A geometric computation of cohomotopy groups in codegree one

Michael Jung and Thomas O 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 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≥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.

Read the full paper in Algebraic and Geometric topology here