Homotopy classes of proper maps out of vector bundles

TL;DR

This paper classifies proper maps from vector bundles over compact spaces to Euclidean spaces, providing a homotopy classification, stability ranges, and connections to framed submanifolds and Pontryagin-Thom theory.

Contribution

It introduces a classification of homotopy classes of proper maps from vector bundles, extending to maps between Euclidean spaces and exploring stability and geometric implications.

Findings

Homotopy classes of proper maps are classified for vector bundles over compact spaces.

A stability range for proper maps between Euclidean spaces is established.

Connections to framed submanifolds and Pontryagin-Thom construction are discussed.

Abstract

In this paper we classify the homotopy classes of proper maps $E\rightarrow \mathbb R^k$, where $E$ is a vector bundle over a compact Hausdorff space. As a corollary we compute the homotopy classes of proper maps $\mathbb R^n\rightarrow \mathbb R^k$. We find a stability range of such maps. We conclude with some remarks on framed submanifolds of non-compact manifolds, the relationship with proper homotopy classes of maps and the Pontryagin-Thom construction.