Parametrized geometric cobordism and smooth Thom stacks

TL;DR

This paper introduces a new theory of parametrized geometric cobordism using smooth Thom stacks, providing a geometric refinement of the Pontrjagin-Thom construction that encompasses various geometric data.

Contribution

It develops a smooth Thom stack framework for parametrized cobordism, extending existing theories to include diverse geometric structures on manifolds and bundles.

Findings

Generalizes the parametrized cobordism of Galatius-Madsen-Tillman-Weiss

Enables inclusion of metrics and connections in cobordism theory

Provides a versatile geometric refinement of the Pontrjagin-Thom construction

Abstract

We develop a theory of parametrized geometric cobordism by introducing smooth Thom stacks. This requires identifying and constructing a smooth representative of the Thom functor acting on vector bundles equipped with extra geometric data, leading to a geometric refinement of the the Pontrjagin-Thom construction in stacks. We demonstrate that the resulting theory generalizes the parametrized cobordism of Galatius-Madsen-Tillman-Weiss. The theory has the feature of being both versatile and general, allowing for the inclusion of families of various geometric data, such as metrics on manifolds and connections on vector bundles, as in recent work of Cohen-Galatius-Kitchloo and Ayala.