Let X be a finite CW complex, and let DX be its dual in the category of spectra. We demonstrate that the Poincaré/Koszul duality between T HH(DX) and the free loop space Σ ∞ + LX is in fact a genuinely S 1 -equivariant duality that preserves the Cn-fixed points. Our proof uses an elementary but surprisingly useful rigidity theorem for the geometric fixed point functor Φ G of orthogonal G-spectra.
Abstract. For any perfect fibration E −→ B, there is a "free loop transfer map" LB + −→ LE + , defined using topological Hochschild homology. We prove that this transfer is compatible with the Becker-Gottlieb transfer, allowing us to extend a result of Dorabia la and Johnson on the transfer map in Waldhausen's A-theory. In the case where E −→ B is a smooth fiber bundle, we also give a concrete geometric model for the free loop transfer in terms of Pontryagin-Thom collapse maps. We recover the previously known computations of the free loop transfer due to Schlichtkrull, and make a few new computations as well.
The topological Hochschild homology THH(A) of an orthogonal ring spectrum A can be defined by evaluating the cyclic bar construction on A or by applying Bökstedt's original definition of prefixTHH to A. In this paper, we construct a chain of stable equivalences of cyclotomic spectra comparing these two models for THH(A). This implies that the two versions of topological cyclic homology resulting from these variants of THH(A) are equivalent.
Abstract. We study the K-theory and Swan theory of the group ring R[G], when G is a finite group and R is any ring or ring spectrum. In this setting, the well-known assembly map for K(R[G]) has a companion called the coassembly map. We prove that their composite is the equivariant norm of K(R). This gives a splitting of both assembly and coassembly after K(n)-localization, a new map between Whitehead torsion and Tate cohomology, and a partial computation of K-theory of representations in the category of spectra.
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