Cyclotomic structure in the topological Hochschild homology of $DX$

TL;DR

This paper establishes a genuine $S^1$-equivariant duality between topological Hochschild homology of the dual of a finite CW complex and its free loop space, revealing new symmetries in their structure.

Contribution

It proves that the Poincaré/Koszul duality between $THH(DX)$ and the free loop space is a genuine $S^1$-equivariant duality, preserving $C_n$-fixed points.

Findings

The duality is genuinely $S^1$-equivariant.

The proof employs a new rigidity theorem for geometric fixed points.

The duality preserves $C_n$-fixed points in the spectra.

Abstract

Let $X$ be a finite CW complex, and let $DX$ be its dual in the category of spectra. We demonstrate that the Poincar\'e/Koszul duality between $THH(DX)$ 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.