Cyclotomic structure in the topological Hochschild homology of $DX$

Abstract

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 $THH\left(DX\right)$ and the free loop space ${\Sigma }_{+}^{\infty }LX$ is in fact a genuinely $S^1$–equivariant duality that preserves the $C_n$–fixed points. Our proof uses an elementary but surprisingly useful rigidity theorem for the geometric fixed point functor ${\Phi }^G$ of orthogonal $G$–spectra.