A Proof of the Conner Conjecture

Annals of Mathematics

A Proof of the Conner ConjectureAuthor(s): Robert OliverSource: Annals of Mathematics, Second Series, Vol. 103, No. 3 (May, 1976), pp. 637-644Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970955 .

Accessed: 22/11/2014 20:48

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals ofMathematics.

http://www.jstor.org

This content downloaded from 64.141.84.23 on Sat, 22 Nov 2014 20:48:27 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=annals

http://www.jstor.org/stable/1970955?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Annals of Mathematics, 103 (1976), 637-644

A proof of the Conner conjecture

By ROBERT OLIVER

The main result to be proved below is that if a compact Lie group G acts on a space X, where X is either paracompact of finite cohomological dimension and with finitely many orbit types, or compact Hausdorff, then X/G is Z- or Z,-acyclic (p any prime, and Z, = Z/pZ) if X is. Combining this with results of Conner [3] gives that the orbit space of any compact Lie group action on Euclidean space is contractible.

All spaces considered here are paracompact, and sheaf cohomology will be used. Since with closed supports and constant coefficients, this agrees with both the Alexander-Spanier and Cech cohomology theories for such spaces, the results on acyclicity in [2], (3] and [5] can be used.

By a "p-group" will be meant any group H with torus identity com- ponent Ho such that HI/HO has p-power order. The standard results of Smith theory for actions on acyclic spaces will be used (as, e.g., in [5], ? IV. 1): any action of a p-group on a paracompact Z,-acyclic space of finite cohomological dimension with finitely many orbit types has Z,-acyclic fixed-point set. Similarly, any action of a torus on a paracompact Z-acyclic space of finite cohomological dimension with finitely many orbit types has Z-acyclic fixed-point set: this follows from Theorem III. 1' in [5].

Up until Theorem 2, it will be assumed without explicitly repeating each time that all group actions are on paracompact spaces of finite cohomological dimension with finitely many orbit types. It follows from [7], Proposition A.11 that the orbit space of any such action has finite cohomological dimension.

Assume first that G is connected, and that Y _ X is a pair of G-spaces, with Y closed in X. For coefficient ring A, the Leray spectral sequence for the orbit map p: X-* X/G has E2-term

E2i' (X, Y) = Hi(X/G; FCj(p, p I Y; A))

(with the notation of Bredon [1]) and converges to H*(X, Y; A). YCj(p, pj Y; A) is the sheaf over X/G generated by the presheaf

U - Hj(p-'U, Ynfp-'U; A)



638 ROBERT OLIVER

and with stalks EXY(p, p I Y; A). = H5(p-'x; A) for x e X/G - Y/G; Xj(p, p I Y)X = 0 for x e Y/G. In particular XYC1(p, p I Y; A) (and thus E,',(X, Y)) is zero for j > n = dim (G).

We define a map wr*: XYCj(p, p I Y; A) Hj(G; A): on the stalk over any point x e X/G- Y/G, w* is the map on cohomology induced by an equivariant map of G onto p-'x (any two such are homotopic). To see that this is con- tinuous, fix some point x and some c e Hi(p-'x, A). By the slice theorem, there is a neighborhood U of x in X/G - Y/G with an equivariant retraction r: p-' U- p-'x. Set ' = r*(c), and for y e U let i,: p-'y c p- U denote the inclusion; then {i*(0): y e U} forms an open neighborhood of c in XfC(p, p I Y; A), and wr*(i*(e)) = wr*(c) for all y e U.

Thus, wr* maps XiC to the constant sheaf H5(G; A) over X/G - Y/G, inducing a homomorphism

wr*: E2i'j(X, Y) -- H'(X/G, Y/G; Hj(G; A)) .

An orientation class $ e Hn(G; A) induces

$*: E2 '(X, Y) > H'(X/G, Y/G; A)

and from this a "transfer" map

t: H'+n(X, Y; A) - E , E2 - H'(X/G, Y/G; A)

All of these maps are functorial. Let p be a fixed prime; from now on, until Theorem 1 has been proved,

all cohomology groups (unless otherwise indicated) will be assumed to be taken with Zp-coefficients. For any G and any G-space X, let X8 denote the union of the fixed point sets of all order p subgroups of G.

