EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS
J.P. MAY
Contents
1. The fundamental groupoid and categorical definitions 3
2. Functors to the category of G-spaces over orbits 5
3. Coherent families of connected covers 7
4. Orientability of spherical G-fibrations 8
5. Coherent families of Thom classes and Thom isomorphisms 10
6. Orientations in ordinary equivariant cohomology 13
7. Concluding remarks 16
References 16
Let G be a compact Lie group and let E*Gbe an RO(G)-graded cohomology
theory on G-spaces. We shall explain a sensible way to think about orientations
and the Thom isomorphism theorem in the theory E*G, offering an alternative to
the approach given by Costenoble and Waner in [7]. Both approaches generalize
the restricted theory given by Lewis and myself in [15], and both grew out of j*
*oint
work of Costenoble, Waner, and myself [5].
In the study of nonequivariant bundles and their orientations, an innocuous
first step is to assume that the base space is (path) connected. The analogous
equivariant assumption is that the G-set of components of the base space is a s*
*ingle
orbit, but this assumption doesn't get us very far. There is an entirely satisf*
*actory
theory of equivariant Thom isomorphisms and Poincare duality under the much
more stringent hypothesis that the base space X be G-connected, in the sense th*
*at
each XH is non-empty and (path) connected. This is developed in detail in [15,
IIIx6 and Xx5]. The basic problem, then, is to generalize that theory to a less
restricted class of base spaces. The obvious approach is to parametrize changes
of fiber representation on the fundamental groupoid. However, doing this direct*
*ly
leads to a fairly complicated, and hard to compute, generalization of equivaria*
*nt
cohomology [5, 6, 7, 8]. We seek a variant approach that allows us to work with*
*in
the framework of RO(G)-graded cohomology theories, so that we can apply rather
than generalize the pre-existing theory of [15].
Our essential idea is that, to obtain a satisfactory general theory, it seems*
* reason-
able to give up the idea that the Thom isomorphism must be a single isomorphism.
Rather, we shall define it to be an appropriate family of isomorphisms. More pr*
*e-
cisely, we shall define an E*G-orientation of a G-vector bundle, or more genera*
*lly of a
spherical G-fibration, to be a suitably coherent family of cohomology classes. *
*Each
class in the family will determine a Thom isomorphism, and these isomorphisms
will be nicely related. If the base space is G-connected, then all of these Th*
*om
1
2 J.P. MAY
classes and Thom isomorphisms will be determined by one member of the family.
In general, all members of the family will be determined by choosing members of
the family indexed on components of fixed point spaces XH which do not contain
any K-fixed points for K larger than H.
We begin in Sections 1-3 by defining the fundamental groupoid ss(X), discussi*
*ng
functors defined on it, and constructing the family of "H-connected covers" of a
G-space X. This construction can be expected to have other uses in equivariant *
*al-
gebraic topology. Many arguments in algebraic topology begin with the statement
"We may assume without loss of generality that X is connected". Our H-connected
covers give a tool that often allows us to give the same start to equivariant a*
*rgu-
ments. However, the reader is warned that there are some prices to be paid, bey*
*ond
the intrinsic complexity. Probably the most significant is that the H-connected
cover of a finite G-CW complex will in general be infinite dimensional. For th*
*is
reason, I have not yet succeeded in obtaining a satisfactory treatment of Poinc*
*are
duality that starts from the Thom isomorphism theorem given here.
We define the notion of an orientable spherical G-fibration in Section 4. Th*
*is
does not depend on our H-connected covers, but we use these in our definition of
an E*G-orientation of a spherical G-fibration in Section 5. Our Thom isomorphism
theorem in E*G-cohomology follows directly from the definition and the work in *
*[15].
We specialize to ordinary RO(G)-graded cohomology with Burnside ring coefficien*
*ts
in Section 6. Nonequivariantly, orientability as defined topologically is equiv*
*alent
to cohomological orientability with integer coefficients. We prove that perhaps
the most natural topological notion of equivariant orientability is equivalent *
*to
cohomological orientability with Burnside ring coefficients. We briefly mention
other examples in Section 7.
The theory here was suggested by joint work with Steve Costenoble and Stefan
Waner [5], and later Igor Kriz, that began in the late 1980's and is still larg*
*ely
unpublished. In that work, we take the more direct approach, confronting head-
on the problem of parametrizing the change of fiber representations of a G-vect*
*or
bundle by the equivariant fundamental groupoid of the base space. Rather than
giving up the idea that an orientation is a single cohomology class, we constru*
*ct
more complicated cohomology theories in which a single class can encode all the
complexity. That approach has been exploited in a series of papers on this and
related topics by Costenoble and Waner [6, 7, 8, 9, 10]. A fully coherent theo*
*ry
of orientation will require a comparison of that approach to the one given here.
Although Costenoble, Kriz, Waner, and I sketched out such a comparison some
years ago, the details have yet to be worked out. It is a pleasure to thank Cos*
*tenoble,
Waner, and Kriz for numerous discussions of this material.
This paper is a very small token of thanks to Mel Rothenberg, my colleague and
friend for the last 32 years. I wish I had a better paper to offer, since this*
* one
should have a sign on it saying "speculative, may not be useful", but it is in *
*one of
the areas that Mel has pioneered (e.g. [18]) and that I have in part learned fr*
*om
him. It has been a privilege to work with him all these years to help make Chic*
*ago
a thriving center of topology.
EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 3
1.The fundamental groupoid and categorical definitions
We here recall our preferred definition of ss(X) and give some categorical la*
*n-
guage that will help us define structures in terms of it. An equivalent defini*
*tion
appears in [11, 10.7], and a definition in terms of Moore loops is given in [17*
*, App].
We assume that G is a compact Lie group, and we only consider closed subgroup*
*s.
Let OG denote the topological category of orbit spaces G=H and G-maps between
them, where H runs through the (closed) subgroups of G. Let hOG be the homotopy
category of OG . Of course, OG = hOG if G is finite. The following observation
describes the structure of hOG for general compact Lie groups G. Recall that if
ff : G=H -! G=K is a G-map with ff(eH) = gK, then g-1Hg K.
Lemma 1.1. Let j : ff -! fi be a G-homotopy between G-maps G=H -! G=K.
Then j factors as the composite of ff and a homotopy c : G=H x I -! G=H such
that c(eH; t) = ctH, where c0 = e and the ctspecify a path in the identity comp*
*onent
of the centralizer CG H of H in G.
