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

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.