LEMMA 1. Assume that any isotropy subgroup of X - Y is finite of order prime to p (i.e., Y Q X8). If

H'+'4(Xy Y) -->H'(XIG, Y/G)

is zero, then H*(X/G, Y/G) - 0.

Proof. If H?7 G is finite of order prime to p, then

1*: H*(G/H) H*(G)

is an isomorphism. Thus,

i s*: a o s*(pe p , Y) H*(G)

is an isomorphism on each stalk, and we get



PROOF OF THE CONNER CONJECTURE 639

If H*(X/G, Y/G) # 0, let i be the largest dimension where it is non- zero. Then E~k (X, Y) = 0 for k > i,

d*E2 ff >H'(XIG, YIG) is an isomorphism, and thus

tHi+-(X~ Y) >Hi(XIG, YIG)

is an isomorphism. Since we are assuming e = 0, H*(X/G, Y/G) must be zero. D

PROPOSITION 1. If G is connected (and non-trivial), and X is Z,-acyclic, then H*(X/G, X8/G) = 0.

Proof. For any x e X8/G, p-'x _ G/H, where H is either positive dimensional or finite of order a multiple of p. In either case,

7r*: H"(GIH) H "(G)

is the zero map, and so

7* : SJC(p IX8) - Hn(G)

is zero. It follows that the transfer e on X8 is zero in all dimensions.

For i ? 0 we have the commutative square

a H'+n-'(Xs) 43H'+n(XS X8)

Hi-i(X.IG) H'(XIG, X.IG) (n > 1 by assumption), and it follows that the transfer map e for (X, X8) is zero in all dimensions. By Lemma 1,

H*(X/G, X8/G) = 0 . D

Referring back to the proof of Lemma 1, one sees that we have also proved the peculiar result: if any connected G acts on a Zp-acyclic space X, then X8, the union of the fixed point sets of order p subgroups, is Zp-acyclic. This is definitely not true in general (e.g., for G = S3 and p = 2).

The conclusion of Proposition 1 can now generalize to actions of arbitrary groups. If G is finite of order prime to p, and X is Zp-acyclic, then X8 = 0 and X/G is Zp-acyclic by the transfer map. Otherwise

PROPOSITION 2. If G is any compact Lie group, either positive dimensional or finite of order a multiple of p, acting on a Zp-acyclic space X, then

H*(X/G, X8/G) = 0 .



640 ROBERT OLIVER

Proof. We may assume that XG # 0 (otherwise replace X by its suspension). Fix a base point x e XG; x e X8 by the assumption on G.

Let G Q G be some connected group containing G as subgroup (e.g., G = U(n) for some n), and define

Y= (G XG X)/(G XG X) .

Since G XG X fibers over GIG with fiber X and section G x, x, Y is Zp-acyclic (the fibration is not closed if X is not compact, but the spectral sequence still applies by [7], Proposition A. 4.) Furthermore, it is easily seen that

Y3 = (G XGXS)/(G XG X);

so Y/G = X/G, YS/G = XS/G, and by Proposition 1,

H*(X/G, X.IG) = H*(Y/G, YJ/G) = 0 *El To extend these results, we study more closely the p-groups in G. If

T is a maximal torus for G, and H any p-subgroup, then (G/N(T))H is non- empty (G/N(T) has Euler characteristic 1) and so H lies in some conjugate to N(T). Thus, the maximal p-groups in G are precisely those conjugate to the extension of T by a p-Sylow subgroup of N(T)/T, and any p-group is contained in such a maximal one.

For any p-group H(z G and any G-space X, define

X(H) UaeG XaHa ; Xf(H) - U {X(K): K; H, K a p-group}

Thus, X'fH) is the set of points x e X such that Gx contains some subgroup conjugate to H, but not as a maximal p-group.

LEMMA 2. If HC- G is a p-subgroup which is not maximal, then N(H)/H contains a subgroup of order p.

Proof. Since H is not maximal, it follows that X(G/H)- 0 (mod p). Thus,

X(N(H)/H) = X((G/H)H) - Z(G/H) 0_ (mod p)

and N(H)/H must be either positive dimensional or finite of order a multiple of p. El

PROPOSITION 3. If X is a Zp-acyclic G-space and H_ G is any p-group, then X(Hf/G is Zp-acyclic if H is a maximal p-group, and

H*(X(H)/G, X'H)/G) 0

if H is not maximal.