Proof.Let j(eH; t) = gtK. Since we can lift this path in G=K to a path in G
starting at g0, we may assume that the gt specify a path in G. Now g-1tHgt K
for all t, so we can define d : H x I -! K by d(h; t) = g-1thgt. Since the adjo*
*int
"d: I -! Map (H; K) is a path through homomorphisms, the Montgomery-Zippen
Theorem [4, 38.1] implies that there are elements kt 2 K such that k0 = e and
d(h; t) = k-1tg-10hg0kt. Define i : K -! Hom (g-10Hg0; K) by i(k)(h0) = k-1h0k.
The image of i may be identified with K=L, where L is the subgroup of elements k
such that i(k) = i(e) is the inclusion of g-10Hg0 in K. It follows that i is a *
*bundle
over its image. We may regard "das a path in Hom (g-10Hg0; K), and we can lift *
*it
to a path k : I -! K with k(0) = e. Thus we may assume that the kt specify a
path in K. Now define ct = gtk-1tg-10. Then j(eH; t) = ctg0K, c0 = e, and_the_ct
are in CG H, as desired. |__|
Definition 1.2.Let X be a G-space. Define the fundamental groupoid ss(X) as
follows. Its objects are the pairs (G=H; x), where H G and x 2 XH ; we think
of this pair as the G-map x : G=H -! X that sends eH to x. The morphisms
(G=H; x) -! (G=K; y) are the equivalence classes [ff; !] of pairs (ff; !), where
ff : G=H -! G=K is a G-map and ! : G=H x I -! X is a G-homotopy from x to
yOff. Here two such pairs (ff; !) and (ff0; !0) are equivalent if there are G-h*
*omotopies
j : ff ' ff0and k : ! ' !0 such that
k(a; 0; t) = x(a) and k(a; 1; t) = y O j(a; t)
for a 2 G=H and t 2 I. If G is finite, then ff = ff0and j is constant. Composit*
*ion
is evident. Define a functor " = "X : ss(X) -! hOG by sending (G=H; x) to G=H
and sending [ff; !] to the homotopy class [ff] of ff. A G-map f : X -! Y induces
a functor f* : ss(X) -! ss(Y ) such that "Y O f* = "X . A G-homotopy h : f ' f0
induces a natural isomorphism h# : f* -! f0*.
To be precise about orientability and orientations, we need some abstract def-
initions and constructions, which are taken from joint work with Costenoble and
Waner [5]. The first encodes the formal structure of the fundamental groupoid.
Definition 1.3.A groupoid over a small category B is a small category C together
with a functor " : C -! B that satisfies the following properties. For an objec*
*t b
4 J.P. MAY
of B, the fiber Cb is the subcategory of C consisting of the objects and morphi*
*sms
that " maps to b and idb.
(i)For each object b of B, Cb is either empty or a groupoid (in the sense that
each of its morphisms is an isomorphism).
(ii)(Source lifting) For each object y 2 C and each morphism fi : a -! "(y) in
B, there is an object x 2 C such that "(x) = a and a morphism fl : x -! y
in C such that "(fl) = fi.
(iii)(Divisibility) For each pair of morphisms fl : x -! y and fl0 : x0- ! y i*
*n C
and each morphism fi : "(x) -! "(x0) in B such that "(fl) = "(fl0) O fi, t*
*here
is a morphism ffi : x -! x0in C such that "(ffi) = fi and fl0O ffi = fl.
Remark 1.4.We say that C has unique divisibility if the morphism ffi asserted to
exist in (iii) is unique. This holds for fundamental groupoids when G is finit*
*e,
but not when G is a general compact Lie group. When it holds, C is exactly a
"categorie fibree en groupoides" over B as defined by Grothendieck [13, p. 166].
Definitions 1.5.Let C be a groupoid over B.
(i)C is skeletal over B if each fiber Cb is skeletal (has a single object in *
*each
isomorphism class of objects).
(ii)C is faithful over B if " is faithful (injective on hom sets).
(iii)C is discrete over B if each Cb is discrete (has only identity morphisms).
Lemma 1.6. C is discrete over B if and only if it is skeletal and faithful ove*
*r B.
Proof.Clearly, if C is skeletal and faithful, then it is discrete. If C is disc*
*rete,_then
it is clearly skeletal, and it is faithful by Remark 1.7(ii) below. *
* |__|
Remarks 1.7.Let C be a groupoid over B.
(i)If C is skeletal over B, then divisibility implies that the object x asser*
*ted to
exist in the source lifting property is unique. If C is skeletal and faith*
*ful over
B, then the morphism asserted to exist in the source lifting property is a*
*lso
unique. If C is faithful over B, then it is uniquely divisible.
(ii)By divisibility, any two morphisms x -! y of C over the same morphism of
B differ by precomposition with an automorphism of x over the identity mor-
phism of "(x). Thus C is faithful over B if and only if the only automorph*
*isms
in each Cb are identity maps.
(iii)If fl : x -! y is a morphism of C such that "(fl) is an isomorphism, then
fl is an isomorphism, as we see by application of divisibility to the equa*
*lity
"(fl)"(fl)-1 = id. If every endomorphism of every object of B is an isomor-
phism, as holds in OG , then every endomorphism of every object of C is an
isomorphism.
Construction 1.8. Let C be a groupoid over B. We construct the discrete
groupoid over B associated to C .
(i)Construct a faithful groupoid C =" with the same objects as C by setting
C ="(x; y) = Im(" : C (x; y) -! B("(x); "(y)):
The quotient functor C -! C =" is the universal map from C into a faithful
groupoid over B.
(ii)Construct a skeletal subgroupoid C 0of C by choosing one object in each
isomorphism class of objects of Cb for each object b of B and letting C 0be
EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 5
the resulting full subcategory of C . The inclusion C 0-! C is an adjoint
equivalence of categories over B; its left inverse is a retraction ae obta*
*ined
from any choice of isomorphisms from each object of each Cb to an object of
Cb0. We call C 0a skeleton of C .
(v) By Remark 1.7(iii), passage from C to C =" creates no new isomorphisms, so
that we can make the same choices of objects for C and for C =" when formi*
*ng
skeleta. Then C 0=" = (C =")0. We call this category the discrete groupoid*
* over
B associated to C .
Lemma 1.9 (Joyal). A discrete groupoid C over B determines and is determined
by an associated contravariant functor : B -! Sets.
