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. Using homotopy theory it is known that \(\pi ^n(M) = H^n(M;{\mathbb {Z}}) \oplus {\mathbb {Z}}_2\). 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 similar result to the mod 2 degree theorem for maps \(M \rightarrow S^{n+1}\). Finally we will observe that the zero locus of a section in an oriented rank n vector bundle \(E \rightarrow M\) 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.