Proof. Consider the map




f:( N(H), (X.') fN(H)) > I'G, X' fG) It is clearly a closed map and surjective; to show that it is one-to-one on the difference space we must show for any x e XH-(x8H))H, with K = GXY that N(H) acts transitively on (G/K)H . But K contains H as a maximal p-subgroup; for any aKe (G/K)H, HC aKa-1, so a-'Ha and H are both maximal p-groups in K. There is thus b e K such that a-'Ha = bHb-1; i.e., ab e N(H) and (ab)K aK.

It follows that f* is an isomorphism of the relative cohomology groups by the tautness properties of sheaf cohomology. If H is a maximal p-subgroup, then X('H) = 0, XH/(N(H)/H) is Z,-acyclic (since N(H)/H is finite of order prime to p), and so X(H)/G is Z,-acyclic.

Assume now that H is not maximal in G, and set Y = X', regarded as an N(H)/H-space. Then Y8 is the set of all x e X such that G. D H and such that

(N(H)/H)X = N(H) G$ = NGx(H)/H X H

contains a subgroup of order p. By Lemma 2, this is the case if and only if H_ Gx and not as a maximal p-subgroup. Thus, Y. = (X'H))H, and

H*(X(H)/G, X(H/G) -H*(Y/(N(H)/H), Y,/(N(H)/H)) = 0 by Proposition 2. ii

The main result now follows immediately:

THEOREM 1. For any compact Lie group G, any action of G on a paracompact Zp-acyclic space X of finite cohomological dimension and with finitely many orbit types has Zp-acyclic orbit space.

Proof. Let {Hi: i = 0, 1, * * *, k} be conjugacy class representatives for all p-subgroups of G which occur as maximal p-groups in isotropy subgroups for X. Assume they are arranged such that if Hi contains some subgroup conjugate to Hj, then i < j. In particular, H0 is a maximal p-subgroup for G.

For i = 0, 1, ..., k, set Xi = U=0 X(Hj). Then XO/G = X(Ho)IG is Zp-acyclic by Proposition 3, and for all i ? 1,

H*(X%/G, Xi-i/G) - H*(X(Hi)/G, XHi/G) - 0

(all subspaces are closed, so excision works). Thus, XkG = X/G is Zp-acyclic. F1

To get similar results in the Z-acyclic case, one carries out the same procedure, but uses tori instead of p-groups. So until Theorem 2 has been



642 ROBERT OLIVER

proved, all cohomology will be with Z-coefficients, and X8, for a G-space X, is now defined to be the set of points with positive dimensional isotropy subgroup. Spaces are still assumed to be paracompact of finite cohomological dimension.

PROPOSITION 4. If G is connected of dimension n > 1, and X is a Z- acyclic G-space, then H*(X/G, X8/G) = 0.

Proof. As in the proof of Proposition 1, we get that

eH * +"(X, Xj) > H*(X/G, X8/G) is zero, since the transfer map for X8 is zero. For any isotropy subgroup Hfor X- X8, the map

H*(G/H) H*(G) has finite kernel and cokernel, and since there are only finitely many orbit types, there exists N such that the kernel and cokernel of

7w*: (C*(p, p I X8) > H*(G)

consists of N-torsion on all stalks and in all dimensions. Passing to cohomology, the kernel and cokernel of

1*r E2t j(X, X8) H'(XIG, XI/G; Hi(G))

are N2-torsion in all dimensions. In particular, if H*(X/G, X8/G) : 0, with k being the largest non-zero dimension, then El" consists of N2-torsion for i > k, and the inclusion E'ke , E2"' has finite exponent. Thus, the image of the transfer map has finite exponent in Hk(XIG, X8/G).

It follows from Proposition 3 (applied to all finite p-subgroups) that H*(X/G, X8/G; Z,) = 0 for any prime p, and thus, that multiplication by any integer is an isomorphism on H*(X/G, X8/G). So the transfer must map onto Hk(X/G, X8/G), which contradicts that it is zero. D2

Now, the exact same proof as for the Zp-case may be carried through. THEOREM 2. If a compact Lie group G acts on a paracom pact Z-acyclic

space X of finite cohomological dimension with finitely many orbit types, then X/G is Z-acyclic. II

In [3], Conner defines an "acyclic model" mod A. for a group G to be a compact A-acyclic fixed point free G-space with finitely many orbit types and with A-acyclic orbit space. A group G has property Q(A) if any connected non-abelian subgroup of G has an acyclic model mod A.

