections, and

x" l q : P c G

is 2gp-acyclic. Since G permutes freely the cells in X - J f, each term in the cellular chain complex C,(X, Jf; 7@)) is a free 7@)[G]-module. We have

H~(X,J(; ;g(p))_=/~i(X; 7/(p))-~0 for i+k,

and so the following exact sequence

0~Hk(X; 7@))--, Ck(X, ~; ~(p))~...--, Co(X, X; 7Z(p~)~O

splits to show that Hk(X; 2g(v)) is 7Z(p)[G]-projective. Conversely, assume Hk(X; 2g(v)) is projective. 2g(,)[G] is a local ring, since G

is a p-group, and so ([9], Lemma 1.2) any finitely generated projective 7z(p)[G]- module is free. If X is simply connected, we have

~k(X)| ~ Hk(X; ~,~)

by the generalized Hurewicz theorem. We can thus choose a basis (over :g(v)[G]) for Hk(X; 7Z(;)) of elements representable in homotopy, and attach orbits of cells G xD k+l accordingly to construct a ]gp-acyclie G-complex X' _~X. By Smith theory, we now get that XP=(X') P is ;gFacyclic for any I=~P _~G.

If X is not simply connected, then the above argument applies to XX. Then S(X e) is 2~p-acyclie for all 14: P_~ G, and so X e is 7/Facyclie. []

One more lemma will be necessary in order to apply Lemma 2.1 to ~ - complexes. If H is a G-poset and ~___H a connected family, then for any m + e e ~ , n(c 0 will denote the product of all primes p such that e=/~ for some fie~(H) with p(fi) a p-group. Set n(m)=0.

Lastly, define 7Z~)=TZ(,(~)) for c~e~, where 7l(,), for any integer n>=0, means 2g localized at the set of primes pin. Note that with this definition 2~(,,)= 2g(o ) =7Z (and 2g(1)=11~). In other words, in ~(~) precisely those p for which X~ need not be 2~v-aeyclie for an arbitrary contractible ~-complex X have been in- verted; hence X~ must be 7l(~-acyclic for such an X (see Proposition 1.2).

Lemma 2.2. Let ~ be a connected family, c ~ , and X a finite @-complex such that for all ~ e ~ with fl<c~, X~ is 7Z~)-acyclic. Then for any prime p]n(~) and any p-subgroup

1 * K/p(~) ~_ Gjp(cO,

(X~) K is 7Zp-acyclic.

G-CW-Surgery and Ko(2gG ) 19

Proof Let (2 be a p-Sylow subgroup of K. Write H=p(c 0. Since H<zK with p- power index, K = Q H, and P = Q c~ H is a p-Sylow subgroup of H. Consider c~p: the unique element of H with p(ae)=P and c%>c~ (see Lemma 1.1).

Since c~e~, there is some fie_Y/with p(6) a p-group, such that c~=$. P being a p-Sylow subgroup, we may assume that P_~p(c~). Then c~<6, c~<c%, and p(c%) ___p(6); so c%<6 by Lemma 1.1. We thus have

SO C~ is the unique maximal element of both, and ~ = a~. The fixed point set of K splits as a wedge:

(XS= V x~, p(/~)= K

and so it suffices to show that X~ is Zp-acyclic for all such ft. Fix such a fi; if {7<fi: ),e~, ~-} is empty, then Xp is a point, and we are done.

Assume otherwise: choose y<f l with 7ef t . Consider the following poset elements and corresponding subgroups:

~Zp

fiQ riO_ Q P(fiO)

fi P K

~, U p(~)

We want to show that riO_ < fi" By definition, riO_<= fiQ, so Q ~ p(fiO_). Also, riO_ < ~p and fiO_e~ imply fiO_<~;=c~, so H~_p(fiO_). Thus, K=HQ~p(fiO_). Furthermore, y<f l and 7<riO_ (since Y<fio and y e ~ ) ; so we get fiO_<=fi by Lemma 1.1.

Thus, fiO_<fi<fio, and so Xtjo~_X~_X~. By Proposition 1.2, Xe(=XpQ. Furthermore, fiO_<c~ and p[n(fiO_), so X~=X~(~ is 7Zv-acyclic by hypothesis. []

Lemma 2.1 and 2.2 can now be combined as follows:

Lemma 2.3. Let ~ be a connected family, e e l , and X a finite ~-complex such that for all f i e~ with fi<c~, X~ is 7l(~}-acyclic. Assume further that IYIi(X~; ~(~)) =0 for all i<k=dim(X~). Then Hk(X~; 2g(~)) is a projective 2g(~)[G~/p(c~)]- module. []

It remains to find conditions for Hk(X; R) to be free when already known to be projective; this can be studied combinatorially only when R=TX(,) for n > l .

Lemma 2.4. Let n>= 1, and assume M is a projective ~(,)[G]-module. Write II~M for Q| and )~QM for its character. Then M is free if and only if Z~t(g)=0 for all l=t=geG.

Proof If M is flee, then so is QM, and so X~M(g)=0 for all l q=gEG. Con- versely, one easily checks that if Z~M(g)=0 for all l=t=geG, then • M must be a free Q[G]-module:. By a theorem of Swan ([22], Corollary 6.4), the p-adic completion M v is 2gp[G]-free for all pin; and so M/pM is Z,[G]-free.

20 R, Oliver and T, Petrie

We may clearly assume that n is a product of distinct primes, and thus that

;g,,~ I~ ff.p; M/nM-~ [I M/pM, etc. pin pin

Let F be the free 2g(,)[G]-module of the same rank as M; then M/pM~-F/pF for all pin, and so M/nM~F/nF. Since M is projective, any isomorphism from M/nM to F/nF is covered by a homomorphism f: M ~ F . The determinant (over 2~(,~) of f is a unit mod n, thus a unit in 2~), and so f is an isomorphism. So M is free. []

Lemma 2.5. Let X be a finite G-complex such that Hi(X; ;g(,))= 0 for i=l= k and Hk(X; ;Z~,)) is 2g(,~[G]-projective. Then Hk(X; 2~(,}) is free if and only if for any cyclic subgroup 1 ~ H~_ G, z(X u) = 1.

Proof. Let

O-~ CN ~ C~_ I--, . ..--, Co-*O

be the rational cellular chain comptex for X: Ci= C~(X; Q). Then

N ( - 1) i Ec3 = ( - 1) k E~/k(X; Q)3 + Q

i=O

in the rational representation ring, and so for any g~G,

N

( - 1) ~ )~ ;~) (g ) + 1 = ~ ( - 1r Zc,(g). i=0

C i has a basis corresponding to the/-cells of X and permuted accordingly by G; it follows that )~c,(G) is just the number of/-cells in XL Thus

ZH~(X;Q)(g) = ( - 1) k [Z(Xg) - 13

for all g~G; by Lemma 2.4, Hk(X; 2g(,)) is free if and only if z(Xg) = 1 for any 1 4=g~G. []

The description of ~(G,,~) can now be proven:

Proposition 2.6. Let ~ ~_ 17 be any connected family. Let X be any finite ~7. complex. Then [X] ~49(G, Y ) if and only if for any c~E [~ and any cyclic subgroup K/p(~) ~_ Gjp(cr z((X~) ~) = 1.

Proof, I. ~ (G ,~ ) was defined to be the set of all elements {Y}~Q(G,g) for all ,~--resolutions Y; ~vhere by definition {Y} = [X] for some ~-complex X with z(X) = 1 and differing from Y only by free orbits. To prove the necessity of the conditions on ~(G, ~) , it thus suffices to show that for any such resolution Y, z((Y~)~)= 1 for all e, K as above with K4= 1.

