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

Abstract

Abstract. We identify the topological Hochschild homology (T HH) 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 T HH 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 T HH(M f ) ≃ M f ∧BX + which yields a convenient description of T HH(M U ). This splitting holds even when the classifying map f : X → BG is only a homotopy commutative A∞ map, provided that the induced multiplication on M f extends to an E∞ ring structure; this permits us to recover Bokstedt's calculation of T HH(HZ).

A simple universal property of Thom ring spectra

“…The goal of this section is to deduce an integral analogue of Theorem , using the framework of this paper. Our approach is different from the arguments given in , as it does not rely on any further explicit homology computations. Unless otherwise stated, we will implicitly work in the ‐complete category of spaces and spectra.…”

“…As in [, Section 9.3], we may glue these maps together to construct as a Thom spectrum as well. Here, we work in the category of all spectra, not just ‐complete ones.…”

“…Proof Using Theorem , it is enough to compute the topological Hochschild homology of . If , the claim thus follows from [, Theorem 3], while the case is covered by [, 1.6].…”

“…Proof. Following ideas of Beardsley [5], we will first construct an intermediate Thom spectrum Mφ p associated to a map φ p : S 1 → BGL 1 (Mg p ) from the base of the fiber sequence (6). The properties of this spectrum allow us then to determine the homotopy groups of Mg p , from which the claim will follow.…”

Toward a Galois theory of the integers over the sphere spectrum

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Recent work [1,2,7,18,44] in higher algebra allows the reinterpretation of a classical description [8,11,31,32] of the Eilenberg-MacLane spectrum HZ as a Thom spectrum, in terms of a kind of derived Galois theory. This essentially expository talk summarizes some of this work, and suggests an interpretation in terms of configuration spaces and monoidal functors on them, with some analogies to a topological field theory.

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