Topological Hochschild Homology of Thom Spectra Which Are E∞-ring Spectra

We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E∞ 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∞ 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) ≃ Mf∧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∞ map, provided that the induced multiplication on Mf extends to an E∞ ring structure; this permits us to recover Bokstedt’s calculation of THH(HZ).