By Lemma 2.1, ye is Zp-acyclic for any p-subgroup 1 4: P ~ G for any prime p. So for any m = t = c ~ and any p[n(~), Y~ is a wedge component of the fixed point set of some non-trivial p-group and thus ~p-acyctic. By Lemma 2.5, z((Y~)~)= 1 for any cyclic subgroup K/p(cr Gjp(cz).

G-CW-Surgery and Ko(TgG ) 21

It remains to consider the case e = m : namely, to show that z(YK)= 1 for any cyclic subgroup I@K~G. But /~,(Y; Q)=/t ,(Y)| is (l~[G]-free (/~,(Y) is projective, so Swan's Theorem 8.1 [221 applies). Again applying Lemma 2.5, z (Yr)=l for any cyclic I:I:K~G.

II. To prove the converse, assume that X is a finite N-complex with )(((X~)K)= 1 for any c ~ and any cyclic subgroup K/p(e)~_GJp(cO. Let c~@m be a minimal element of ~ such that X~ is not ~(~)-acyclic. Set k = dim(X~).

Attach e-cells (G/p(~))xD i (1<-iNk) to X to kill all homotopy in X~ of dimension less than k. Attaching an a-cell corresponds to attaching a cell D ~ to X~ via a map Si-I--,X, representing the homotopy element to be killed, and then extending the attachment equivariantly to the full orbit. Since G= is the subgroup of those g with g~=c~, the cells attached to X= correspond precisely t o (aJp (~ ) ) x D i.

These cell attachments produce an N-complex X'~_X, where X', is k- dimensional and (k-1)-connected. Consider Hk(X'~; 7g(~)) as a 7z(~)[G~/p(a)]- module. It is projective by Lemma 2.3 (and the minimality assumption on c~); with the hypotheses on Euler characteristics, Lemma 2.5 shows that it is free. Choosing a basis and adding more a-celts G/p(cOxD k+l accordingly now produces a complex X"_. X with X'~' Z~)-acyclic.

By construction, X"/X is a Gc~-complex. We have x(X~)= 1 by hypothesis and )((X;')=I by acyclicity, so IX"] = I X ] by Lemma 1.4. This process can thus be continued with other elements of ~ , constructing an N-complex Y _~X" with [Y] = IX], and such that Y= is 7Z(,)-acyclic for all m @ e ~ - .

Now attach free orbits of cells G x D ' to Y to push all homology into the top dimension. Again by Lemma 2.3, the remaining homology group is projective, and so the resulting complex Y' is an N-resolution. Since )~(Y)= 1 and Y' differs from Y by free orbits, we have

[X]=[Y]={Y ' }e~(G,N) .

The following technical temma will be needed for calculations in the appendix. The proofs are contained in the proof of Proposition 2.6 (part II).

Lemma 2.7. Let N be a connected family. Let X be any finite N-complex. a ) / f [X]E~(G, N), then there is an N-resolution Y~ X such that {Y}--IX]

and Y/X is an ~-complex. b) Assume N = ~ , let ~ e N be a minimal element, and assume X~ is 7~)-

acyclic. Then there is an N-resolution Y~_X such that {Y} =0 and Y==X~ (i.e., Y/X is an (N-GcO-complex). []

It remains to repeat these constructions with G-maps, rather than in- dividual G-complexes. For any f: X--,I ~, Z f and Mf denote the mapping cylinder and mapping cone, respectively. K, ( f ) and n, ( f ) denote the relative homology and homotopy groups:

K,(f)=H,(Zf,X)-~IZI,(MI); ~,(f)=g,(Zf,X).

Since the spaces dealt with are not always simply connected, the following version of the relative Hurewicz theorem will be needed:

22 R. Oliver and T. Petrxe

Lemma 2.8. Let f: X--+Y be a map between path connected spaces. For k>=2, if h i ( f ) = 0 for all i<k, then K i ( f ) = 0 for all i<k and the Hurewicz map

~k( f )~Kk( f ) is a surjection.

Proof See [211, Theorem 7.5.4. []

The main result of this section now shows how "G-CW-surgery" is used to construct G-maps whose mapping cones are G-resolutions. Remember that for any map f: X ~ Y,,

is defined to be the minimal family such that Mf is an f f fcomplex.

Proposition 2.9. Let f: X--* Y be a map between finite G-complexes,/7 = H(Y), W ~_/7 any connected family containing ~'~I and f f ~_ ~ any family containing ~ . Assume that

[Mf]6A(G, ~ ) + a ( G , ~').

Then there is an extension f ' : X'-~ Y of f inducing an isomorphism on ~cl , such that X ' /X is an ~"-complex, M f , is an ~-resolution, and {Mf,}sA(G, ~).

Proof The proof is carried out in three steps. First, f is extended to a map fo: X o ~ Y such that [MIo]~A(G,@ ). In Step 2, fo is extended to a map f: X ~ Y such that for any m =# e ~ - , f~: X~_-~ Y= is a 7Zlss equivalence. Lastly, free orbits of cells are attached to X, extending f to f ' : X ' ~ Y such that My, is an ~-resolution. These extensions must, of course, be constructed so that X / X o and Xo/X are ~ '-complexes and so that {Ms, } = [Mr] = [Mio ].

Step 1. Choose an element 3= ~ n~[e]~f2(G,~') (n~TZ) such that s e ~ '

[MI] - ~A(G, ~).

We now want to attach cells to X and extend f to a map f0: X 0 ~ K such that Xo/X is an ~-'-complex and [Xo/X ] = 3.

One way to do this is to choose, for all c~j~', arbitrary finite CW- complexes K~ with z(K=)= n~, and define

Xo = x ]3 ( LI G/p(~) x K~).

The extension to fo is defined by choosing arbitrary maps from K s to Y~_c ypcs) for each e, and extending them equivariantly to G/p(cO x K s. Then Xo/X is a finite W'-complex,

[Xo/X] = ~ n~[~] = 3 ,

and so [MIo]=[MI] -~eA(G ,~ ) by Proposition 1.3 (c).

Step 2. We now show that fo can be extended to a map f: X ~ Y such that X / X o is an ~-complex, [Ms] = [Myo], and such that f= is a ~(s)-homology

G-CW-Surgery and Ko(7/ G ) 23

equivalence for all m =# a 6 3 . Define

Z o = { a e ~ l ( f o ) a : ( X o ) ~ Y ~ is not a 7/(a)-homology

equivalence for some fic~.~, fi<a}.

The proof will be by induction on the order of 2 o. If S o_ {m}, we are done. If not, let a be a minimal element, and consider

the map (f0)~. An element of rq((f0)~) corresponds to a map from S '-1 to (Xo) ~ and an extension to D ~ in Y~, which is just the information needed to attach an /-cell to (X0) ~ and extend the map, sending it into Y~. As usual, any such cell attachment can be extended equivariantly to an "a-surgery"-attaching an a-cell (G/p(a)) x D i to X o and extending fo.

We can thus perform a-surgeries to kill off all lower homotopy, starting with ~x((fo)~), and ending with an extension f~ : X , ~ Y such that for some k, rc,((fl)~)=0 for i<k and dim(m~i~)~)=k. Then (Lemma 2.8)

Hk(M~y~) ~) = Hk((Mj. )~) = Kk((fl)~)

is the unique non-zero reduced homology group of the mapping cone. If Hk((ml,)~; :g(,)) is free as a ~m[Gjp(a)]-module, one can choose a basis, represent the elements by relative homotopy classes (Lemma 2.8 again), and add a-cells (G/p(a))• D k+l accordingly. The result is an extension of f~ to a map f2: X 2 ~ Y with (f2)~ a ~(,)-homology equivalence.