Proof.Given C , define as follows. For an object b of B, (b) is the set of obj*
*ects
of Cb. For a morphism fi : a -! b of B and an object y of Cb, (fi)(y) is the
unique object of Ca that is the source of a map covering fi. Given , define C as
follows. Its objects are pairs (b; y), where b is an object of B and y 2 (b). *
* A
morphism (a; x) -! (b; y) is a morphism fi : a -! b of B such that (fi)(y) =_x.
Composition, and the functor " : C -! B, are evident. |__|
Now return to the fundamental groupoid.
Notations 1.10. Let ss0(X) denote the discrete groupoid over hOG associated to
the fundamental groupoid ss(X). The quotient functor ss(X) -! ss(X)=" identi-
fies [ff; !] and [ff0; !0] whenever [ff] = [ff0], so that functors defined on s*
*s(X) factor
through ss(X)=" if their values on morphisms are independent of paths. The cate-
gory ss0(X) is obtained from ss(X)=" by choosing one point in each component of
each fixed point space. The notation ss0(X) is justified since the associated *
*con-
travariant functor hOG - ! Setscan be identified with the evident functor that
sends an orbit G=H to the set of components ss0(XH ).
2. Functors to the category of G-spaces over orbits
In the next section, we show how to construct a system of interrelated "H-
connected covers" associated to a given G-space X. The interrelationships will *
*be
encoded in terms of functors defined on ss(X). We describe the target category *
*and
the abstract nature of the functors we will be concerned with in this section.
Let U be the category of compactly generated, weak Hausdorff spaces and let
GU be the category of G-spaces.
Definition 2.1.Define GU =OG to be the category of G-spaces over G-orbits.
The objects of this category are G-maps O : X -! G=H and the morphisms are
commutative diagrams of G-maps
f
X _______//_Y
O|| ||
fflffl| fflffl|
G=H __ff_//G=K:
There is an evident notion of a homotopy between such morphisms and a resulting
homotopy category hGU =OG . Let " : hGU =OG -! hOG be the evident augmen-
tation functor; it forgets the G-spaces and remembers the G-orbits.
6 J.P. MAY
We think of a G-map O : X -! G=H as having the total space X and base space
G=H, although we do not require O to be a fibration. Let V = O-1(eH) X. Then
V is an H-space, and the action of G on X induces a G-map : G xH V - ! X
that is easily verified to be a bijection. To avoid point-set pathology, we agr*
*ee to
restrict attention to completely regular (e.g., normal) G-spaces. For such X, *
*is a
homeomorphism of G-spaces. The following remark gives a more concrete, but less
canonical, description of the category hGU =OG .
Remark 2.2.For a commutative diagram of G-maps
f
G xH V _____//G xK W
O || ||
fflffl| fflffl|
G=H ___ff__//_G=K
with ff(eH) = gK, define "f: V -! W by "f(v) = g-1f(v). Then "fis an H-map,
where H acts on W by hw = g-1hgw, and
(2.3) f(j; v) = (jg; "f(v))
for j 2 G. We call f the G-map associated to the pair (f"; g). Suppose that maps
f0 and f1 over homotopic G-maps ff0 and ff1 are associated to pairs (f"0; g0) a*
*nd
(f"1; g1). Then (f0; ff0) and (f1; ff1) are homotopic if and only if there is a*
* path "ft
connecting "f0to "f1in the space of H-maps V -! W . The point is that, by Lemma
1.1, the homotopy ff0 ' ff1 can be written in the form fft(eH) = ctg0K, where ct
is a path in CG H starting at e, and the conjugate H-action on W is then the sa*
*me
throughout the homotopy.
Definition 2.4.Let C be a groupoid over hOG . A C -space is a functor Y : C -!
hGU =OG over hOG . Given a map A : C -! C 0of groupoids over hOG , a map
OE : Y -! Y 0from a C -space Y to a C 0-space Y 0is a natural transformation
OE : Y -! Y 0O A over hOG .
There is a less conceptual but perhaps more easily understood version of this*
* def-
inition in terms of our concrete description of GU =OG . We shall work througho*
*ut
in terms of this alternative version.
Lemma 2.5. A C -space Y determines and is determined by the following data.
(i)An H-space Z(x) for each object x in the fiber CG=H of C .
(ii)A homotopy class of H-maps Z(fl; g) : Z(x) -! Z(y) for each morphism
fl : x -! y of C and element g of G such that "(fl)(eH) = gK, where
"(fl) : G=H -! G=K; here H acts on Z(y) by ha = (g-1hg)a for a 2 Z(y).
These data must satisfy the following properties:
(iii)In (ii), Z(fl; gk) ' k-1Z(fl; g) for k 2 K.
(iv)Z(id; e) ' idand Z(fl0; g0) O Z(fl; g) ' Z(fl0O fl; gg0) when fl0O fl is d*
*efined.
Given A : C - ! C 0, a map OE : Y -! Y 0from a C -space Y to a C 0-space Y 0
determines and is determined by H-maps i = ix : Z(x) -! Z0(Ax) for objects
x 2 CG=H such that Z0(Afl; g) O ix ' iy O Z(fl; g) for pairs (fl; g) as in (ii).
Proof.Given the specified data, set Y (x) = G xH Z(x) and let Y (fl) be the hom*
*o-
topy class of the G-map associated to the pair (Z(fl; g); g); property (iii) en*
*sures
that Y (fl) is independent of the choice of g. Conversely, given Y , let Z(x) *
*Y (x)
EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 7
be the H-space over the orbit eH and let Z(fl; g) : Z(x) -! Z(y) be the composi*
*te
of Y (fl) and multiplication by g-1. Similarly, given i, let OEx : Y (x) -! Y 0*
*A(x)
be the G-map associated to the pair (ix; e) and, conversely, given OE, let_ix_b*
*e the
restriction Z(x) -! Z0(Ax) of OEx. |__|
3.Coherent families of connected covers
A standard tool in equivariant algebraic topology is to study G-spaces by mea*
*ns
of their diagrams of fixed point spaces. On the diagram level, it is quite triv*
*ial to
give a notion of an H-connected cover. The following definition encodes that tr*
*ivial
starting point of our work.
Definition 3.1.An OG -space is a continuous contravariant functor OG - ! U ,
and a map of OG -spaces is a natural transformation. Let OG U denote the cat-
egory of OG -spaces. For a G-space X, define the fixed point OG -space X by
(X)(G=H) = XH . For a fixed point x 2 XH , let (XH ; x) denote the component
of x in XH . Define the H-connected cover of X at x to be the sub OH -space
(X; x) of the OH -space X such that (X; x)(H=J) = (XJ; x) for J H.
