A homotopy coherent Pontryagin-Thom isomorphism

Abstract

Classically, the Pontryagin-Thom isomorphism asserts that the multiplicative cohomology theory given by (structured) geometric cobordism is isomorphic to the cohomology theory determined by an associated Thom spectrum. We construct a presentably symmetric monoidal stable $\infty$-category of homotopy invariant sheaves with transfers on smooth manifolds whose unit is precisely (structured) geometric cobordism. We show the endomorphism ring of the unit can be canonically identified with the associated $\mathbb{E}_\infty$-Thom ring spectrum, i.e., we provide an $\mathbb{E}_\infty$-lift of the Pontryagin-Thom isomorphism.