4
A. ADEM, R. L. COHEN, W. G. DWYER
Let
g
be the Lie algebra of G and ad the vector bundle over B associated to
the adjoint action of
G
on
g.
We will denote the total space
X
x
a
g
of ad by
E.
Let
tr' :
Ex V
-+
E-+ B be the composite of ad with the product projection. By
using the zero-section of ad it is possible to lift the map p : X
-+
B to a map p' :
X
-+
E. This induces an embedding
p'
x e : X
-+
Ex
V
with normal bundle, say,
v'. It is immediate that the Thorn space B"' is Ev Bad, and it is well-known (see
(BS])
that the corresponding Thorn space
xv'
is equivariantly homeomorphic to
Ev (X+)· The umkehr map we are seeking is the Thom-Pontryagin collapse map
Ev Bad
-+
Ev (X+)· One observes that this collapse map is equivariant, The
adjoint map Bad
-+
nvEv(X+) consequently factors through a map Bad
-+
(OvEv (X+
))G
which stabilizes to
The map
t'X
may be composed with the inclusion of the fixed point set in the
homotopy fixed point set to give a map
The natural map Q(X+)-+ Qa(X+) is a weak G-equivalence, in other words,
it is an equivariant map that is a (nonequivariant) homotopy equivalence. Such
weak G-equivalences become equivalences when the homotopy fixed point functor
is applied.
DEFINITION
2.1:
If
X is a finite, free G--CW complex, the transfer map tx
(X/G)ad-+ (Q(X+))hG is the composite
where the first map is
t~
and the second map is the inverse of the above natural
homotopy equivalence of homotopy fixed point sets.
We now need to extend this definition. Suppose first that
X
is a free
G-
CW complex which is not necessarily finite. In this case one restricts at first
to the finite G-subcomplexes
I
of X, constructs maps
tK
as above, and then
checks that these transfers fit together and extend to a transfer map tx. This
procedure is standard and we leave its verification to the reader (see (Cl] or
(LMS] for details). If
X
is a G-space which is not a G-CW complex, we will
tacitly assume that
X
has been replaced by a G--CW approximation (LMS],
eg. by the equivariant version of the realization of its singular complex. If
X
is a G-space which is not free, we can make the above construction with the
equivalent free G-space EG x X. The upshot of this is to obtain for any G-space
X
a transfer map
Suppose now that E is an infinite loop space with an action of G, that is,
an infinite loop space (M2] together with an action of G that respects all of
the infinite loop structure. In this case the infinite loop structure gives a map
Q(E+)
--+
E which is G-equivariant and induces a map (Q(E+))hG
--+
EhG;
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