We can lift this essentially nonequivariant structure to the equivariant leve*
*l by
means of a construction due to Elmendorf [12]; see also [16, Vx3 and VIx6]. We
shall gradually make sense of and prove the following result in this section.
Theorem 3.2. Let X be a G-space. There is a ss(X)-space X such that, for x :
G=H -! X, X (x) is the H-connected cover of X at x. There is a natural map
of ss(X)-spaces " : X - ! X, where X is regarded as a constant ss(X)-space. For
J H, "Jx: X (x)J - ! XJ is the composite of a canonical weak equivalence
X (x)J -! (XJ; x) and the inclusion (XJ; x) -! XJ.
Here we are thinking of ss(X)-spaces in terms of the data specified in Lemma
2.5. We recall the main properties of Elmendorf's construction.
Theorem 3.3. There is a functor : OG U -! GU and a natural transformation
" : -! Idsuch that, for an OG -space T , each "(G=H) : (T )H -! T (G=H) is
a homotopy equivalence. If X is a G-CW complex, then
[X; T ]G ~=[X; T ]OG:
For an OG -space T , evaluation of " at G=e gives a G-map
"(G=e) : (T )G -! T (G=e):
When T = X, so that T (G=e) = X,
"(G=e)J = "(G=J) : (X)J -! XJ:
Thus "(G=e) : X -! X is a weak G-equivalence for any G-space X, and "(G=e)
is a G-homotopy equivalence if X is a G-CW complex.
Definition 3.4.The H-connected cover of X at x 2 XH is the H-space
X (x) = (X; x):
Thus we have homotopy equivalences "(H=J) : X (x)J -! (XJ; x) for J H.
Applying to the inclusion of OH -spaces (X; x) X and composing with
"(G=e) : X -! X, we obtain an H-map "x : X (x) -! X such that "Jxis the
composite of the homotopy equivalence "(H=J) and the inclusion (XJ; x) -! XJ.
8 J.P. MAY
It remains to discuss the functoriality and naturality of this construction, *
*which
is the crux of the matter. We recall that the functor : OG U -! GU is given by
a categorical two-sided bar construction:
T = B(T; OG ; OG):
Here OG : OG -! U is the covariant functor that sends the object G=H of OG to
the space G=H. The construction is suitably functorial in all three of its vari*
*ables.
An alternative description of T may make the functoriality clearer. Define a sm*
*all
topological G-category C (T; G) as follows. The object G-space of C (T; G) is t*
*he
disjoint union of the G-spaces T (G=H)xG=H, where G acts on the orbit factors. A
morphism ff : (t; c) -! (t0; c0) is a G-map ff : G=H -! G=H0 such that ff*(c) =*
* c0
and ff*(t0) = t, where the subscript and superscript *'s indicate the evaluatio*
*n of
covariant and contravariant functors. There is an evident topology and G-action
on the set of morphisms such that the source, target, identity, and composition
functions are continuous G-maps. Up to canonical homeomorphism of G-spaces,
T = BC (T; G):
For a homomorphism : G -! G0, a G-functor C (T; G) -! C (T 0; G0) induces a
G-map T -! T 0, where G acts on the targets by pullback along ; similarly, a
G-natural transformation induces a G-homotopy.
Let [ff; !] : x -! y be a morphism in ss(X). Let ff : G=H -! G=K be given
by ff(eH) = gK and let c(g) : H -! K be the conjugacy injection that sends h to
g-1hg. By Lemma 1.1, if we change ff in its homotopy class, then we replace g by
cg for some c in the identity component of CG H. Therefore, although c(g) depen*
*ds
on the choice of g in its coset, it does not depend on the choice of ff in its *
*homotopy
class. The homomorphism c(g) determines a functor OH - ! OK that sends H=J
to K=g-1Jg, and we also have the H-map H=J -! K=g-1Jg that sends hJ to
(g-1hg)(g-1Jg). Using the functoriality of the two-sided bar construction, there
results an H-map
X ([ff; !]; g) : X (x) -! X (y):
The properties specified in (iii) and (iv) of Lemma 2.5 are satified. Here (ii*
*i) is
not obvious since a homotopy is required, but it is easy to check that the two
maps specified there are obtained by passage to classifying spaces of categories
from naturally equivalent functors and are therefore homotopic.
Intuitively, this transport along paths ensures that our H-connected covers a*
*re
related by the evident commutative diagrams to inclusions of components of fixed
point spaces. The naturality of the construction with respect to G-maps X -! Y
is checked similarly.
4.Orientability of spherical G-fibrations
Nonequivariantly, there is only one sensible definition of an orientation of *
*a vector
bundle, but this is a calculational fact that does not extend to the equivariant
setting. The point is that
Z=2 ~=ss0(O(n)) ~=ss0(P L(n)) ~=sso(T op(n)) ~=ss0(F (n))
for all n 1, including n = 1. Nothing like this holds equivariantly. There are
(at least) eight different reasonable orientation theories for G-vector bundles*
*, cor-
responding to the linear, piecewise linear, topological, and homotopical catego*
*ries
EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 9
and their stable variants. Similarly, there are six orientation theories for P*
*L G-
bundles, four for topological G-bundles, and two for spherical G-fibrations. We
shall focus on the stable spherical G-fibration case, but the modifications for*
* the
other cases are easily imagined. A general framework is given in [5]. We begin *
*in
this section with the simpler notion of orientability. Even this depends on the*
* type
of G-bundle or G-fibration we consider. By a G-fibration, we understand a map
that satisfies the G-covering homotopy property (G-CHP).
Definitions 4.1.(i) Let OG \GU =OG be the category of G-spaces : X - !
G=H with sections oe : G=H -! X and section-preserving maps of G-spaces over
orbits. For an H-representation V , let SV be the one-point compactification *
*of
V . We have a G-fibration G xH SV -! G=H with section given by the points at
infinity. Define the category Fn of n-sphere G-fibrations to be the full subcat*
*egory
of OG \GU =OG whose objects are the G-fibrations with section that are fiber G-
homotopy equivalent to some G xH SV .
(ii) A homotopy between maps in Fn is a section-preserving homotopy; compare
Remark 2.2. The homotopy category hFn is a groupoid over the category hOG .
