A refinement of the Pontryagin-Thom theorem for unstable Thom spectra and its applications

TL;DR

This paper refines the Pontryagin-Thom theorem to give a clearer geometric understanding of unstable Thom spectra's homotopy groups, enabling new computations of cobordism groups and applications to Spin manifolds.

Contribution

It provides a more explicit geometric interpretation of the homotopy groups of unstable Thom spectra and applies this to compute new cobordism groups and analyze Spin manifolds.

Findings

Explicit geometric interpretation of unstable Thom spectra's homotopy groups

Computed previously unknown cobordism groups using Smith homomorphism

Characterized cobordism groups related to Spin manifolds with boundary

Abstract

The Pontryagin-Thom construction provides a fundamental link between cobordism groups and the homotopy groups of Thom spectra. Our main result refines this theorem, providing a more explicit geometric interpretation of the homotopy groups of unstable Thom spectra. Building on this result, we show that previously unknown cobordism groups can be expressed as homotopy groups of unstable Thom spectra. Furthermore, using the Smith homomorphism, we compute these groups. As applications, we determine the values of $n$ for which there exists a Spin manifold with boundary $S^n$ admitting a line subbundle orthogonal to the boundary, and provide a precise characterization of the cobordism group introduced by Bais, May Custodio, and Torres.