  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.