(iii) Define the stable homotopy category shFn of n-sphere G-fibrations over or*
*bits
to have the same objects as hFn and stable homotopy classes of maps. Then shFn
is also a groupoid over hOG , and we have a canonical map i : hFn -! shFn of
groupoids over hOG .
To control the colimits implicit in (iii), let U be the direct sum of countab*
*ly many
copies of each irreducible orthogonal representation of G; since any representa*
*tion
of H G extends to a representation of G on a possibly larger vector space, U
is also the sum of countably many copies of each irreducible representation of *
*H.
For a G-representation W , G xH SW ~=G=H x SW over G=H, and we have the
fiberwise smash products X ^ W of spherical G-fibrations of dimension n and such
trivial G-fibrations. The set of stable maps X -! Y is the colimit over W U of
the set of maps of spherical G-fibrations X ^ W -! Y ^ W .
Restricting objects and morphisms appropriately, we obtain analogous defini-
tions for vector bundles (or better, their fiberwise one-point compactification*
*s),
piecewise linear bundles, and topological bundles. We need a lemma before we can
explain what it means for a spherical G-fibration p : E -! B to be orientable.
Lemma 4.2. An n-sphere G-fibration p : E - ! B determines a functor p* :
ss(B) -! hFn over hOG . A map
f
D _____//E
q || |p|
fflffl|fflffl|
A __d__//B
of n-sphere G-fibrations determines a natural isomorphism f* : q* -! p* O f* of
functors ss(A) -! hFn over hOG . If
D x I_h___//E
qxid|| |p|
fflffl| fflffl|
A x I_j___//B
10 J.P. MAY
is a homotopy between maps of spherical G-fibrations (f0; d0) and (f1; d1), then
f*1= p*j# O f*0.
Proof.This is an exercise in pulling back spherical G-fibrations along G-maps_x*
* :
G=H -! B and using the G-covering homotopy property. |__|
Definition 4.3.A spherical G-fibration p : E -! B is orientable (in the stable
sense) if the composite of p* : ss(B) -! hFn and i : hFn -! shFn has the
property that ip*[ff; !] = ip*[ff0; !0] for every pair of morphisms [ff; !] and*
* [ff0; !0]
such that [ff] = [ff0]. Intuitively, over a given [ff], the stable homotopy cla*
*ss of the
map of fibers over orbits induced by a path between orbits is independent of the
choice of path.
In the language of Construction 1.8, orientability requires ip* to factor thr*
*ough
the universal faithful groupoid associated to ss(B). We must distinguish between
orientability in the stable sense and orientability in the unstable sense since*
* it is
possible to have ip*[ff; !] = ip*[ff0; !0] but p*[ff; !] 6= p*[ff0; !0].
The following lemma is easily verified no matter how we define orientability.
Lemma 4.4. Each G xH SV is an orientable spherical G-fibration.
Nonequivariantly, when defining orientations of bundles, we implicitly compare
fibers to Rn with its standard two orientations. This amounts to choosing a ske*
*leton
of the category of n-dimensional vector spaces. Equivariantly, we must orient t*
*he
G xH SV and use their orientations as references, and we must start by fixing a
skeleton of shFn. We have already discussed how to do this in Construction 1.8.
Definition 4.5.Let SphFn denote the discrete groupoid over hOG associated to
shFn and let ae : shFn -! SphFn be the canonical equivalence of categories.
Explicitly, for each H G, choose one homomorphism f : H -! O(n) in each
conjugacy class and let Vf = Rn with H acting through f. Choose one SV in each
stable homotopy class of such H-linear n-spheres. The objects of SphFn are the
resulting n-sphere G-fibrations G xH SV . For each n-sphere G-fibration X over
G=H, we have an isomorphism : X -! G xH SV in shFn, and these chosen
isomorphisms determine ae.
Definition 4.6.Let p : E -! B be an n-sphere G-fibration. Define p# to be the
composite of p* : ss(B) -! shFn and ae : shFn -! SphFn. We continue to write
p# for its restriction to a skeleton skss(B) of ss(B).
Now recall Notations 1.10. The following immediate observation gives a con-
ceptual characterization of orientability in terms of the relationship between *
*the
fundamental groupoid and the component groupoid of B.
Lemma 4.7. The n-sphere G-fibration p : E -! B is orientable if and only if
p# : skss(B) -! SphFn factors through the associated discrete groupoid ss0(B).
5. Coherent families of Thom classes and Thom isomorphisms
Let E be a commutative ring G-spectrum, in the classical homotopical sense:
we have a unit map S -! E and a product E ^ E -! E satisfying the usual
unity, associativity, and commutativity diagrams in the stable homotopy category
of G-spectra of [15]; see also [16, XIIIx5]. We are interested in the RO(G)-gra*
*ded
cohomology theory E*Grepresented by E. Evaluated on G-spaces, we understand
EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 11
the unreduced theory, writing "E*Gfor the reduced theory on based G-spaces. We
shall make use of the precise treatment of RO(G)-grading given in [16, XIIIxx1,*
*2].
We may regard E as an H-ring spectrum for any H G, and we write E*Hfor the
theory on H-spaces represented by E. For an H-space Y and G-representation ae,
EaeG(G xH Y ) ~=EaeH(Y ):
We begin with a generalization of the notion of a cohomology class of a G-spa*
*ce.
Definition 5.1.Let C be a groupoid over hOG , let q : C -! SphFn be a map
of groupoids over hOG , and let Y : C -! GU =hOG be a C -space. For an object
x 2 CG=H , write q(x) = G xH SV (x), and describe Y as in Lemma 2.5 in terms of*
* a
system of H-spaces Z(x). An E*G-cohomology class indexed on q of the C -space
Y consists of an element (x) 2 EVH(x)(Z(x)) for each object x 2 CG=H . The (x)
are required to be compatible under restriction in the sense that
Z(fl; g)*((y)) = (x);
where fl : x ! y is a morphism of C and g is an element of G such that "(fl)(eH*
*) =
gK; compare Lemma 2.5(ii). Here
Z(fl; g)* : EVK(y)(Z(y)) -! EVH(x)(Z(x))
is the composite of restriction
EVK(y)(Z(y)) -! EVH(y)(Z(y))
along c(g) : H -! K and the map
EVH(y)(Z(y)) -! EVH(x)(Z(x))
induced by the H-map Z(fl; g) : Z(x) -! Z(y) and the inverse of the stable
H-equivalence f" : SV (x)-! SV (y)such that (f"; g) determines the stable G-
equivalence q(fl) : G xH SV (x)-! G xK SV (y); compare Remark 2.2.
