# Stable Pontryagin-Thom construction for proper maps

## TL;DR

This paper proves two conjectures related to the stabilization and classification of proper maps between manifolds and Euclidean spaces, establishing a Pontryagin--Thom type correspondence in the stable range.

## Contribution

It provides a proof for a conjecture on the stabilization of homotopy classes of proper maps and constructs a Pontryagin--Thom type bijection for these maps.

## Key findings

- Homotopy classes of proper maps stabilize as dimension increases.
- A Pontryagin--Thom type bijection exists for proper maps in the stable range.
- Explicit construction of the bijection is provided.

## Abstract

We will present proofs for two conjectures stated in arXiv:1808.08073. The first one is that for an arbitrary manifold $W$, the homotopy classes of proper maps $W\times\mathbb{R}^n\to\mathbb{R}^{k+n}$ stabilise as $n\to\infty$, and the second one is that in a stable range there is a Pontryagin--Thom type bijection for proper maps $W\times\mathbb{R}^n\to\mathbb{R}^{k+n}$. The second one actually implies the first one and we shall prove the second one by giving an explicit construction.