To see that Hk((MI~):" 7/(~)) is free, note first that (MI,)~=(MIo) p is 7/(~)- acyclic for all f l<a with f i e ~ (a was minimal in So). So the module is projective, by Lemma 2.3. Since

[Mfo ] cA (G, ~ ) ___ ~(G, ~,~),

z((Mco)~)=x((MI1)~)=I for any non-trivial cyclic subgroup K/p(a)~_GJp(a) (Proposition 2.6), and Lemma 2.5 now applies to show that Hk((My,)~, ; ;g(~)) is free.

Thus, f2: X 2 ~ Y can be constructed with [X2/Xo] a Ga-complex and (f2)~ a 2g(~)-homology equivalence. Note that (Myo) ~ and (My,)~ both have Euler characteristic 1: the first since [Myo]e~(G,~ ) and the second by acyclicity. So x((X2/Xo) ~) = 1, [X2/Xo] = 0 by Lemma 1.4, and [My2 ] = [Myo].

Defining 22 in a way analogous to So, we see that S2~_So-Ga. Thus IS21 <lS0[, and by the induction hypothesis, f= extends to a map f: X ~ Y such that X/X= is an ~-complex, [My]=[My2],and f~ is a 7Z.(~)-homology equivalence for all m ~ = f l ~ . So f~/X o is also an ~'-complex, and we are done.

Step 3. Now attach cells in free orbits G • D i to )( and extend f (i.e., perform "m-surgeries"), killing off all lower homotopy of s until an extension f ' : X ' ~ Y has been constructed such that

ni(f') = 0 for all i < k = dim(My,).

In particular, M I, has homology only in dimension k, and so Lemma 2.3 applies to show that Hk(My, ) is a projective 7Z[G]-module. So M,, is an ~ - resolution.

24

Since M f, differs from My only by free ([My]cA(G, N)), we have

{M~,} = [ M f ] e A ( G , ~ ) . []

R. Oliver and T. Petrie

orbits, and x(Mf)=I

3. The Main Theorem and Applications

In order to finish the proof of Theorem 3.2, some technical results are needed. Due to their length, the proofs of these results are deferred to Sect. 4,

The idea behind these theorems is that when constructing quasi-equivalences, 0N-resolutions representing elements in Bo(G , ~ ) are used to remove the projective homology groups remaining after Proposition 2.9 has been applied. The first result guarantees that one need perform c~-surgeries only for ~ :

Theorem 4.3. For any connected G-family ~ , Bo(G, ~,~) =Bo(G, ~).

The second result extends the first, further reducing Bo(G, ~) :

Theorem 4.5. Let .~ be any connected family. For any ~e,~, define

= {p(/~)l/~E~,/3__> ~}.

Then, for all ~, ~ is a connected family of subgroups of G~ (the isotropy subgroup of c~ in .~ ), and

Bo(G, Y ) = ~ ; Ind~(B~ ~)).

These results suffice to complete G-CW-surgery when certain fixed point sets are non-empty. In other cases, an additional restriction on ~ is needed:

Definition. A connected family ~ is called simply generated if for any ~eBo(G,@), there is a two-dimensional 0~'-resolution X with {X} =0 and 7~(X) =4.

We have been unable to determine whether every connected family is simply generated. We are able to show that it holds under the following conditions:

Theorem 4.12. Let ~ _ H be a connected family. Assume ~ has the property: for any c z ~ and any fl>~ (flEFf) such that p(fl) is a SyIow subgroup of p(~), p(~) < G~. Then ~ is simply generated.

Corollary 4.13. Assume that G is abelian. Then any connected G-family @ is simply generated.

Corollary 4.14. Let G be any finite group, and Y a connected G-family such that for all c ~ , p(cz) is a p-group for some p. Then ~ is simply generated.

The following lemma is the means for applying simple generation in the proof of Theorem 3.2:


Lemma 3.1. Let f: X ~ Y be a map between G-complexes, and ~c_17(y) a family such that X~ is non-empty and connected for ~e~ . 77wn, for any 2- dimensional J~-complex A with zero-skeleton A~ there is a finite G-complex X'~_ X and an extension f ' : W--+ Y of f such that X ' /X is an ~-complex having the G-homotopy type of A.

Proof First fix a point x~eX~ for each ~ e ~ , so that xg~=gx~ for all geG. Corresponding to each 1-cell of type ~ in A, attach an orbit G/p(c% • D 1 to X by gluing both endpoints of (g xD l) to Xg~; extend f by sending all of (g xD ~) to f(xg~). This produces a space Jf and an extension f such that J(/X can be identified, as an ~-complex, with A ~ (the 1-skeleton of A).

It remains to construct the extension (X',f ') by "lifting" the 2-cells of A to J(. In other words, for any 2-cell of type ~ G/p(c 0 x D 2 in A, we must be able to lift the attachment map for (e x D E) from A~ to zY~, SO that its image in Y~ is contractible. But this is clearly possible: since X~ is connected, X~ has the homotopy type of X~ v A~ with A~ mapped to a point in Y~. []

Theorem 3.2. Let f: X - ~ Y be a map between finite G-complexes, 17=H(Y), and .No_ 17 any connected family containing ~,~. Let ~"c_ ~ be any subfamily containing ~ , and assume that either (1) ~ is simply generated, or (2) for all ~ e ~ , (X~)G~+0. Then there is a finite G-complex X'~_X and a quasi-equivalence f ' : X'--* Y extending f with X ' /X an ~"-complex, if and only if

[M:]eA(G,~)+[2(G,~') in f2(G,17).

In particular, if nl(Y)= 1, this condition is necessary and sufficient for being able to extend f to a pseudo-equivalence.

Proof I f f extends to a q.e. f ' : X ' ~ Y with X' /X an ~'-complex, then M:, is a contractible ~-complex. So

[M:,]eA(G,~) (by 1.6), [X'/X]ef2(G,~'),

and [Mf] = [M:,] + [X'/X]. The condition on [M:] is thus necessary. Consider the converse. Applying Proposition 2.9, we may assume that Mr

is an ~-resolution and {M:}eA(G,~); in particular, 7G(M:)eBo(G,~). There are two cases.

(1) If ~ is simply generated, there is a 2-dimensional ~-resolution A with 7G(A)=~;~(My) (Bo(G,~)=Bo(G,~) by Theorem 4.3). By Lemma 2.1, A~ is 2g~)- acyclic for any m # c ~ e ~ ; so A~ is connected for any ~e~" and we may assume A ~ is a point. Similarly, (M:)~ is 2g(~)-acyclic for all r , + ~ e ~ , and so X~ is non- empty and connected for all ~e~- (Y~ is non-empty and connected, by definition).

Lemma 3.1 applies to give a finite G-complex J(_~X and an extension f: X ~ Y off , such that X / X is an ~-complex and

H,(My, M f) = H3(My , My) ~- H2(A).

So I2I,(My) is 2gG-projective (we may assume dim(Mr)_>_ 3), and

(-1)i[I7t,(Mf)]=O in /s


Now, m-surgeries can be performed on X (i.e., adding free orbits of cells G x D i) to produce a q.e. f ' : X'-- ,Y extending f.

(2) Assume now that (X~)G~+0 for all c ~ , and fix points x ~ X ~ ~. Using the notation of Theorem 4.5, there are (G,,~)-resolutions A~ for ~ e ~ , such that

~ Ind~(7~(A~))= 7G(Mr

For each ~, let q~A~ denote the basepoint, attach G x G A~ to X by identifying (g x %) with gx~, and extend f by sending g x A~ to f(gx~). Let X and f denote the space and extension thus constructed.