The simultaneous functoriality in the grading and the space that we have used
is explained in [16, XIIIxx1,2]. Essentially, this is just an exercise in the u*
*se of the
suspension isomorphism in RO(G)-graded cohomology.
We shall apply this definition with C = ss(B), taking q to be the functor p# :
ss(B) -! SphFn associated to an n-sphere G-fibration p. The relevant ss(B)-space
is the Thom ss(B)-space T(p) given by the following definition.
Definition 5.2.Let p : E -! B be an n-sphere G-fibration. Define the (based)
Thom G-space T p to be the quotient space E=oeB. For example, if we start with
a G-vector bundle , then its Thom space is obtained from the fiberwise one-point
compactification of by identifying all of the points at infinity. We have the *
*map
of ss(B)-spaces ": B -! B of Theorem 3.2. Define the Thom ss(B)-space T(p) by
letting T(p)(x) = T (p(x)), x 2 BH , be the Thom H-space of the pullback p(x) of
p along ": B(x) -! B.
The point of the definition is that the H-space B (x) is H-connected and, as
we now recall, orientation theory for n-sphere G-fibrations over G-connected ba*
*se
spaces is well understood.
We first define orientations of spherical G-fibrations over orbits, then defi*
*ne
orientations of spherical G-fibrations over G-connected base spaces, and finally
give our new definition of orientations of general spherical G-fibrations.
12 J.P. MAY
Definition 5.3.The Thom G-space of : G xH SV -! G=H is G+ ^H SV , and
E"VG(G+ ^H SV ) ~="EVH(SV ) ~=E0H(pt):
An E*G-orientation, or Thom class, of is an element 2 "EVG(G+ ^H SV ) that
maps under this isomorphism to a unit of the ring E0H(pt).
Definition 5.4.Let p : E -! B be an n-sphere G-fibration over a G-connected
base space B. For any x 2 BG , p-1(x) is a based G-space of the homotopy type
of SV for some n-dimensional representation V of G, and V is independent of the
choice of x. Moreover, for all x : G=H -! B, the pullback of p along x is fiber
G-homotopy equivalent to G xH SV . An E*G-orientation, or Thom class, of p is
an element 2 "EVG(T p) that pulls back to an orientation along each orbit incl*
*usion
x.
Definition 5.5.Let p : E -! B be an n-sphere G-fibration. An E*G-orientation,
or Thom class, of p is an E*G-cohomology class indexed on p# : ss(B) -! SphFn
of the Thom ss(B)-space T(p) such that, for each x 2 BH , (x) 2 EVH(x)(T (x)) is
an orientation of the pullback p(x) of p along " : B (x) -! B. We say that p is
E*G-orientable if it has an E*G-orientation.
Here, for x 2 BH , V (x) is the fiber H-representation at x, so that SV (x)is*
* stably
G-homotopy equivalent to p-1(x), the equivalence being fixed by the specificati*
*on
of p# . Observe that the equivalence fixes a stable H-map
(5.6) i(x) : SV ' p-1(x) -! T(p)(x):
The following observation should help clarify the force of the compatibility co*
*ndition
required of our orientations on H-connected covers.
Lemma 5.7. Let p be an n-sphere G-fibration. The following diagram commutes
for a morphism [ff; !] : x -! y in ss(B) with ff(eH) = gK:
i(y)* V (y) 0
EVK(y)(T (p)(y))___//"EK(SV (y))~= EK (pt)
T (p)([ff;!];g)*|| p#([ff;!];g)*||c(g)||
fflffl| fflffl| fflffl|
EVH(x)(T (p)(x))i(//x)*_"EVH(x)(SV~(x))=E0H(pt):
Proof.The map p# ([ff; !]; g)* is defined exactly as was T(p)([ff; !]; g)* in D*
*efinition
5.1, and the left square is a naturality diagram. The right square commutes by_a
direct unravelling of definitions. |__|
Remark 5.8.If the horizontal arrows are isomophisms, then the left vertical arr*
*ow
is determined by the right vertical arrow and the compatibility reduces to a qu*
*estion
of compatible units in the rings comprising the Mackey functor E0 with E0(G=H) =
E0H(pt). As we shall see in the next section, this is exactly what happens when*
* p is
orientable and we specialize to ordinary cohomology with Burnside ring coeffici*
*ents.
Compatible Thom isomorphisms follow immediately from [15, Xx5], where a
generalization of the following theorem is proven.
EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 13
Theorem 5.9. Let p : E -! B be an n-sphere G-fibration over a G-connected
base space B and let 2 "EVG(T (p)) be a Thom class. Then cupping with defines
a Thom isomorphism
= (p) : Eae(B) -! "Eae+VG(T (p))
for all ae 2 RO(G).
Again, we refer to [16, XIIIxx1,2] for precision about the grading.
Theorem 5.10. Let p : E -! B be an n-sphere G-fibration and let {(x)} be a
Thom class of p. Then the (x), x 2 BH , give rise to Thom isomorphisms
(p(x)) : E*H(B (x)) -! E*+VH(x)(T (p)(x));
where the H-space SV (x)is stably equivalent to p-1(x). Moreover, the following
diagrams are commutative for [ff; !] : x -! y, where ff(eH) = gK and ae 2 RO(K):
(p(y)) ae+V (y)
EaeK(B (y))____//"EK (T (p)(y))
B ([ff;!];g)*|| |T(p)([ff;!];g)*|
fflffl| fflffl|
EaeH(B (x))(p(//x))_"Eae+VH(x)(T (p)(x)):
Here the vertical arrows are as specified in Definition 5.1.
6. Orientations in ordinary equivariant cohomology
We have formalized the intuitive geometrical notion of orientability in Defin*
*ition
4.3 and have expressed this notion categorically in Lemma 4.7. It is natural to*
* hope
that this notion coincides with the notion of orientability with respect to a s*
*uitable
cohomology theory.
