John R. Klein
Wayne State University, Detroit, MI 48202 USA klein@math.wayne.edu
Let 
be the geometric realization of a simplicial
group. One then has the ``group ring over the sphere
spectrum'' 
, which is the spectrum with
G-action whose j-th space is 
,
where G acts by left translation.
The dualizing spectrum 
is
obtained by taking the homotopy fixed points
of G acting on .
If X is a connected based space, we may identify X with BG
up to homotopy for some G (using a suitable group model for
the loop space of X). In this way, we can associate a to
any given connected based space 
. One reason for considering
is that it detects Poincar'e duality:
Fact. Assume X is homotopy finite. Then
X is a Poincar'e duality space of dimension n if
and only if is the (-n)-sphere spectrum.
This can be applied to show that the assignment 
(with 
) is multiplicative for fibrations of
connected homotopy
finite spaces. As a corollary, one has a homotopy theoretic
solution to a problem posed by C.T. C. Wall:
This last statement can be used in turn to give a new proof (actually a slight strengthening) of a result of W. Browder:
Other applications.
This circle of ideas has connections
with the transfer and Tate cohomology:
right translation together with inversion
provides another G-action on which induces a G-action
on . If E is any spectrum with G-action, then
it is possible to concoct in an elementary way a ``norm'' map
for any G. This map is a weak equivalence
whenever E is a homotopy finite spectrum with
G-action, or whenever BG is homotopy finite. It
is not a weak equivalence in general.
If is such that 
is the total singular complex of a compact Lie group, then
it turns out that is equivariantly weak equivalent
to the suspension spectrum of 
the one
point compactification of the adjoint representation of G.
In this case, one obtains the usual norm map whose cofiber
is the Tate construction for G acting on E.