On the topological Hochschild homology of $DX$

Cary Malkiewich

Published 2015-05-25Version 1

We begin a systematic study of the topological Hochschild homology of the commutative ring spectrum $DX$, the dual of a finite CW-complex $X$. We prove that the "Atiyah duality" between $THH(DX)$ and the free loop space $\Sigma^\infty_+ LX$ is an $S^1$-equivariant duality that preserves the $C_n$-fixed points, in addition to the ring structure and Adams operations. We then prove a stable splitting on $THH(D\Sigma X)$, and use this to calculate $THH(DS^{2n+1})$ and $TC(DS^1)$. Our approach uses a new, simplified construction of $THH$ due to Angeltveit et al., building on the work of Hill, Hopkins, and Ravenel. We also extend and elucidate this new model of $THH$, using a simple but powerful rigidity theorem for the geometric fixed point functor $\Phi^G$ of orthogonal $G$-spectra.