Nonequivariantly, the relevant theory is integral cohomology. The real reason
this works is that orientability is a stable notion and Z coincides with the ze*
*roth
stable homotopy group of spheres. Equivariantly, the analogue of Z is the Burns*
*ide
ring A(G), which is the zeroth equivariant stable homotopy group of spheres. As
was first explained by Bredon [3], ordinary equivariant cohomology theories are
indexed on coefficient systems, namely contravariant functors M : hOG - ! Ab ,
where Ab denotes the category of abelian groups. We have the Burnside ring
coefficient system A such that A(G=H) = A(H). As was proven in [14], the
ordinary cohomology theory indexed on M extends to an RO(G)-graded theory
if and only if the coefficient system M extends to a Mackey functor. See [15,
Vx9] or [16, IXx4] for a discussion of Mackey functors in the context of compact
Lie groups. The Burnside ring coefficient system does so extend, hence we have
the ordinary RO(G)-graded cohomology theory H*G(-; A). It is represented by an
Eilenberg-Mac Lane G-spectrum HA [16, XIIIx4], and HA is a commutative ring
G-spectrum.
We abbreviate HA*G-orientability to A-orientability and H*G(X; A) to H*G(X).
We proceed to relate orientability to A-orientability, beginning with the case *
*of
G-fibrations over G-connected base spaces.
Theorem 6.1. Let p : E -! B be an n-sphere G-fibration, where B is G-connected.
Let x 2 BG , let V be the fiber G-representation at x, and consider the map
i : SV ' p-1(x) T (p). The following statements are equivalent.
14 J.P. MAY
(i)p is orientable.
(ii)p is A-orientable.
(iii)i* : "HVG(T (p)) -! "HVG(SV ) ~=A(G) is an isomorphism.
Proof.By G-CW approximation, we may assume without loss of generality that
B is a G-CW complex with a single G-fixed base vertex x. Let Bq be the q-
skeleton of B, let Eq = p-1(Bq), and let pq be the restriction of p to Eq. Let
Cq = T (pq)=T (pq-1). Observe that SV ' T (p0). If c : G=Hc x Dq -! Bq is the
characteristic map of a q-cell of B, then the pullback of p along c is trivial *
*and is
thus equivalent to G=Hcx Dq x SV . Moreover, the equivalence is determined by a
choice of path connecting x to c(e; 0). These equivalences determine an equival*
*ence
between the wedge over all q-cells c of the G-spaces (G=Hc)+ ^ Sq ^ SV and the
quotient G-space Cq.
Consider cohomology in degrees V + i, where i is an integer. We have
H"VG+i((G=H)+ ^ Sq ^ SV ) ~=Hi-qH(pt):
This is zero unless i q and it is A(H) when i = q, by the dimension axiom. We
conclude by long exact sequences and lim1exact sequences that
"HV -i(T (pq)) = "HV -i(T (p)) = 0
for i 1 and there is an exact sequence
* ffi
0 -! HVG(T p)-i!HVG(SV )-! HVG+1(C1):
Q
Here ffi may be viewed as a map A(G) -! A(Hc) of A(G)-modules, where the
product runs over the 1-cells c. A 1-cell c is specified by a loop at x in BHc.*
* The
component of ffi in A(Hc) can be interpreted geometrically as the difference be*
*tween
the identity map of SV and the stable H-equivalence of SV obtained by the action
of this loop on SV . The three statements of the theorem are each equivalent_to*
* the
assertion that ffi = 0. |__|
Observe the relevance of our definition of orientability in the stable sense.*
* The
conclusion would fail if we defined orientability in the unstable sense.
Before generalizing this result, we recall a standard fact about conjugation
homomorphisms between Burnside rings. Let ff : G=H - ! G=K be given by
ff(eH) = gK and consider c(g) : H - ! K. Since c(k)* : A(K) -! A(K) is
the identity for k 2 K, by inspection of the the standard inclusion of A(K) into
a product of copies of Z (e.g. [15, Vx2]), we see that c(g)* : A(K) -! A(H) is
independent of the choice of g in its coset gK. It is also independent of the c*
*hoice
of ff in its homotopy class, by Lemma 1.1. We write c(g)* = c(ff)*.
Theorem 6.2. Let p be an n-sphere G-fibration. The following statements are
equivalent.
(i)p is orientable.
(ii)Each p(x) is orientable.
(iii)Each p(x) is A-orientable.
(iv)p is A-orientable.
Moreover, an HA-orientation of p is specified by a collection of units (x) 2 A*
*(H)
for points x 2 BH of the discrete groupoid ss0(B) that satisfy the compatibili*
*ty
condition c(ff)*((y)) = (x) for a map fl : x -! y of ss0(B) with "(fl) = ff.
Equivalently, is specified by an automorphism of the functor p# : skss(B) -!
SphFn over hOG .
EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 15
Proof.Since the notion of orientability of p depends only on the behavior of the
pullbacks of p along paths and paths lie in connected components, the equivalen*
*ce of
(i) and (ii) is immediate from the properties of H-connected covers given in Th*
*eorem
3.2. The equivalence of (ii) and (iii) is part of the previous theorem, and it *
*is trivial
that (iv) implies (iii), by consideration of pullbacks. Thus assume (iii) and c*
*onsider
the diagram of Lemma 5.7 with E = HA. Its horizontal arrows are isomorphisms
by the previous theorem, and Remark 5.8 applies to give the specified descripti*
*on
of an A-orientation in terms of units of Burnside rings. In particular, we may *
*take
(x) to be the identity element for all x, and this shows that p is A-orientable.
Finally, the group of automorphisms of an object GxH SV of SphFn is canonically
isomorphic to the group of stable H-equivalences of SV and thus to a copy of the
group of units of the Burnside ring A(H). For our functor p# : skss(B) -! SphFn,
the compatibility condition on units required of an A-orientation can be interp*
*reted_
as the naturality condition required of an automorphism of functors. |*
*__|
Let A-Or(p) denote the set of A-orientations of an orientable n-sphere G-
fibration p.
Corollary 6.3.By multiplication of units or, equivalently, by composition of au-
tomorphisms of the functor p# : skss(B) -! SphFn over hOG , A-Or(p) acquires a
structure of commutative group.
Nonequivariantly, there are both topological and cohomological notions of an
orientation, and these notions coincide. Equivariantly, we have explained a coh*
*o-
mological notion of an orientation. There is also a topological notion, defined*
* in [5].
However, these two notions do not coincide. To explain this, we sketch the defi*
*nition
given in [5]. Working in the category of groupoids over hOG , consider maps into
SphFn. In [5], we construct and characterize a particular map ae : Cn -! SphFn
such that Cn is faithful over hOG and any map from a faithful groupoid over hOG
into SphFn factors up to isomorphism through at least one map into Cn; this is a
weak universal property of ae, which, intuitively, is a kind of universal orien*
*tation.
