Regular homotopy preserves the formal Gauss class

The adjoint $\widehat{H}:[0,1]\to C^{\mathrm{\infty }}(M,N)$, $\widehat{H}(t)=H_t$, is continuous for the weak compact-open $C^{\mathrm{\infty }}$ topology directly from the definition: a basic weak neighbourhood of $H_{t_0}$ is determined by finitely many charts, compact pieces $K_i\subseteq M$ and tolerances ${\epsilon }_i$ on derivatives of order at most $r_i$, and compactness with continuity gives a closed time interval $J$ about $t_0$ relative to $[0,1]$ on which $H(K_i\times J)$ lies in every prescribed target chart. For each of the finitely many occurring multi-indices $\alpha$ the local expression $D^{\alpha }({\psi }_i\circ H_t\circ {\phi }_i^{-1})(x)$ is continuous on the compact set ${\phi }_i(K_i)\times J$, hence uniformly continuous there, so for $t$ close to $t_0$ every one of the finitely many derivatives stays within its tolerance; therefore $\widehat{H}$ is continuous. Its image lies in $\mathrm{Imm}(M,N)$ because every slice of $H$ is an immersion by [F1].

Composition with the continuous derivative map [F3] gives the continuous path $t\mapsto D(H_t)=(H_t,d{H}_t)$ in $\mathrm{FImm}(M,N)$, since the composite of continuous maps is continuous and $\widehat{H}$ takes values in $\mathrm{Imm}(M,N)$ by step 1.1.

A continuous path has all its points in a single path component [F5], so the path component of $(H_t,d{H}_t)$ in $\mathrm{FImm}(M,N)$ is independent of $t$; hence every invariant of formal data that is constant on path components takes the same value at $(H_0,d{H}_0)$ and $(H_1,d{H}_1)$, which is the necessity half of the classifications: regularly homotopic immersions have homotopic formal data.

The two instances: for $M=S^m$, $N={\mathbb{R}}^n$ with $n\ge m+2$, the path component of the formal data is detected by the difference class of the basepoint-evaluation lemma through the section-space description of $\mathrm{FImm}(S^m,{\mathbb{R}}^n)$, so the class in ${\pi }_m(V_m({\mathbb{R}}^n))$ is constant along a regular homotopy; for $M=S^1$, $N={\mathbb{R}}^2$ the corresponding invariant of a path component is the winding number, which for the derivative $(f,df)$ is the rotation number of the immersion, as established in Formal immersions of the circle in the plane are classified by the winding number. The sphere difference-class conclusion is The basepoint evaluation of the Stiefel section space is a fibration. The compact-source identification of path components with regular homotopy classes used in the converse direction is [F4] with its countable-choice hypothesis; the invariance proved here uses steps 1.1–3.1 and inherits the structural countable-choice hypothesis of [F3]. No approximation or additional choice is used in this direct argument.