Now, Mf has the G-homotopy type of

M f v S ( V G• o~E~

In particular, f t , (My) is ~TG-projective, and

( - 1) i [/t,(My)] = 0~/s O). t = 0

So as in case (1), m-surgeries can be performed on ( X , f ) to produce an extension (X', f ' ) such that f ' is a quasi-equivalence.

If zl(Y)=0, then clearly any q.e. is a pseudo-equivalence. []

Remark. Note that if YP is connected for all P ~ and XG+0, then Condition (2) above - (X~)G~0 for ~ E ~ - always holds ( X ~ X G for r162 There are many situations where the simpler condition X ~ + 0 can replace (1) and (2) in Theorem 3.2. It's difficult, though, to find a general theorem using that condition; everything would be much simpler if we could show all connected families to have simple generation.

The remainder of this section deals with corollaries to Theorem 3.2. The first group (3.3 to 3.7) involve changes in the isotropy group structure under quasi-equivalence, while 3.8 to 3.10 involve changes in fixed point sets.

Corollary 3.3. (Compare [18]). Let Y be a finite G-complex. Then there is a finite G-complex X' with Iso(X')___~ and a q.e. f ' : X ' ~ Y,, if and only if

[Y+]~A(G,H)+E2(G,~(H)), (H=H(Y)) .

Proof. Apply 3.2 with X = 0, ~ =/7, and i f ' = / 1 = ~(H). Note that f f is simply generated by 5.14. So 3.2 does apply yielding X' and f ' with X '+ a/7-complex; hence

Iso(X')_= p(/~)_= ~. [ ]

By a "G-homotopy sphere" will be meant a finite G-complex with the homotopy type of a sphere.

Corollary 3.4. Let V be a complex representation of G. 7hen there is a G- homotopy sphere X with Iso(X)_~@ and a p.e. f: Z~S (V) .

G-CW-Surgery and Ko(TZG) 27

Proof Apply 3.3 with Y=S(V). Since V is a complex representation, yn is an odd-dimensional sphere for all H, and so [Y+] =0 in Y2(G, II(Y)). []

Simple examples show that 3.4 is false in general for real representations ([S(V) +] is not usually zero in this case).

Lemma 3.5. Suppose ~(Y)c__II=II(Y) is connected and simply generated, and [Y+] =0 in Y2(G, II). 7%en for any ~ ' such that ~ ( y ) c y ' ~ . y ( y ) , there is a G- complex X' with a q.e. f ' : X'---, Y such that X '+ is an ST'-complex.

Proof Take X = 0 and ff=~,~(Y), and apply Theorem 3.2. []

Lemma 3.6. Let Y be as in 3.5, II=H(Y), and let JF~_5~ be a family of subgroups such that

p (~r ~ _ Iso(Y).

77wn there is a G-complex X' with Iso(X')= J f and a q.e. f ' : X' ~ Y.

Proof Let X =I_[ G/H, where the union ranges over all conjugacy classes of He0Y{. Let f : X ~ Y be any map (~_cIso(Y)). Set ~,~=~(Y), ~ ' = p - 1 6 ~ ) n ~ - ; so

[M N ~A(G, ~-)+ O(G,.~-').

Thus, 3.2 applies to yield X' and f ' such that X'/X is an ~'-complex. Since Iso(X)=~, and p(J~')~H, Iso(X')=J~& []

Corollary 3.7. Let G be abelian, and V a complex representation of G, Let 2If ~_ 5~(G) be any family of subgroups such that

p ( ~ (S(V))) ~_ ~ c Iso(S(V)).

Then there is a G-homotopy sphere Z with Iso(Z)=~r having a p.e. f: Z-*S(V).

Proof Let Y=S(V), so [Y+] =0. if(Y) is connected (Y is a smooth G-manifold) and simply generated (by 4.13: G is abelian). So 3.6 applies. []

It should be noted that Iso(S(V)) is closed under intersections for any representation V of G. Corollary 3.7 shows one way of constructing G-homotopy spheres whose isotropy subgroups don't have this property; an analogous result in the smooth G-category is shown in [16]. A description of the structure of Iso(2) when Z is a G-homotopy sphere is an important step in describing the G-homotopy types of actions on spheres. The results of this paper will be used by tom Dieck and Petrie in a forthcoming paper treating this problem.

We now consider how the fixed point sets of the spaces in a quasi- equivalence can differ when the group G is not of prime power order. This is related to the results in [11] (corollary to Theorem 3) describing the possibil- ities for the fixed point sets of actions on finite contractible complexes. With quasi-equivalences, the answer similarly depends only on Euler characteristics.

Proposition 3.8. Assume G is not of prime power order. Let Y be any finite G- complex, and let F~ .... , F k be the connected components of F = Y~. Then there is a subgroup n r ~ g k such that: given any finite complex F' and a map f: F '~F,


there is a finite G-complex X with X G = F ' and a quasi-equivalence f: X ~ Y with f 6 = f if and only if

(z(F1)-z(F~), ...,Z(Fk)--z(F~))Enr, (F,.' = f - 1 (Fi)).

Proof Apply Theorem 3.2 to the map

f ' = incl o f : F ' ~ Y;

letting ~ and Y'={~H[p(cO, t=G }. Let ~l,...,C~k~H be the elements with p(e~)=G and I~il =El.

Define a homomorphism 0: f2(G, H ) ~ k by

0(IX]) = (z(X~I) - 1, ..., z(X~) - 1)

for any finite H-complex X; then Ker(O)=f2(G,~') . Set nr=~(A(G,H)). Apply- ing Theorem 3.2, f ' extends to a pseudo-equivalence without changing the fixed point set if and only if

[Mf,] E~2(G, Y ) + A (G, H) iff

0([M:,3)~ ~,(A (O,/7)) = nr.

But ~'([M:,])=(x(F1)-Z(F~), ...,Z(Fk)--z(F/J). []

When F is connected, this takes a much simpler form. This case can, of course, be proven directly, avoiding much of the machinery of this paper (see 1-13). Here, n G is the number defined in 1'-113:

Corollary 3.9. Assume G is not of prime power order, and let Y be any finite G- complex such that F = Y~ is connected. Then, given any finite complex F' and any map f: F ' ~ F , there is a finite G-complex X with X G = F ' and a quasi- equivalence f: X ~ Y extending ~ if and only if

z(F') - z(F) (mod nG).

Proof Referring to the notation of Proposition 3.8, we have k= 1, and so n r _~;g. Write n r = n g (n>0); we must show that n = n G. Referring to Proposition 1.6 (and the definitions of n, nG) we have

n~ = n r = {z(Z G) - l I / a finite contractible H(Y)-complex} nG2g= {z(Z G) - l l Z a finite contractible G-complex}.

Clearly, nGI n. Conversely, set H = H ( Y ) , and let Z be an arbitrary finite contractible G-

complex with z(Z6)= 1 +n G. We define a H-complex structure on Z as follows. Let c~6H be the unique element with p(c0=G (ya is connected). Choose a basepoint q s Z G. By Lemma 1.1 for any H ~ G , there is a unique fi__>c~ with p(fl) --H. So a H-complex structure on Z is defined by setting, for any fl~H:

{qZP(~) if fi=_>~ Z~= if f i ~ .

Thus, nG=Z(ZG)- l ~nT~, and so n=n G. []

G-CW-Surgery and K0(ZG) 29

The following example now illustrates more concretely how the machinery of G-posets is needed when the fixed point set is not connected.

