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Topological Hochschild homology of Thom spectra and the free loop space

Andrew J Blumberg, Ralph L Cohen and Christian Schlichtkrull

Geometry & Topology 14 (2010) 1165–1242

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 Hp and H.

topological Hochschild homology, Thom spectra, loop space
Mathematical Subject Classification 2000
Primary: 19D55, 55N20
Secondary: 18G55, 55P43, 55P47, 55R25
Received: 4 November 2008
Revised: 1 March 2010
Accepted: 2 April 2010
Published: 23 May 2010
Proposed: Paul Goerss
Seconded: Bill Dwyer, Haynes Miller
Andrew J Blumberg
Department of Mathematics
Stanford University
Stanford, CA 94305
United States
Ralph L Cohen
Department of Mathematics
Stanford University
Stanford, CA 94305
United States
Christian Schlichtkrull
Department of Mathematics
University of Bergen
Johannes Brunsgate 12
5008 Bergen