A geometric computation of cohomotopy groups in co-degree one

To appear in "Algebraic and Geometric Topology".

By Michael Jung and Thomas Rot

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≥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≥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.

Journal link: https://msp.org/soon/coming.php?jpath=agt

Preprint link: https://arxiv.org/abs/2307.03805