The sphere immersion groups are algebraic-topology computations, not differential-topology constructions

The homotopy-theoretic inputs used on this page — ${\pi }_2(\mathrm{S}\mathrm{O}(3))=0$ (The second homotopy group of SO(3) vanishes) via the quaternion double cover and the covering isomorphism on ${\pi }_n$ for $n\ge 2$; ${\pi }_1(\mathrm{S}\mathrm{O}(2))\cong \mathbb{Z}$ via circle degree (Formal immersions of the circle in the plane are classified by the winding number); the Stiefel connectivities ${\pi }_0(V_m({\mathbb{R}}^n))=0$ for $n\ge m+1$ and ${\pi }_1(V_m({\mathbb{R}}^n))=0$ for $n\ge m+2$ (Stiefel manifolds are connected in positive codimension and simply connected in codimension at least two), and the higher homotopy exact sequences of the fibrations $V_m({\mathbb{R}}^n)\to S^{n-1}$ — are algebraic-topology computations owned by the prerequisite pages on higher homotopy groups and cofiber sequences, fibrations and homotopy exact sequences, covering spaces and lifting, and the Hurewicz, Whitehead, Freudenthal and CW-approximation theorems. Here they are consumed as the values of ${\pi }_m(V_m({\mathbb{R}}^n))$ and ${\pi }_2(\mathrm{S}\mathrm{O}(3))$ in the sense of Higher homotopy group by based cubes; Smale's classification of sphere immersions in Euclidean space states the classification in terms of them.

This page contributes only the differential-topological reduction: the Smale–Hirsch weak homotopy equivalence (from the predecessor page), the identification of formal data with Stiefel sections, and the clutching difference class that turns the classification into these groups. No new homotopy-theoretic machinery is minted here, and conversely the groups ${\pi }_m(V_m({\mathbb{R}}^n))$ remain algebraic-topology inputs; each use is recorded in the dependencies of the items above.