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The dualizing spectrum of a topological group. (English) Zbl 0982.55004

In generalizing the notion of Poincaré duality for a discrete group \(\Gamma\), B. Eckmann and R. Bieri introduced the dualizing module \(D_\Gamma:= \hom_{D(\mathbb{Z}[\Gamma])} (\mathbb{Z}, \mathbb{Z}[\Gamma])\), where hom is taken internally within the derived category of left \(\mathbb{Z}[\Gamma]\)-modules. This construction can be generalized to the dualizing spectrum \(D_G= \hom_{D(S^0[G])}(S^0, S^0[G])\) for the topological group \(G\). The results of this paper show that \(D_G\) is an extremely elegant tool both for proving new theorems and reproving old. Among the most striking are Theorem A showing that the (finitely dominated) classifying space \(BG\) is a Poincaré duality space iff \(D_G\) has the weak homotopy type of a sphere iff \(D_G\) is homotopy finite, and Theorem B showing that \(D_G\) behaves multiplicatively for certain kinds of extension. Associated with \(D_G\) there is also a “norm map” relating invariants to coinvariants. This definition is also motivated by more classical group cohomology. Among the applications are a new proof of the conjecture of C. T. C. Wall that the total space \(E\) of a fibration is a Poincaré space iff the fibre \(F\) and base \(B\) are, and of W. Browder’s theorem that a connected \(H\)-space is a \(PD\)-space. (Both results assume finite domination.) The proofs are all homotopy theoretic. In a particularly interesting final section, the author describes \(D_G\) explicitly for a large number of examples.

MSC:

55P91 Equivariant homotopy theory in algebraic topology
20J05 Homological methods in group theory
55N91 Equivariant homology and cohomology in algebraic topology
55P42 Stable homotopy theory, spectra
57P10 Poincaré duality spaces
55P25 Spanier-Whitehead duality
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
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