Corollary 3.10. Let G=Dv, the dihedral group of order 2p (p an odd prime). Let Y be any finite G-complex, Y1, ..., Yk the connected components of y~v, F= Ya, and Fi=Yic~F. Then, given any F' and a map f: F '~F, there is a finite G- complex X and a q.e. f: X~Y,, with X ~ = F ' and f G = ~ if and only if z(F{) =)/(F,) for all i(Fi'=f-l(Fi)).

Proof Referring to the notation in [12], G e ~ 1 and so 4~(G)=0. It follows from Proposition 1 and Theorem 3 in [12] that for any G-resolution Z, 7G(Z)eBo(G) =D(~G). Since D(2gG)=0 ([19], p. 328), the obstructions for all G-resolutions vanish. In particular, A (G, ~-)= r ~,~) for any connected G-family ~ .

The result now follows from Proposition 3.8, and the description of r ~ ) given by Proposition 2.6. []

This shows that Corollary 3.9 is definitely false if one removes the condition that F be connected. Quinn [18] has defined a concordance relation between finite G-complexes: two complexes X a and X 2 are concordant if there is a third finite G-complex Y and G-maps fi:Xi--*Y which are homotopy equivalences. One can try to generalize Corollary 3.9 by asking the following:

Question. Let G be a group not of prime power order. Let Y be a finite G- complex, F = yG, and let F' be any finite complex with

z(F') =- z(F) (mod nG).

Is there a finite G-complex X concordant to Y with XG=F'? The condition on Euler characteristics is clearly necessary. Corollary 3.9

answers the question affirmatively when ~ ( Y ) = 0 and F is connected.

4. Calculations in the Projective Class Group

We now finish by proving the two technical theorems on the decomposition of Bo(G,Y ) needed in Sect. 3: Theorems 4.3 and 4.12 below. Again, the goal is to show that elements in B 0 are represented by G-resolutions with as simple a structure as possible.

To simplify the statements of some of the lemmas, we define a simple Y - resolution to be an ~-resolution X such the {X} =0 in Y2(G,~); these are, of course, the only G-resolutions which occur in defining B o. The following lemma will be needed for the geometric manipulations involved with decom- posing B 0.

Lemma 4.1. Let ~ be a G-family. Then for any simple ~-resolutions X 1 ~_X2, there is an Y-resolution Y~_X2/X 1 such that (Y--Xz/X1) is free, and

~'AY) = 7G(x2)- ~Ax1).

In particular, if X2/X 1 is an : '-complex for some connected family Y ' ~ Y , then

~G(X1) ~ '~G(X2) (mod Bo(G, ~.~')).

30 R, Oliver and T. Peme

Proof From the relative exact sequence, we get that H~(X2/XI) is 2gG- projective for all i, and that

(--1)'EI~I(X2/X1)]=TG(X2)--7~(X1) in /(o(TZG). ~=0

We are done if X2/X 1 has homology only in one dimension; if not, let k be the lowest dimension with Itk(X2/XO+O. Attaching free orbits of cells G x D ~+1 to X2/X 1 to kill ftk(X2/X1) produces a complex Yo~_X2/X1, where ,Ok+l(Yo) is isomorphic to Hk+I(X2/X 1) plus a projective complement to 14k(X2/X O. In other words,

~=0 ~=0

and continuing this procedures a G-resolution Y~-Yo with ?o(Y)=TG(Xz) -To(X1), and (Y-X2/X1) free. In particular, if .~' o~ is a connected subfamily and X2/X 1 an Y"-complex, then so is i1,, and thus 7o(Y)sBo(G, ff'). []

Let :d be any category of pointed G-complexes, closed under wedge pro- ducts and suspension, such that the mapping cone 1V/: of any map f in ~ also lies in c#. To simplify later calculations, we define the concept of a "universal group" C for cal. C is defined to be the semigroup of all :d-homotopy equivalence classes of objects in :d, with addition given by wedge product; modulo the relation that

[Mf] = [ Y] - [X]

for any map f: X---, Y in :d. So the zero element is represented by a point, and [ZX] = - I X ] for any X in c#.

Consider functions of the form

~,: Ob(Cg)~A

(A any abelian group), such that

(1) v(X)=0 if X is ~-homotopy equivalent to a point, (2) 7(u ) for any pair (Y,X) in c~ (7 "preserves cofibration

sequences"). Such functions frequently arise naturally as obstruction maps, and C is clearly universal among them: any such 7 factors uniquely through C.

As an example, the universal group for the category of all H-complexes, for any G-poset H, is easily seen to be the Burnside ring f2(G, F/). In the case of the category No(G, ff) of all "~-complexes X with [ X ] = 0 in Q(G, ff), the following lemma shows that one actually gets trivial universal group.

Lemma 4.2. Let "J be any G-family. Then Do(G,:P)=0, where D O denotes the universal group for D o.

Proof Let X be an arbitrary ..~-complex; then X v Z X lies in No(G,W) (even though X might not). Define a map

f: X v ZX->X v Z X

G-CW-Surgery a n d Ko(7/G ) 31

by letting f rZX be the identity, and sending X to the basepoint. The mapping cone My is Y-homotopy equivalent to X v ZX, and so

[ X v Z X J = [ M f ] = [ X v Z X J - [ X v Z X ] = O in Do(G,Y ).

Now assume that X and Y are any two complexes in ~o(G,Y) such that X~ and Y~ have the same number of cells in each dimension for any cceY. Letting X (k) and y(k) denote the k-skeleta (k=>0), this means that X(k)/X (k-~) and y(k)/y(k-~) are equivalent as Y-complexes for all k. It follows that

IX] - [y] = Ix v z Y] = ~ [(x(k)/x (~-') v z(Y(~)/y(~-')] =o, k=O

and so [X] = [Y] in Do(G, Y). In other words, the class of a space in Do(G, Y) depends only on numbers of cells in each dimension, and not on their attaching maps.

So any element of Do(G,Y) is represented by an Y-complex X which is a wedge product of spheres (i.e., each cell is attached to the basepoint). Since IX]=0 in Y2(G,Y) by assumption, x(X~)=I for all ~ Y , and one easily constructs Y-complexes u and Y2 such that

X v ( Y~ v Z yl ) ~= Y2 v ~ Y2 .

So IX]=0 in Do(G,Y), and D0(G,Y)=0. []

This is now used to prove one of the main results of this section:

Theorem 4.3. For any connected G-family Y, Bo(G, Y)=Bo(G,~).

Proof. We define a map

7: Ob(~o(G, Y))~/s ~ G)/Bo(G, ~)

as follows. For any X in ~o(G,Y), there exists (Lemma 2.7) a simple Y- resolution Y~X such that Y/X is an ~-complex. Set 7(X)={ya(Y)} (brackets denoting the class rood Bo(G, ~)).

If Y'___X is another such Y-resolution, use Lemma 2.7 again, applied to the complex YwxY', to construct a simple Y-resolution Z containing Y and Y' such that Z/Y and Z/Y' are ,~-complexes. By Lemma 4.1,

7~(Y) = 76(Z) ~ 7o( Y') (mod So(G, ~)).

So 7 is well defined. Clearly 7(X)=0 when X is contractible (X is itself a G-resolution). If X 1

_cX 2 is any pair of complexes in 9o(G,Y), apply Proposition 2.6 again to construct simple Y-resolutions Y1c_Y2, with Y2mX2 and X 1=X2~ gl, such that Yk/X1 and Y2/X2 a r e .~-complexes. Then X 2 / X 1 ~_ Y2/Y1, (Y2/Y1)/(X2/X1) is an Y-complex, and by Lemma 4.1 again,

7(X2/X1) = {TG(Y2) - - 7o (Y1)} = 7 ( X 2 ) - 7 ( X 1 ) .


