## Geometry & Topology

### 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 $f:X→BF$, 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 $Hℤ$.

#### Article information

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

Dates
Revised: 1 March 2010
Accepted: 2 April 2010
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732216

Digital Object Identifier
doi:10.2140/gt.2010.14.1165

Mathematical Reviews number (MathSciNet)
MR2651551

Zentralblatt MATH identifier
1219.19006

#### Citation

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. https://projecteuclid.org/euclid.gt/1513732216

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