Topological Hochschild homology of Thom spectra and the free loop space

TL;DR

This paper computes the topological Hochschild homology of Thom spectra arising from loop maps, linking it to the free loop space and enabling explicit calculations for classical spectra.

Contribution

It introduces a new identification of topological Hochschild homology for Thom spectra as a Thom spectrum over the free loop space, with explicit computations for many classical spectra.

Findings

Identifies topological Hochschild homology as a Thom spectrum over the free loop space.

Provides explicit calculations for cobordism and Eilenberg-Mac Lane spectra.

Establishes symmetric monoidal models for spaces over BF.

Abstract

We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f: 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, MSO, MU, etc., and the Eilenberg-Mac Lane spectra HZ/p and HZ.