Homotopy classes of proper maps out of vector bundles

Thomas O. Rot

doi:10.1007/s00013-019-01372-z

Homotopy classes of proper maps out of vector bundles

Thomas O. Rot^*

^*Corresponding author for this work

Mathematics

Research output: Contribution to Journal › Article › Academic › peer-review

Abstract

In this paper, we classify the homotopy classes of proper maps E→ R^k, where E is a vector bundle over a compact Hausdorff space. As a corollary, we compute the homotopy classes of proper maps Rⁿ→ 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.

Original language	English
Pages (from-to)	107-117
Number of pages	11
Journal	Archiv der Mathematik
Volume	114
Early online date	30 Aug 2019
DOIs	https://doi.org/10.1007/s00013-019-01372-z
Publication status	Published - Jan 2020

Funding

I would like to thank Alberto Abbondandolo, Hansjörg Geiges, Gijs Heuts, and Federica Pasquotto for discussions on the content of this paper. This research was supported by NWA startimpuls - 400.17.608. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Funders	Funder number
Northwest Airlines	400.17.608

Keywords

Framed cobordism
Pontryagin–Thom construction
Proper homotopy theory

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AB - In this paper, we classify the homotopy classes of proper maps E→ Rk, where E is a vector bundle over a compact Hausdorff space. As a corollary, we compute the homotopy classes of proper maps Rn→ Rk. 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.

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KW - Pontryagin–Thom construction

KW - Proper homotopy theory

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Homotopy classes of proper maps out of vector bundles

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