Title: The dualizing spectrum of a topological group
Author: John R. Klein
AMS subjclass: Primary: 55P91, 55N91, 55P42, 57P10.
Secondary: 55P25, 20J05,18G15.
Address: Dept. Of Mathematics, Wayne State University, Detroit, MI 48202
e-mail: klein@math.wayne.edu
Abstract: To a topological group G, we assign a
naive G-spectrum D_G, called the "dualizing spectrum" of G.
When the classifying space BG is finitely dominated, we show
that D_G detects Poincare duality in the sense that
BG is a Poincare duality space if and only if D_G is a
homotopy finite spectrum. Secondly, we show that
the dualizing spectrum behaves multiplicatively on certain topological
group extensions. In proving these results we introduce a new tool:
a "norm map" which is defined for any G and for any naive G-spectrum E.
Applications include:
(1) a homotopy theoretic solution to a problem posed by Wall which says that
in a fibration sequence of finitely dominated spaces, the total space
satisfies Poincare duality if and only if the base and fiber do.
(2) An entirely homotopy theoretic construction of the Spivak fibration
of a finitely dominated Poincare duality space.
(3) A new proof of Browder's theorem that every finite H-space satisfies
Poincare duality.
(4) We show how to define a variant of Farrell-Tate cohomology for any
topological or discrete group G, with coefficients
in any naive equivariant cohomology theory E. We prove a vanishing result
for this theory.
In an appendix, we identify the homotopy type of D_G for certain
kinds of groups. The class includes all compact Lie groups, torsion free
arithmetic groups and Bieri-Eckmann duality groups.
(This paper has already been accepted for publication in Math. Annalen.)