#### Volume 14, issue 2 (2010)

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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
##### Abstract

We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps $f:X\to BF$, 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 $L\left(BX\right)$. 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 $Hℤ$.

##### Keywords
topological Hochschild homology, Thom spectra, loop space
##### Mathematical Subject Classification 2000
Primary: 19D55, 55N20
Secondary: 18G55, 55P43, 55P47, 55R25