A simple universal property of Thom ring spectra

Abstract

We give a simple universal property of the multiplicative structure on the Thom spectrum of an $n$-fold loop map, obtained as a special case of a characterization of the algebra structure on the colimit of a lax $\mathcal{O}$-monoidal functor. This allows us to relate Thom spectra to ${\mathbb{E}}_n$-algebras of a given characteristic in the sense of Szymik. As applications, we recover the Hopkins--Mahowald theorem realizing $H{\mathbb{F}}_p$ and $H\mathbb{Z}$ as Thom spectra, and compute the topological Hochschild homology and the cotangent complex of various Thom spectra.

Updated to correct a small error found by Qi Zhu in Corollary 3.9. Thanks, Qi!

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Cited by in corpus (20)

• On a nilpotence conjecture of J.P. May
• Computing homological residue fields in algebra and topology
• Eilenberg-MacLane spectra as equivariant Thom spectra
• On the homotopy type of L-spectra of the integers
• Purity in chromatically localized algebraic $K$-theory
• Exotic Multiplications on Periodic Complex Bordism
• Some Higher Coalgebra
• Perfectoid rings as Thom spectra
• Higher chromatic Thom spectra via unstable homotopy theory
• Characters and transfer maps via categorified traces
• Thom spectra, higher $THH$ and tensors in $\infty$-categories
• The Ando-Hopkins-Rezk orientation is surjective
• Commutative ring spectra
• Toward a Galois theory of the integers over the sphere spectrum
• Adjoining roots in homotopy theory
• Nilpotence in E_n Algebras
• Dualizable objects in stratified categories and the 1-dimensional bordism hypothesis for recollements
• A monoidal Grothendieck construction for $\infty$-categories
• Topological Hochschild homology of X(n)
• A Thom Spectrum Model for $C_2$-Integral Brown--Gitler Spectra