So 7 factors through Do(G ,if) , and is zero by Lemma 4.2. The image of ? is

just Bo(G, if)/Bo(G, ~) ,

and so Bo(G,~)=Bo(G,~ ). []

It remains to show that for certain families Y, all elements of Bo(G,~) are representable by 2-dimensional G-resolutions. This will be done by finding a decomposition for B0, using again the idea of universal groups.

For any G and any n > 1, let YU,(G) denote the category of all finite pointed Z(,)-acyclic G-complexes which are free away from the basepoint. Let W,(G) denote the universal group for Yl/~,(G). The next lemma describes how W,(G) arises in describing the B 0 groups.

Remember that for any connected family ~- and any m # c ~ , n(~) was defined as the product of all primes p such that ~ = fi for some fi > ~ with p(fi) a p-group. When different families are involved, this will also be denoted n~(~) for clarity.

Lemma 4.4. Let 17 be a G-poset, ~ = S a connected family in 11, and c ~ a minimal element. Set H = p(e) and n = n(~). Then

a) For any complex X in Yg~(GjH), there exists a simple if-resolution Y such that X = Y~.

b) There is a surjective homomorphism

7: W.(G2n)---So(G, J)/Bo(G, ~-G~)

such that for any simple g-resolution Y,,

7(L) = {7~(Y)}

(brackets denoting the class mod Bo(G, ~ - GcO). c) Define ~ = {P(f i ) l f i~, fi>-_ ~}. Then ~ is a connected family of subgroups

of G~, and no(G, ~ ) = Bo(G, f f - Ga) + Ind~(Bo(G ~, G))-

Proof (a) Let X be a complex in ~tU,,(GjH), and let x e X be the basepoint: thus X is 2g(~)-acyclic and GJH acts freely on X - x. Define

=(G x ~ X)/(G x ~x);

this has an obvious structure as Y-complex ( w i t h ) ~ = X ) . By Lemma 2.7, there is a simple if-resolution Y_~_~ such that Y~ = X~ = X.

(b) We first show that there is a well-defined map

7: Ob(~#/~(GjH))~ Bo( G, ~)/Bo(O, ~ - G ~)

such that 7(Y~)={TG(Y)} for any simple ~--resolution Y. With (a) shown, it suffices to prove that

7G(Y) ---- 7~(Y') (rood B0(G, ~ - G r

for any two simple if-resolutions Y and Y' with Y~ = 11'. Given Y and Y', apply Lemma 2.7 to the complex YUGyY' , constructing a simple Y-resolution Y

G-CW-Surgery and Ko(TZG ) 33

containing Y and Y', so that f-~= Y~= Yd- Then ~/Y and Y/Y' are both ( 2 - Gc0-complexes and

?G(Y) -= ?o(f') - "/G(Y') (mod Bo(G, 2 - G ~))

by Lemma 4.1. Clearly, 7(X)=0 for any contractible complex X in ~r That ?

preserves cofibration sequences is shown by again applying Lemmas 2.7 and 4.1, exactly as in the proof of Theorem 4.3, So we get a well defined map

~,: W,,(GJH)~Bo(G, 2)/Bo(G, 2 - G~),

which is suriective by definition of B0(G, 2 ) . (c) Consider the set //~={]~/71/~=>~}. It is G~-invariant, so the set

=p(Yn/7~) is G~-invariant, and is a family of subgroups of G~. By Lemma 1.1, /7~ is in one-to-one correspondence with the subgroups of

H=p(c O. 2 being connected, 2c~/7~ is a connected family in H~ (regarded as an H-poset). So ~,~ is a connected family of subgroups of H, and thus connected as a family of subgroups of G~.

Let Y be any (G~, ~)-resolution with basepoint y. Then

Y= (G x o Y)/(G x G~Y)

is an 2-resolution with f'~ = Y~. We have

and so 7G(Y)=Ind~(?a,(X)). Clearly, n~(H)=n~(a)=n. Applying (b) to the families 2 and ~,~ with

minimal elements a and H, respectively, we get a commutative triangle

W.(G J H ) - s - {H})

Bo(G, 2)/Bo(G, 2 - Gcz).

The vertical map is thus onto, and

Bo(G, 2 ) = Bo(G, 2 - G ~) + Ind~(Bo(G ~, ~)). []

The formula in Lemma 4.4 (c) now reduces the study of Bo(G,2), for a Nmily 2_~/ / , to the case w h e r e / / i s the poset of subgroups of G, and 2--- contains a unique maximal subgroup normal in G:

Theorem 4.5. Let 2 be any connected G-family. For any c ~ , define

= (p(fi) l f l ~ , fl > c~).

Then, for all ~, ~ is a connected family of subgroups of G,, and

Bo(G, 2 ) = ~, Ind~=(Bo(G ~, ~)).

34 R. Ohver and T Petrie

Proof. Theorem 4.3 reduces this to the case ~ = ~ . The result then follows from Lemma 4.4 (c), and induction on Igl. []

The next step is to describe W,(G) in terms of K-theory. The following technical lemma will be needed for constructing G-maps.

Lemma 4.6. Let (X,x) and (Y,y) be pointed finite G-complexes such that G acts freely on X - x . Assume that X and Y are both (k-1)-connected (for some k> 2) and that H~(X)~O for i>k. 7hen, if (p: Hk(X)--*Hk(Y ) is any (7ZG-linear) homomorphism, there is a (basepoint preserving) G-map f: X ~ Y such that Hk(f) ~-~0.

If, in addition, G acts freely on Y - y , H~(Y)=0 for i>k, and q) is an isomorphism, then f is a G-homowpy equivalence.

Proof. This is a simple application of equivariant obstruction theory. We use the notation in [5].

Since G acts freely on X - x , the obstruction groups have the form

H~(X, x; (oj(Y)) = H i [Homea(C,(X, x), nfl Y))].

The chain complex {C,(X,x)} being ~G-free with one non-zero homology group, Hk(X ) must be projective, and so

0 i+k H~(X,x; ~flY))_~ Hom~G(Hk(X),~fly)) i=k.

So Theorem 2.11 in [5] applies, and G-homotopy classes of based maps from X to Y are in one-to-one correspondence with elements of Hom~G(Hk(X), Hk(Y)). Let f: X ~ Y be the map corresponding to (p.

If H,(Y)=0 for i > k and ~ is an isomorphism, the same procedure can be used to construct an inverse to f, and show that the composites are homotopic to the identity. So f is a G-homotopy equivalence. []

For any n > 1, we define j~,(TZ G) to be the category of 2g G-modules of finite homological dimension which are finite of order prime to n. Again, 7Z(,) denotes the integers localized at the set of primes dividing n (namely, inverting all primes pXn). There is a localization exact sequence

K~(71G)~KI(7Z,(,)[G] )_ ~ . K o ( ~ [ ~ ( ~ G ) ) ~ Ko(~G)---,Ko(2g(,)[G])

(see [4], Chapter IX, Theorem 6.3) with homomorphisms t? and ~ defined as follows:

1. For any eeEnd~a(/~G k) such that c~(,) is an automorphism of/g(,,)[G] k,

0([TZ(n) [ G] k, ~(n)]) = [Coker (c~)] ~ Ko( ~/n( ~ G ) ).

2. For any M in JP/,(;gG), let re: P ~ M be a surjection of a projective 7ZG- module onto M; then Ker(u) is projective ([20], Theorem 4.12) and

r = [P] - [Ker(70].


