A geometric computation of cohomotopy groups in codegree one

Jung, Michael; Rot, Thomas O

doi:10.2140/agt.2025.25.3603

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A geometric computation of cohomotopy groups in codegree one

Michael Jung and Thomas O Rot

Algebraic & Geometric Topology 25 (2025) 3603–3626

DOI: 10.2140/agt.2025.25.3603

Abstract

Using geometric arguments, we compute the group of homotopy classes of maps from a closed $(n + 1)$ -dimensional manifold to the $n$ -sphere for $n \geq 3$ . Our work extends results of Kirby, Melvin and Teichner for closed oriented 4-manifolds, and of Konstantis for closed $(n + 1)$ -dimensional spin manifolds, considering possibly nonorientable and nonspinnable manifolds. In the process, we introduce two types of manifolds that generalize the notion of odd and even 4-manifolds. Furthermore, for $n \geq 4$ , we discuss applications of rank $n$ spin vector bundles and obtain a refinement of the Euler class in the cohomotopy group that fully obstructs the existence of a nonvanishing section.

Keywords

cohomotopy, Pontryagin–Thom construction, pin structures, homology with local coefficients, vector bundles, Euler class

Mathematical Subject Classification

Primary: 55Q55

Secondary: 55N25, 57R15, 57R22

References

Publication

Received: 9 August 2023

Revised: 25 March 2024

Accepted: 24 May 2024

Published: 1 October 2025

Authors

Michael Jung
Department of Mathematics Vrije Universiteit Amsterdam Amsterdam Netherlands

Thomas O Rot
Department of Mathematics Vrije Universiteit Amsterdam Amsterdam Netherlands

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Volume 25, issue 6 (2025)