A simple universal property of Thom ring spectra

TL;DR

This paper establishes a universal property for the multiplicative structure of Thom spectra derived from n-fold loop maps, connecting them to $ ext{E}_n$-algebras and enabling new computations and realizations.

Contribution

It introduces a simple universal property characterizing Thom spectra's algebraic structure, linking them to $ ext{E}_n$-algebras of specific characteristics.

Findings

Recovered the Hopkins--Mahowald theorem for $H\mathbb{F}_p$ and $H\mathbb{Z}$ as Thom spectra

Computed topological Hochschild homology of various Thom spectra

Analyzed the cotangent complex of Thom 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.