A parametrized Pontryagin--Thom theorem

TL;DR

This paper presents a space-level enhancement of the Pontryagin--Thom theorem, linking maps from manifolds to Thom spaces with a moduli space of submanifolds, advancing the theoretical understanding of these topological structures.

Contribution

It introduces a novel space-level version of the Pontryagin--Thom theorem, providing a deeper connection between maps and submanifold moduli spaces.

Findings

Established a space-level equivalence between mapping spaces and moduli spaces of submanifolds.

Extended the classical Pontryagin--Thom theorem to a more refined, space-level context.

Enhanced the theoretical framework for studying manifolds and Thom spaces.

Abstract

We prove a space-level enhancement of the Pontryagin--Thom theorem, identifying the space of maps from a manifold to a Thom space with a moduli space of submanifolds.