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 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( H ℤ).