Geometry & Topology

Topological Hochschild homology of Thom spectra and the free loop space

Andrew J Blumberg, Ralph L Cohen, and Christian Schlichtkrull

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We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f:XBF, 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.

Article information

Geom. Topol., Volume 14, Number 2 (2010), 1165-1242.

Received: 4 November 2008
Revised: 1 March 2010
Accepted: 2 April 2010
First available in Project Euclid: 20 December 2017

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Zentralblatt MATH identifier

Primary: 19D55: $K$-theory and homology; cyclic homology and cohomology [See also 18G60] 55N20: Generalized (extraordinary) homology and cohomology theories
Secondary: 18G55: Homotopical algebra 55P43: Spectra with additional structure ($E_\infty$, $A_\infty$, ring spectra, etc.) 55P47: Infinite loop spaces 55R25: Sphere bundles and vector bundles

topological Hochschild homology Thom spectra loop space


Blumberg, Andrew J; Cohen, Ralph L; Schlichtkrull, Christian. Topological Hochschild homology of Thom spectra and the free loop space. Geom. Topol. 14 (2010), no. 2, 1165--1242. doi:10.2140/gt.2010.14.1165.

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