Smale's classification of sphere immersions in Euclidean space

For the quotient identification in [F7], every orthonormal $m$-frame extends to an orthonormal basis by finite-dimensional Gram–Schmidt. Two matrices have the same first $m$ columns exactly when they differ on the right by $\mathrm{diag}(I_m,Q)$ with $Q\in O(n-m)$. Thus the first-column map induces a continuous bijection $O(n)\mathrm{/}O(n-m)\to V_m({\mathbb{R}}^n)$; it is a homeomorphism because the source is compact and the target Hausdorff. When $n\ge m+1$, [F6] gives $T{S}^m\oplus {\epsilon }^{n-m}\cong {\epsilon }^n$. Inclusion of the tangent summand is a smooth monomorphism; polar-normalizing it by [F1] gives an isometric section of $E$, so $\mathrm{\Gamma }(E)\mathrm{\ne }\mathrm{\varnothing }$.

Clause 1: if $n\ge m+2$, then by [F3] the fibre $V_m({\mathbb{R}}^n)$ is simply connected, and $\mathrm{\Gamma }(E)\mathrm{\ne }\mathrm{\varnothing }$ by step 1.1; hence [F2] gives the non-canonical bijection ${\pi }_0\mathrm{\Gamma }(E)\cong {\pi }_m(V_m({\mathbb{R}}^n))$, the non-canonicity coming from the choice of trivialisation and of the section used to identify the difference classes. Passing to immersions: the derivative map is a weak homotopy equivalence by [F5], so it induces a bijection ${\pi }_0\mathrm{Imm}(S^m,{\mathbb{R}}^n)\cong {\pi }_0\mathrm{FImm}(S^m,{\mathbb{R}}^n)$, and [F1] identifies the latter with ${\pi }_0\mathrm{\Gamma }(E)$; the resulting bijection between regular homotopy classes of immersions and ${\pi }_m(V_m({\mathbb{R}}^n))={\pi }_m(O(n)\mathrm{/}O(n-m))$ is realised by the clutching difference class of the tangent framings.

Clause 2: if $n=m+1$, then $V_m({\mathbb{R}}^{m+1})\cong \mathrm{S}\mathrm{O}(m+1)$ by [F3], using the unique final normal vector that completes a frame to a positive orthonormal basis. In particular the fibre is path connected, so [F2] and step 1.1 give the surjection ${\pi }_m(F)\to {\pi }_0\mathrm{\Gamma }(E)$. Its only possible identifications are the evaluation-loop action. Given a loop of evaluated frames $e_t$ based at $e_0$, write $C(e)$ for the uniquely completed oriented matrix and put $A_t=C(e_t)C(e_0{)}^{-1}$. These matrices define a loop in $\mathrm{S}\mathrm{O}(m+1)$ with $A_0=A_1=I$ and $A_t{e}_0=e_t$. For every section $s$ with $s(x_0)=e_0$, the sections $s_t(x)=A_{t}s(x)$ lift that loop and return to the same section $s$. Thus every evaluation loop acts trivially on every component of the fixed-value section space. The exact-sequence component map is therefore injective as well as surjective, giving the asserted non-canonical bijection ${\pi }_m(\mathrm{S}\mathrm{O}(m+1))\cong {\pi }_0\mathrm{\Gamma }(E)$. By [F5] it also classifies regular homotopy components of immersions; in particular vanishing of this group gives a single component.

Clause 3: for $m=1$, $n=2$, clause 2 applies with $\mathrm{S}\mathrm{O}(2)=S^1$; the winding invariant of [F8] is a surjection ${\pi }_0\mathrm{\Gamma }(E)\to \mathbb{Z}$ that is also injective by the classification of formal immersions of the circle, so ${\pi }_0\mathrm{\Gamma }(E)\cong \mathbb{Z}$ and [F5] gives ${\pi }_0\mathrm{Imm}(S^1,{\mathbb{R}}^2)\cong \mathbb{Z}$ with the rotation number as invariant. For $m=2$, $n=3$, clause 2 applies with $V_2({\mathbb{R}}^3)\cong \mathrm{S}\mathrm{O}(3)$ and ${\pi }_2(\mathrm{S}\mathrm{O}(3))=0$ by [F4], so $\mathrm{\Gamma }(E)$ is path connected and all immersions $S^2\to {\mathbb{R}}^3$ are regularly homotopic. The Smale–Hirsch input carries its countable-choice hypothesis.