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# Homology, Homotopy and Applications

## Volume 19 (2017)

### Number 1

### Topological Hochschild homology of $K/p$ as a $K_p^\wedge$ module

Pages: 253 – 280

DOI: https://dx.doi.org/10.4310/HHA.2017.v19.n1.a13

#### Author

#### Abstract

For commutative ring spectra $R$, one can construct a Thom spectrum for spaces over $BGL_1R$. This specialises to the classical Thom spectra for spherical fibrations in the case of the sphere spectrum. The construction is useful in detecting $A_\infty$-structures: a loop space (up to homotopy) over $BGL_1R$ yields an $A_\infty$-ring structure on the Thom spectrum. The topological Hochschild homology of these $A_\infty$-ring spectra may be expressed as Thom spectra.

This paper uses the identification of topological Hochschild homology of Thom spectra to make computations. Specifically, we take $R$ to be the $p$-adic $K$-theory spectrum and consider a certain map from $S^1$ to $BGL_1R$, so that the Thom spectrum is equivalent to the $\textrm{mod}\, p$ $K$-theory spectrum. We make computations at odd primes.

#### Keywords

Thom spectra, topological Hochschild homology, $K$-theory

#### 2010 Mathematics Subject Classification

Primary 55P42. Secondary 55N15, 55P43.

Published 6 June 2017