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

TL;DR

This paper introduces a geometric invariant for maps into spheres and zero loci of sections of vector bundles, revealing a new $bZ_2$-valued obstruction related to the Euler class and framing counts.

Contribution

It provides a geometric description of the $bZ_2$ component of the cohomotopy group $pi^n(M)$, linking it to embedded circles with specific framings and zero loci of vector bundle sections.

Findings

The $bZ_2$ invariant can be computed by counting framed embedded circles in $M$.

Zero loci of sections define elements in $pi^n(M)$ with the $bZ_2$ part as an invariant.

Vanishing Euler class implies this $bZ_2$ invariant is the obstruction to nowhere vanishing sections.

Abstract

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.