By a theorem of Swan ([22], Theorem A), P(.) is 2g(.)[G]-ffee for any projective 77G-module P. It follows that the above sequence can be shortened to an exact sequence

K~(2gG)__,K~(2g(n)[G.]) ~_, Ko(Jd,(;EG))__ o ,/~o(2gG)~0 '

Proposition 4.7. Let G be any finite group, and n>= 1. Then there is an isomorphism

co: W~(G)~ Ira(0),

with the following property: for any complex X in "[r with only one non-zero reduced homology group Hk(X) ,

c o ( [ x ] ) = ( - 1) k [ H k ( X ) ] + K o (~ / . ( ;~ G)).

Proof For any complex Y in ~,,(G), we define its "length" d(Y) to be the difference of the maximum and minimum dimensions i for which/]i(Y)#0.

Step 1. We first show that W,(G) is generated by elements IX] with E(X)=0. It clearly suffices to show for any Y in YU,(G) with E(Y)>0 that we can write [Y] = [X] + IX'], with d(X)=0 and d(X')<d(Y). Replacing Y by Z I7, if necessary, we may assume it to be simply connected.

Fix such a Y,, and let k be the lowest dimension with /]k(Y)~=0. Let X be any simply connected complex in ~/gs with [(X)=0, such that there is a surjection

qo : Hk(X)--> Hk(Y ).

For instance, if r is the exponent of f ig(Y)(so (r,n)=l), then (Zr[G]) ~ surjects onto Hk(Y) for some t, and X can be taken to be a wedge product of Zr- Moore spaces permutted freely by G.

By Lemma 4.6, there is a map f: X ~ Y with Hk(f)= q0; then

Hk(Mf) ~- Coker(~o) = 0,

and so f(Mr Since [Y]= IX] + [MI], we are done.

Step 2. For any complex X in ~r with basepoint x, its cellular chain complex

O--).CN(X,x; 7~(n))-'-)'C N I(X,x; 7~(n))-----)....--+Co(X,x; 7](n))---)'0

(N=dim(X)) is an exact sequence of free, based (up to +g) 2g~,}[G]-modules. Using the procedures described in Sects. 3 and 4 of [8], we define a "torsion" element

T,(X)@KI(7Z,n ) [G])/{ +g},

and thus an element

~o(X) = O(z(X)) aK o (.G(2~ G)).

If X is contractible, then z(X) lies in the image of KI(2~G), and so eg(X)=O.

36 R. Oliver and T. Peme

If (Y,X) is any pair of spaces in ~#~,(G), then

z ( Y / X ) = r ( Y ) - r ( X ) and co(Y/X)=co(Y)-co(X)

by Theorem 3.1 of [83. So co induces a well-defined homomorphism

co: W,(G)-, Im(8).

If X is in ~/~,(G) and Hi(X)=O for i # k , let x e X denote the bascpoint, and consider the cellular chain complex

O_.~CN(X,x ) ON ) CN_I (X ,x ) oN_i) (~1 ... ~ Co(X,x)--,O.

The torsion r(X) is defined in [-8] by splitting the localized sequence up into short exact sequences, and taking their torsion individually. Since almost all of these are exact before localizing, their torsion lies in the image of KI(;gG), and we get

z(X) =_ ( - 1) k [Im(Sk + 0(,)~--~Ker(Sk)(,)] (mod Im(K 1 (2g G)) ).

Referring to the definition of 8,

co(X) = ( - 1) k 8([Im(8 k + 1)(,)~--~Ker(Sk)(,)])

= ( - 1) k [Ker(Sk)/Im(Sk + 1)] = (-- 1) k [Hk(X)J-

Step 3. It remains to prove co an isomorphism by constructing an inverse map

co*" Im(8)-+ W,(G).

Im(8)=Ker(q)) is easily seen to be the Grothendieck group on all finite 2gG- modules M of order prime to n and having a free resolution

For any such M, the resolution can be used to construct a complex X M in ~#~,(G), with cells in dimensions 2 and 3 only (apart from the basepoint), such that

fit, (XM) = Hz(XM) ~- M.

Set co*(M)= [XM].

The class [XM] in W,(G) is well defined, for a given module M, by Lemma 4.6. Likewise, for any short exact sequence

O-* M'--* M ~ M" ~O

there is a map f: XM,--*X u such that [MI]=[XM,, ] (again applying Lemma 4.6); so co* is a well defined homomorphism on Ira(0). That cooco*=id is clear; that co*oco=id follows from the fact that W,(G) is generated by complexes of length zero (Step 1). []

G-CW-Surgery and Ko(7/ G ) 37

As an immediate corollary, we get:

Lemma 4.8. Assume that n 1 ... . . G are positive integers, and n=(nl, ...,G) their g.c.d. Then Wn(G ) is generated by its subgroups Wn,(G ) (1 <iN k).

Proof. Let g be the 1.c.m. of the n~, and consider the commutative triangles:

Ko(~.(;z G))

Ko(JC/n,(~G) ) o, ,,i~o(~G )

/ Ko(Jg.(7ZG))

For any m > 1, we have the decomposition

Ko(Jm(~G)) = | Ko(p-torsion 7LG-modules of finite projective dimension). pZ,n

Thus, the vertical maps are monomorphisms (inducing inclusions W~(G) c W,,(G)~_ W~(G)), and Ko(ddn(;~G))is generated by the Ko(Jd,,(TZG)).

Now, using the fact that ~ maps Ko(dde(:gG)) onto Ko(gG), we get that Ker(0) is generated by the Ker(0i) for 1_<iN k. Applying Lemma 4.7, W,(G) is generated by the Wn,(G ). []

Now, for any H_c G, we write for short

BH(G) = B0(G, {1,gHg- 1}).

In other words, it is the group of projective obstructions arising from G- resolutions all of whose orbits, away from the basepoint, are of type G or G/H.

For any pair H<~G, we define an idempotent 6H~QG:

1 ~/=V~ 2 h.

I1~1 h e l l

Lemma 4.9. Assume H-~ G. Then a) BH(G ) = Ker [/(o(ZG)-~/~o(7ZG @u))]. b) Any element of BH(G ) has the form ?~(X) for some 2-dimensional G-

resolution X (having only orbits of type G and G/H away from the basepoint).

Proof. Write a=7ZG (g~), and let Ind~o denote the induced map from /(0(7ZG) to/ (o(~G @H))" Step 1. We first show that BH(G)c_Ker(Ind~). Let X be an arbitrary G- resolution with basepoint x and orbit types only G and G/H, and consider the cellular chain complex {C,(X, x)}. If k= dim(X), then the sequence

O~a| Ck(X , x)~. . . ~ a . Co(X , x)~O


is exact by Lemma 5 of [12], where for any 2gG-module M,

a. M = (a|176176

Each group Ci(X,x ) is a sum of copies of 7Z[G] and ;g[G/H]. Clearly, a- 7Z [G] = a; furthermore,

a. 7/[G/H] ~- Z [ G / H ] ~- ~ . a

is a projective summand of a. So each term in the above exact sequence is projective, summands g [ G / H ] occur equally often in even and odd dimensions (z(XF/)= 1), and a| is stably free. Thus,

7G(X) e Ker(Ind~ G).

Step 2. We finish proving the lemma by directly constructing, for any #eKer(Ind~G), a 2-dimensional G-resolution X (with the right orbit types) having 7G(X) = ~.

Write Z u = ~ h=lHl.aH, and consider the fiber square h ~ H

?ZG i= - , ZIG/R]

l 1 ;2 G/Z~ " , 7 z [ G / H I (n = IHI).

We can identify a = a H a G ( 1 - - a u ) a with ~[G/H]G~G/2~, and so

Uer(Ind~G)= Ira[0 ' KI(2g,[G/H])~I~o(7ZG)].

