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

Abstract

AbstractThe set of unrestricted homotopy classes $$[M,S^n]$$ [ M , S n ] where M is a closed and connected spin $$(n+1)$$ ( n + 1 ) -manifold is called the n-th cohomotopy group $$\pi ^n(M)$$ π n ( M ) of M. Using homotopy theory it is known that $$\pi ^n(M) = H^n(M;{\mathbb {Z}}) \oplus {\mathbb {Z}}_2$$ π n ( M ) = H n ( M ; Z ) ⊕ Z 2 . We will provide a geometrical description of the $${\mathbb {Z}}_2$$ Z 2 part in $$\pi ^n(M)$$ π n ( M ) analogous to Pontryagin’s computation of the stable homotopy group $$\pi _{n+1}(S^n)$$ π n + 1 ( S n ) . This $${\mathbb {Z}}_2$$ 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 $$M \rightarrow S^{n+1}$$ M → S n + 1 . Finally we will observe that the zero locus of a section in an oriented rank n vector bundle $$E \rightarrow M$$ E → M defines an element in $$\pi ^n(M)$$ π n ( M ) and it turns out that the $${\mathbb {Z}}_2$$ 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 $${\mathbb {Z}}_2$$ Z 2 invariant is the final obstruction to the existence of a nowhere vanishing section.