Topological Hochschild homology of Thom spectra and the free loop space

Abstract

We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps $f:X\to BF$, where $BF$ denotes the classifying space for stable spherical fibrations. To do this, we consider symmetric monoidal models of the category of spaces over $BF$ and corresponding strong symmetric monoidal Thom spectrum functors. Our main result identifies the topological Hochschild homology as the Thom spectrum of a certain stable bundle over the free loop space $L\left(BX\right)$. This leads to explicit calculations of the topological Hochschild homology for a large class of ring spectra, including all of the classical cobordism spectra $MO$, $MSO$, $MU$, etc, and the Eilenberg–Mac Lane spectra $H\mathbb{Z}∕p$ and $H\mathbb{Z}$.