Homotopy classes of proper maps out of vector bundles

Abstract

In this paper we classify the homotopy classes of proper maps $E\to {\mathbb{R}}^k$, where $E$ is a vector bundle over a compact Hausdorff space. As a corollary we compute the homotopy classes of proper maps ${\mathbb{R}}^n\to {\mathbb{R}}^k$. 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.