Sphere self-maps are homotopic exactly when their degrees agree

The unit sphere is the regular level $\mathrm{\mid }x{\mathrm{\mid }}^2=1$, with nonzero differential $2⟨x,\cdot ⟩$ and tangent space $x^{\perp }$. Its standard smooth structure is supplied by the regular-level theorem. The stereographic inverse charts are $u\mapsto (2u\mathrm{/}(1+\mathrm{\mid }u{\mathrm{\mid }}^2),\pm (\mathrm{\mid }u{\mathrm{\mid }}^2-1)\mathrm{/}(1+\mathrm{\mid }u{\mathrm{\mid }}^2))$, with the two omitted poles understood, and their transition is $u\mapsto u\mathrm{/}\mathrm{\mid }u{\mathrm{\mid }}^2$; all expressions are smooth on their domains. The boundary orientation is defined by requiring $(x,v_1,\dots ,v_m)$ to be positive in ${\mathbb{R}}^{m+1}$. (A regular level set is an embedded submanifold, Induced boundary orientation).

Given: ${\mathrm{A}\mathrm{C}}_{\omega }$, an integer $m\ge 1$ and the unit sphere $S^m\subseteq {\mathbb{R}}^{m+1}$ with its standard smooth structure and its outward-normal-first orientation (Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$, the local calculation, the local calculation).

For $m\ge 1$ the sphere $S^m$ is compact, path-connected and connected, and it is a closed connected oriented smooth $m$-manifold (For $n\ge 1$, every Euclidean closed ball and every Euclidean sphere of positive radius is compact, For $n\ge 2$, the sphere $S^{n-1}$ is path-connected and connected, the local calculation, Smooth manifolds and their smooth charts).

For a nonempty closed connected oriented smooth $m$-manifold $M$ with $m\ge 1$, degree induces a bijection from free homotopy classes $[M,S^m]$ to $\mathbb{Z}$: two continuous maps are homotopic exactly when their degrees agree, and every integer is realized (The Hopf degree theorem for oriented domains, Degree of a proper smooth map by compact-support cohomology).