Given some ~eKer(Ind~G), write it as ~ = a ( [ f ] ) for some feAut(;g,[G/H]'), Let

f: ;g [ G / H ] ' ~ 7 / [G /H ] '

be some lifting to a monomorphism, and consider the diagram

ZG' '~ , Z [ G / H ] ' ~ : Z[G/H]'

(ZG/Z . ) ' j; , z [ ~ / H ] ' ~- : Z, , [G/H] ' .

Then (see [9], p. 28), a([f])=[P]-[;gG'], where P is the fiber product of (7ZG/s and 7Z, [G/H]' over j] and : o f - f o? Since f is one-to-one, we have J 2 - - 2 J "

p = {x~Gtli~2(x)cIm(f)} . t

= Ker [;g G' ,2 , 2g [G/H]'--+Coker(f)]

. t ~_ -

=Ker[gG'| t ~ J', 2g[G/H]'].

G-CW-Surgery and Ko(TI G) 39

Now construct a 2-dimensional G-complex X with basepoint, having t orbits each of type

G/H x D1, G/H x D 2, and G x D2;

with boundary map realizing (it2+f). Then IZl.(X)=H2(X)~P, so X is a G- resolution with

7a (X)=a ( [ f ] )= {. []

The goal here is to show, for as many connected families ~,~ as possible, that any element of Bo(G, ff) is representable by a 2-dimensional ~,~-resolution. In view of Lemma 4.9 (b), it suffices to show that

Bo(G, ~,~)= ~ Ind~(Bp(,)(G,)); c ~ e ~

in other words, that Bo(G,~) is generated by elements 7G(X) where X is an {m,G~}-resolution for ~ e ~ . (In fact, it is easily seen that the above decomposition is necessary for having 2-dimensional complexes.)

We first show the following simple case of such a decomposition:

Lemma 4.10. Let ~ = { 1 , H , K } , where H ~ K are both normal in G. Then

Bo(G, Y) = B~(G) + BK(G ).

Proof This time, set a=7/G@u,o/~ >. Since a. TZ[G/H] and a.7/[G/K] are both projective a-modules, we get

Bo(G, y ) c_ Ker [Ind~G :/~0(TfG)o/~0(a)],

exactly as in the proof of Lemma 4.9 (Step 1). Since Bu(G ) and BK(G ) clearly lie in Bo(G , @), it suffices to show that

Ker(Ind~ o ) = BH(G ) + BK(G).

Consider the following commutative square:

Ko(71G ) ,1 , Ko(TIG<aK> )

li~ ljl =Ko(7IG/X~)@Ko(7I[G/K])

Ko(TZG <cr,>) j2 , K0(a )

Ko(7LG/Y_,u)QKo(2~ [G/H]) Ko(7fG/ZH)OKo(7/[G/H]/ZIr @Ko(~ [G/K3).

We have BK(G)=Ker(i 0 and BH(G)=Ker(i2) , by Lemma 4.9, and so it suffices to show that i 2 maps Ker(ia) onto Ker(jz ). Fix some

~e Ker(j2) _c Ko(7Z [G/H]).

40 R. Oliver and T. Peme

By Lemma 2 of [131, there exist projective 2~[G/H].modules P1 and ~ with ~=[P1]-[P2], and a monornorphism f: P t - ' ~ whose cokernel is a finite ~[G/K]-modute of order prime to iGJ. Then T has finite homological dimension as a ;gG-module; let

be a projective resolution (over ~ZG). Set ~= [ ~ ] - [~ ] ~Ko(ZG). Since T ~ :E[G/H]| there is an exact sequence

Z T O,

and so i2(~):~ (Schanuel's lemma). Thus, ]1 i~(~)=0, and so ~ goes to zero in Ko(TZ [G/K]). On the other hand,

(~ G/SK)| T=O

(T being a ~,[G/K]-modute of order prime to [KI), ~ goes to zero in Ko(2~G/SK) , and so i1(~)=0. Thus, Ker( i j maps onto Ker(jJ. []

This can now be applied to show:

Lemma 4.11, Let Y = ~ be a connected family of subgroups of G, such that there is a unique maximal element H<~ G in .~. Assume further that for all plIHt, H~ is normal in G (where Hp is a p-Sytow subgroup of H). 7hen

B0(G , g ) = Bo(G, ~ - {H}) + BH(G).

Proof. Set n=n(H), the product of all primes p such that H=H~, and let Pl .... ,Pk be the primes dividing IHI but not dividing n. For all i, set

n i = (np t"*" P~)/P~"

By Lemma 4.4 (b), ;~G induces a surjection

?: N,(G/H)-+Bo(G, ~)/Bo(G, W" - {H}).

In view of Lemma 4.8 (W,(G/H) being generated by the W,~(G/H)), it suffices to show for all i that

y(W,,(G/H)): [Bo(G , Y - {H}) + BH(G)]/Bo(G, ~ - {H}). (1)

Fix some i, and let X be any complex in ~,(G/H). Set -7~={I,H~,H}, and note that n~:(H)[ G (in other words, p,Xn:%(H)). By

Lemma 4.4 (a), there is a simple (G,~)-resolution ~ with Y : = X . Applying Lemma 4.10:

?,o(Y)eSo(G, ~ ) = Bn~ (G ) + Bn(G ) ~ Bo(G, ~ - {H}) + BH(G),

;~([X]) = {7~(Y)}, and so (1) holds. []

This can finally be applied to give the second main result of this section:

G-CW-Surgery and Ko(ZG ) 41

Theorem 4.12. Let Y~_I I be a connected G-family. Assume ~ has the property." for anj, c ~ and any fl>c~ (fl~H) such that p(fl) is a Sylow subgroup of p(~), p(fl) .~ G~. 7hen

Bo(G,Y)= ~ Ind~(Bp(~)(G~)).

In particular, ~ is simply generated: all elements of Bo(G , y ) are representable as ?G(X) with dim(X)= 2.

Proof Since Bo(G,~)=Bo(G,J ~ ) (Theorem 3.3), we may assume that ~ = ~ . Fix a minimal element e e l , and set

= {p( /~) l /~Y, /~>_- ~}.

The hypotheses of Lemma 4.11 apply to ~ as a family of subgroups of G~, and we get

Bo(G, @) = B o(G, ~ - G ~) + Ind~(B0(G ~, o~)) (Lemma 4.4)

= Bo(G, Y - a ~) + Indo6 EBo(G~, ~ - {p(cO} ) + Bp(~)(G~)]

(Lemma 4.11) = Bo(G, Y - G ~) + Ind~(Bm)(G~))

(note that Ind~(Bo(G~,~-{p(e)}) )C_Bo(G,Y-Ge)) . The theorem follows by induction ( ] ~ - G ~ ] < l g l ) . []

Two particular cases where these conditions hold are:

Corollary 4.13. Assume that G is abeIian. Then any connected G-family ~ is simply generated.

Proof The hypothesis "p( f l )~G~" of Theorem 4.12 holds trivially in this case. []

Note. With a little more work, Corollary 4.13 can be shown to hold for any nilpotent group G.

Corollary 4.14. Let G be any finite group, and ~ a connected G-family such that for all c ~ , p(e) is a p-group for some p. Then ~ is simply generated.

Proof For any c~e~, if p(c0 is a p-group then the only fl>c~ with p(fl) a Sylow subgroup of p(c0 is fl=cc So/~=fl, and p(fl)=p(a).~G~. []

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(1960)

Received July 30, 1980; received in final form May 27, 1981

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G-CW-surgery andK0(ℤG)