Fix an orientable n-sphere G-fibration p : E -! B for the rest of the section*
*. The
functor p# : skss(B) -! SphFn factors through the discrete groupoid ss0(B), and
we now agree to write p# for the resulting functor defined on ss0(B). The topol*
*ogical
notion of an orientation is a pair (i; j) consisting of a functor i : ss0(B) -!*
* Cn over
hOG together with a natural isomorphism j : p# -! ae O i. Let Or(p) denote the
set of such orientations of p. Precomposing with automorphisms of p# for fixed *
*i,
we obtain a free right action of A-Or(p) on Or(p). Call the orbit set Or(p)=A; *
*it
can be identified with the set of those functors i : ss0(B) -! Cn that can be p*
*art
of an orientation (i; j). Let F (p) be the set of all functors i : ss0(B) -! Cn*
* over
hOG such that p# and ae O i agree on objects. In general, not all such functors*
* are
components of orientations, and we have an inclusion ff : Or(p)=A -! F (p).
Let Ax : hOG -! Ab be the contravariant functor that sends G=H to the group
of units of A(H) and continue to write Ax for its composite with " : C -! hOG
for any groupoid C over hOG . By analyzing the obstruction to the construction
of j such that (i; j) is an orientation, one arrives at the following propositi*
*on. We
omit the proof, as it is not very illuminating. The essential ingredients are *
*the
cited weak universal property of ae and the fact that SphFn is a uniquely divis*
*ible
groupoid over hOG .
16 J.P. MAY
Proposition 6.4.There is an exact sequence of pointed sets
* -! Or(p)=A-ff!F (p)-fi!H1(ss0(B); Ax ) -! *:
Thus H1(ss0(B); Ax ) measures the difference between topological and cohomo-
logical orientations: if fi is a bijection, the notions are equivalent.
7. Concluding remarks
Whenever one has cohomological orientations of a class of G-vector bundles
that are sufficiently natural in G, one will have cohomological orientations in*
* the
sense that we have defined. Since this paper was written around the deadline for
submissions to this volume, I have not had time to check details of the followi*
*ng two
examples, but they are surely correct. Here it makes sense to use the variant o*
*f the
theory appropriate to unstable G-vector bundles rather than to stable G-fibrati*
*ons.
Clearly orientations in the former sense give rise to orientations in the latte*
*r sense.
The methods of [2] should give the following result; compare [1].
Example 7.1. Complex G-vector bundles admit canonical KU*G-orientations. Real
G-vector bundles with Spin structures and dimension divisible by eight admit
canonical KO*G-orientations.
Tautological orientations should give the following result.
Example 7.2. Complex G-vector bundles admit canonical MU*G-orientations. Real
G-vector bundles admit canonical MO*G-orientations.
At the most structured extreme, as in the nonequivariant case, we have the
following observation.
Example 7.3. A spherical G-fibration is ss*G-orientable if and only if its pull*
*backs
to H-connected covers are stably fiber homotopy trivial with suitably compatible
trivializations.
To obtain a Poincare duality theorem along the present lines, one would have
to prove an Atiyah duality theorem for the H-connected covers of smooth compact
G-manifolds M. That is, if M embeds in V with normal bundle , one might hope
that the H-spaces T()(x) and M (x) are V -dual for x 2 MH . Although M (x)
is infinite dimensional, one has complete homotopical control on its fixed point
spaces, which are homotopy equivalent to smooth manifolds. I have not explored
this question.
References
[1]M.F. Atiyah. Bott periodicity and the index of elliptic operators. Quart. J*
*. Math. 19(1968),
113-140.
[2]M.F. Atiah, R. Bott, and A. Shapiro. Clifford modules. Topology 3(1964), Su*
*pplement 1,
3-38.
[3]G. Bredon. Equivariant cohomology theories. Lecture Notes in Mathematics Vo*
*l 34. Springer-
Verlag, 1967.
[4]P.E. Conner and E.E. Floyd. Differentiable periodic maps. Ergebnisse der Ma*
*th. N. S. 33.
Academic Press. 1964.
[5]S.R. Costenoble, J.P. May, and S. Waner. Equivariant orientation theory. Un*
*distributed
preprint, 1989.
[6]S.R. Costenoble and S. Waner. Equivariant orientations and G-bordism theory*
*. Pacific J.
Math. 140(1989), 63-84.
EQUIVARIANT ORIENTATIONS AND THOM ISOMORPHISMS 17
[7]S.R. Costenoble and S. Waner. The equivariant Thom isomorphism. Pacific J. *
*Math.
152(1992), 21-39.
[8]S.R. Costenoble and S. Waner. Equivariant Poincare duality. Michigan Math. *
*J. 39(1992),
325-351.
[9]S.R. Costenoble and S. Waner. The equivariant Spivak normal bundle and equi*
*variant surgery.
Michigan Math. J. 39(1992), 415-424.
[10]S.R. Costenoble and S. Waner. The equivariant Spivak normal bundle and equi*
*variant surgery
for compact Lie groups. Preprint, 1996.
[11]T. tom Dieck. Transformation groups. Studies in Mathematics Vol. 8. Walter *
*de Gruyter.
1987.
[12]A. Elmendorf. Systems of fixed point sets. Trans. Amer. Math. Soc. 277(1983*
*), 275-284.
[13]A. Grothendieck. R^evetementsetale et groupe fondemental. Lecture Notes in *
*Mathematics
Vol. 224. Springer-Verlag. 1971.
[14]L.G. Lewis, J.P. May, and J.E. McClure. Ordinary RO(G)-graded cohomology. B*
*ull. Amer.
Math. Soc. 4 (1981), 128-130.
[15]L.G. Lewis, J.P. May, and M. Steinberger (with contributions by J. E. McClu*
*re). Equivariant
stable homotopy theory. Lecture Notes in Mathematics Vol. 1213. Springer-Ver*
*lag, 1986.
[16]J. P. May, et al. Equivariant homotopy and cohomology theory. CBMS Regional*
* Conference
Series in Mathematics, Number 91. American Mathematical Society. 1996.
[17]J.P. May. G-spaces and fundamental groupoids. Appendix to "An equivariant N*
*ovikov con-
jecture" by J. Rosenberg and S. Weinberger. Journal of K-theory 4 (1990), 50*
*-53.
[18]M. Rothenberg and J. Sondow. Nonlinear smooth representations of compact Li*
*e groups.
Pacific J. Math. 84(1979), 427-444.