Topological Hochschild homology of Thom spectra which are E ∞ -ring spectra

Abstract

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 and is in fact a left adjoint. We prove a splitting result THH(M f) ≃ eq 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).