  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.
