THH of Thom spectra that are E_\infty ring spectra

TL;DR

This paper computes the topological Hochschild homology (THH) of Thom spectra associated with E_ classifying maps, providing new splitting results and extending calculations to cases with homotopy commutative structures.

Contribution

It identifies THH of Thom spectra as E_ ring spectra and proves a splitting result, extending calculations to broader classes of Thom spectra.

Findings

THH of Thom spectra can be described as an indexed colimit.

A splitting theorem for THH(Mf) is established, simplifying calculations.

The results recover and extend Bokstedt's THH(HZ) calculation.

Abstract

We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E_\infty classifying map X -> BG, for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative S-algebra (E_\infty ring spectrum) R can be described as an indexed colimit together with a verification that the Lewis-May operadic Thom spectrum functor preserves indexed colimits. We prove a splitting result THH(Mf) \htp Mf \sma BX_+ which yields a convenient description of THH(MU). This splitting holds even when the classifying map f: X -> BG is only a homotopy commutative A_\infty map, provided that the induced multiplication on Mf extends to an E_\infty ring structure; this permits us to recover Bokstedt's calculation of THH(HZ).