A geometric computation of cohomotopy groups in codegree one

Using geometric arguments, we compute the group of homotopy classes of maps from a closed .nC1/dimensionalmanifold 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 .nC1/-dimensionalspin 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.