EQUIVARIANT ALGEBRAIC K-THEORY
by Z. Fiedorowicz, H. Hauschild, and J.P. May
There are many ways that group actions enter into algebraic K-theory and there
are various theories that fit under the rubric of our title. To anyone familiar with
both equivariant topological K-theory and Quillen's original definition and calcula-
tions of algebraic K-theory, there is a perfectly obvious program for the definition
of the equivariant algebraic K-theory of rings and its calculation for finite
fields. While this program surely must have occurred to others, there are no pub-
lished accounts and the technical details have not been worked out before. That
part of the program which pertains to the complex Adams conjecture was outlined in a
letter to one of us from Graeme Segal, and the real analog was assumed without proof
by tom Dieck [10,11.3.8]. Negatively indexed equivariant K-groups were introduced
by Loday [19].
From a topological point of view, one way of thinking about Quillen's original
definition runs as follows. Let ~ be a topological group, perhaps discrete. One
has a notion of a principal K-bundle and a classifying space B~ for such bundles.
When H is discrete, a principal ~-bundle is just a covering (possibly with dis-
connected total space) with fibre and group ~. Given any increasing sequence of
groups N with union H, we obtain an increasing sequence of classifying spaces
n
BH with union B~. We then think of B~ as a classifying space for stable
n
bundles. When the ~ are discrete, B~ may have desirable homology groups but
n
will have trivial higher homotopy groups. In the cases of interest, we can use the
plus construction to convert B~ to a Hopf space (= H-space) (BN) + with the same
homology. When ~n = GL(n,A), we define ~q(B~) + = Kq(A) for q > 0.
There is a more structured way of looking at (B~) +. In practice, we have sum
maps {9 :~ x ~ ÷ ~m+n and a corresponding Whitney sum of bundles. While this can
m n
he used to give B~ a product, it is generally not a Hopf space, although it is so
24
in the classical bundle theory cases obtained from ~ = U(n) and ~ = O(n).
n n
When ~ = ~ ~ is a permutative category under @ , B~ = II B~ inherits a
n~0 n n~0 n
structure of topological monoid and we can form ~BBQ, the loop space on the class-
ifying space of B~ . Homological analysis of the natural map ~ :B~ ÷ ~BB~
shows that the basepoint component ~0BB~ is equivalent to (B~) +. Deeper
analysis shows that BB~ also has a classifying space, and so on, so that ~BB~
and thus (BH) + are actually infinite loop spaces.
Let G be a finite group. We mimic the outline just given. For a topological
group ~, one has a notion of a principal (G,~)-bundle p:E ÷ B. This is just a
principal ~-bundle and a G-map between G-spaces such that the action by each g c G
is a map of ~-bundles. That is, the actions of G and H on the total space commute
(we think of ~ as acting on the right and G on the left). One has a classifying G-
space B(G,~) for such (G,~)-bundles. This space carries information about the repre-
sentations of G in ~. In particular, one has the following basic fact.
Proposition 0. I. Let H be a subgroup of G. For a homomorphism p :H + ~, define
~O to be the centralizer of 0, namely
{~Io(h)~ = ~p(h) for all h c H}.
Then the fixed point subspace B(G,~) H has the homotopy type of Jl B~ O ' where the
m
union runs over a set R+(H,~) of representatives for the representations of H in ~.
That is, R+(H,~) consists of one p in each conjugacy class of homomorphisms
H ÷ ~. Note that, with H = e, this says that the underlying nonequivariant homo-
topy type of B(G,~) is just B~.
Again, we are interested in increasing sequences {~n } with union ~. While
the third author has conbtructed an equivariant plus construction for G-connected G-
spaces X, namely those X for which all X H are path connected, we shall not use it.
Instead, we shall prove that, in the cases of interest to us, J~n B(G,~n ) is G-
equivalent to a topological G-monoid B ~(G). This means that B~ (G) is a G-space
and a topological monoid such that its unit is a fixed point and its product is a G-
28
map. Since the standard classifying space functor takes G-monoids to G-spaces and
the loop space functor takes G-spaces to G-spaces, this will allow us to construct
the G-space ~BB~(G) together with a natural G-map
~:B ~(G)
+
flBB~ (G).
These ideas lead us to the following definition.
Definition 0.2. For a discrete ring A, let
K(A,G) = ~BB~(A,G)
be the G-space obtained by setting ~n = GL(n,A) in the discussion above.
algebraically closed field A of characteristic unequal to 2, let
KO(A,G) = ~BBO(A,G)
be the G-space obtained by setting ~n = O(n,A). Let
K(G) = ~BB~(G) and KO(G) = ~BB O(G)
be the G-spaces obtained by setting ~n = U(n) and Rn = O(n).
