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has been replaced by a G--CW approximation (LMS],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;

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;