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 W X ! 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.BX /. 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, M SO, MU , etc, and the Eilenberg-Mac Lane spectra H Z=p and H Z.