For an
The appropriate definition of KO(A,G) for general commutative rings A requires
use of more general orthogonal groups and will be given in section 2.
We shall give the details behind this definition in sections 1 and 2, first
giving precise models for the general classifying spaces B(G,R) and verifying
Proposition 0. I and then giving different models for the particular B(G,R n) relevant
to the definition. The second model is needed to obtain the required monoid struc-
tures since the first model is not product-preserving in H. The most important tool
in equivariant homotopy theory is the reduction of equivariant problems to non-
equivariant ones by passage to fixed point spaces, and we study the fixed points of
the simplest examples of the G-spaces introduced in Definition 0.2 in section 3.
For based G-spaces X and Y (with G-fixed basepoints), let IX,Y] G denote the set
of G-homotopy classes of based G-maps X + Y. In particular, for H C G, it is stan-
dard and natural to define
26
H(y) = [(G/H)+ A sq,Y] G = [sq,Y H] = ~ (yH),
q q
where S q is the q-sphere with trivial G-action. Here and henceforward, X+ denotes
the union of a G-space X and a disjoint G-fixed basepoint. A G-map f:X + Y is said
to be a weak G-equivalence if each fH:xH ÷ yH is a weak equivalence. If X and Y
have the homotopy types of G-CW complexes, as holds for all G-spaces we shall
consider, such an f is necessarily a G-homotopy equivalence [7,53]. It is also
natural to consider the "homotopy groups"
~(Y) = [sV,Y]G ,
where S v is the one-point compactification of a real representation V of G.
As we shall see in section 5, equivariant topological K-theory of G-bundles
over compact G-spaces X is represented in the form
KG(X) = [X+,K(G)]G and KOG(X) = [X+,KO(G)]G,
hence the homotopy groups above are all examples of (reduced) K-groups when Y = K(G)
or Y = KO(G). We regard the corresponding invariants of K(A,G) and K0(A,G) as
equivariant algebraic K-groups. We write
KGA
= ~GK(A,G) and
q q
and similarly in the orthogonal case.
KGA = ~GK(A,G)
v v
However, we are really more interested in the
G-homotopy types K(A;G) than in these invariants. While the general linear case is
the central one algebraically, the orthogonal case is important in applications from
algebra to topology. We develop formal properties of these definitions in section
G
4. In particular, we discuss the naturality in A of K(A,G), prove that K,(A) is a
commutative graded ring if A is commutative, and prove the projection formula. We
also verify that [X,K(A,G)] G is naturally a module over the Burnside ring A(G).
Algebraically, the obvious next step is to introduce the equivariant Q-
construction on exact categories, give the equivariant version of Quillen's second
definition of algebraic K-theory, and prove the equivalence of the two notions.
This can all be done, and an exposition will appear in Benioff [5]. With this
27
approach, Quillen's devissage theorem applies directly to the computation of fixed
point categories and thus of equivariant algebraic K-groups.
Another obvious step is to introduce the appropriate notion of a permutative G-
category and prove that K(A,G) is an infinite loop G-space. We have carried out
this step and will present it in [12]. We prefer to be elementary in this paper and
so will only make a few remarks about this in passing.
We shall concentrate here on another obvious step, namely the equivariant
analogs of Quillen's basic calculations in [28] and [29]. We shall first prove the
following result.
Theorem 0.3. Let ~ be the algebraic closure of the field of q elements, where q
q
is a prime which does not divide the order of G. Then there are Brauer lift G-maps
B:K(~q,G) ÷ K(G) and B:KO(~q,G) ÷ KO(G)
whose fixed point maps B H induce isomorphisms on mod n cohomology for all integers n
prime to q.
The outline of the proof is obvious. We simply use our study of classifying
spaces to write down an explicit map and check that on fixed point sets it reduces
to a product of maps of the form studied by Quillen. We give the argument in
section 6 after first developing general facts about the relationship between
representation rings and topological equivariant K-theory in section 5.
As we discuss briefly in section 7, this result and the equivariant Dold
theorem mod k of Hauschild and Waner [13] can be used to prove the following
equivariant version of the Adams conjecture.
Theorem 0.4. Let k be prime to the order of G and let s be minimal such that
k s E ~i modulo the order of G. Then for any stable real G-vector bundle $ over a
compact G-connected base space, there exists an integer e > 0 such that kes~ and
kes~k(~) are stably fibre G-homotopy equivalent.