The set of unrestricted homotopy classes \(\pi^n(M)=[M, \mathbb{S}^n]\), where \(M\) is a closed and connected spin \((n + 1)\)-manifold is called the \(n\)-th cohomotopy group of \(M\). Using homotopy theory it is known that \(\pi^n(M) = H^n(M,\mathbb{Z})\oplus \mathbb{Z}_2\).
L. S. Pontryagin [Transl., Ser. 2, Am. Math. Soc. 11, 1–114 (1959; Zbl 0084.19002)] computed the (stable) homotopy group \(\pi_{n+1}(\mathbb{S}^n)\) for \(n \ge 3\) using differential topology. This paper generalizes his idea and provides a geometrical description of the \(\mathbb{Z}_2\) summand in \(\pi^n(M)\). Namely, the \(\mathbb{Z}_2\) number can be computed by counting embedded circles in \(M\) with a certain framing of their normal bundle.

Finally, it is observed 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 \(E\). As an application, the author provides a simple proof of the well-known fact that the maximal number of linear independent vector fields on \(\mathbb{S}^{4n+1}\) is equal to \(1\).