A counting invariant for maps into spheres and for zero loci of sections of vector bundles

The set of unrestricted homotopy classes $[M,S^n]$ where $M$ is a closed and connected spin $(n+1)$-manifold is called the $n$-th cohomotopy group ${\pi }^n(M)$ of $M$. Moreover it is known that $\pi^n(M) = H^n(M;\mathbb Z) \oplus \mathbb Z_2$ by methods from homotopy theory. We will provide a geometrical description of the ${\mathbb{Z}}_2$ part in ${\pi }^n(M)$ analogous to Pontryagin’s computation of the stable homotopy group ${\pi }_{n+1}(S^n)$. This ${\mathbb{Z}}_2$ number can be computed by counting embedded circles in $M$ with a certain framing of their normal bundle. This is a analogous result to the mod $2$ degree theorem for maps $M\to S^{n+1}$. Finally we will observe that the zero locus of a section in an oriented rank $n$ vector bundle $E\to M$ defines an element in ${\pi }^n(M)$ and it turns out that the ${\mathbb{Z}}_2$ part is an invariant of the isomorphism class of $E$. At the end we show, that if the Euler class of $E$ vanishes this ${\mathbb{Z}}_2$ invariant is the final obstruction to the existence of a nowhere vanishing section.