The author has shown in [6] that any connected non-abelian compact Lie group has a smooth fixed point free action on some disk, which is an




acyclic model mod Z by Theorem 2. Alternatively, the Hsiangs show in [4] that any such G has a linear fixed point free action on some sphere S" and a degree 0 equivariant map f from S' to itself. The inverse limit of the system

f S f f

is a compact finite dimensional Z-acyclic space with a fixed point free G-action having only finitely many orbit types, and is thus also an acyclic model for G. So any compact Lie group has property Q(Z).

Using Conner's techniques, we can now show that for actions on compact spaces, most of the hypotheses in Theorem 1 can be removed.

THEOREM 3. If G is a compact Lie group and A = Z, Z., or Q, then any action of G on a compact A-acyclic Hausdorff space has A-acyclic orbit space.

Proof. For G finite, this is proved in [2], Chapter III. For G = S', since the universal coefficient theorem holds for coho-

mology on compact spaces (see, e.g., [1], Theorem II, 18.3), it suffices to consider the cases A = Z. or Q. For any compact A-acyclic G-space X, set X8 = Xzp if A = Zp, or X, = XG if A = Q; X8 is A-acyclic in both cases (this follows from Theorem III. 1 in [5]). As in the proof of Lemma 1, the Leray spectral sequence for the map

p: (X X.) >(XIG, X.IG)

has E2-term

E2K'j - H'(X/G, Xj/G; Hj(G; A)).

Since H*(X, Xi) = 0, we have H*(X/G, X,/G; A) = 0. If A = Q we are done; if A = ZV this shows that the restriction

H*(XZPn/G; Z,) > H*(Xzpn+l/G; Z,)

is an isomorphism for all n. Since X is compact,

H*(X/G; Z,,) lim H*(Xz p/G; Z,,) - H*(XG; Z,)

H*(XG; Zp) lim H*(XZPn; Z,)

and so XG and X/G are both Z.-acyclic. Thus, it is enough to show, for any connected non-abelian group G,

-that the theorem holds for G if it holds for all proper subgroups of G. Let X be an arbitrary compact A-acyclic G-space, and let Y be an acyclic model (mod Z) for G. Then all fibers in the maps



644 ROBERT OLIVER

p1: (X x Y)/G - X/G,

P2: (X x Y)/G > Y/G

are A-acyclic, so pi and P2 induce isomorphisms of A-cohomology by the Vietoris mapping theorem, and X/G is A-acyclic. FII

The last two theorems follow directly from Theorems 3.10 and 3.12 and Corollary 5.5 in [3].

THEOREM 4. Let X be a locally compact, finite dimensional, separable metric space with G-action, such that any compact invariant subspace contains only finitely many orbit types. Then

( 1 ) If X is n-dimensional and cl(c" (mod A), for A = Z. Zp, or Q. then X/G is clac (mod A).

( 2 ) If X is locally contractible, then so is X/G. D THEOREM 5. Any action of a compact Lie group on a Euclidean space

has contractible orbit space. [1 MATEMATISK INSTITUT, AARHUS, DENMARK

REFERENCES

[1] G. BREDON, Sheaf Theory, McGraw-Hill, 1967. [2] , Introduction to Compact Transformation Groups, Acad. Press, 1972. [3] P. CONNER, Retraction properties of the orbit space of a compact topological trans-

formation group, Duke Math. J. 27 (1960), 341-357. [41 W.-C. HSIANG and W.-Y. HSIANG, Differentiable actions of compact connected classical

groups I, Amer. J. Math. 89 (1967), 705-786. [5] W.-Y. HSIANG, Cohomology Theory of Topological Transformation Groups, Springer,

1975. [6] R. OLIVER, Smooth compact Lie group actions on disks, to appear. [7] D. QUILLEN, The spectrum of an equivariant cohomology ring: I, Ann. of Math. 94

(1971), 549-572.

(Received November 17, 1975)



Documents

A Proof of the Conner Conjecture