A geometric computation of cohomotopy groups in co-degree one

TL;DR

This paper computes cohomotopy groups of maps from closed (n+1)-manifolds to spheres using geometric methods, extending previous results to non-orientable and non-spinnable cases, and introduces new manifold types.

Contribution

It generalizes cohomotopy group computations to broader classes of manifolds and introduces two new manifold types related to odd and even 4-manifolds.

Findings

Computed homotopy classes of maps for non-orientable and non-spinnable manifolds

Introduced two new manifold types generalizing odd and even 4-manifolds

Refined the Euler class as an obstruction for non-vanishing sections of spin vector bundles

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 from Kirby, Melvin and Teichner for closed oriented 4-manifolds and from Konstantis for closed $(n+1)$-dimensional spin manifolds, considering possibly non-orientable and non-spinnable manifolds. In the process, we introduce two types of manifolds that generalize the notion of odd and even 4-manifolds. Furthermore, for the case that $n \geq 4$, we discuss applications for rank $n$ spin vector bundles and obtain a refinement of the Euler class in the cohomotopy group that fully obstructs the existence of a non-vanishing section.