Smoothing continuous families of genuine immersions

Put $a=e\circ \phi$. Extend its parameter coordinates to a Euclidean neighbourhood of $P=P_0\times [0,1{]}^d$ by the tubular projection on $P_0$ and clamping each interval coordinate. This extension $\widehat{a}$ is continuous with every $x$-derivative jointly continuous. For a nonnegative bump ${\beta }_{\delta }$ of mass one define $v_{\delta }(p,x)=\int {\beta }_{\delta }(p-y)\widehat{a}(y,x)dy$, taking $\delta$ below a uniform neighbourhood radius of the compact $P$. The result is smooth jointly in $(p,x)$, and every $x$-derivative passes under the integral. Values and first $x$-derivatives converge uniformly to those of $a$ on finitely many compact source chart pieces covering $M$.

The compact family of pairs $(a(p,x),d_{x}a(p,x))$ lies in the open set of ambient first jets $(z,A)$ for which $z\in T$ and $d{r}_{z}A$ is injective. At an original pair, $d{r}_a{d}_{x}a=d_{x}a$ is injective. Nonvanishing minors and continuity of $dr$ therefore give a uniform positive allowed error on the finitely many compact parameter-source pieces. Choose $\delta$ so that both value and first-derivative errors of $v=v_{\delta }$ are smaller than that error. This uses the distance of the compact family from the complement of the permitted jet neighbourhood, rather than a distance from the whole, possibly noncompact, $e(N)$ to the edge of $T$. No bound on $\mathrm{\parallel }dr\mathrm{\parallel }$ away from $e(N)$ is asserted.

Choose a smooth cutoff $\chi :P\to [0,1]$ supported in $W$ and equal to one near $Q$, by applying [L3] in $P_0\times {\mathbb{R}}^d$; for empty $Q$ set $\chi =0$. The map $b=\chi a+(1-\chi )v$ is smooth, since $a$ is smooth on the support of $\chi$. For $s\in [0,1]$ set $w_s=(1-s)a+sb$ and ${\phi }_s=e^{-1}r(w_s)$. Because $\chi$ depends only on the parameter, both $w_s-a$ and $d_x{w}_s-d_{x}a$ are $(s(1-\chi ))$ times the respective mollification errors. Thus the permitted first-jet conditions of step 2.1 hold throughout, and every ${\phi }_s(p,\cdot )$ is an immersion. The homotopy starts at $\phi$, ends at the smooth map $e^{-1}r(b)$, and is fixed where $\chi =1$.

All source derivatives of ${\phi }_s$ are jointly continuous by the integral formula and smooth composition; [F1] gives a continuous homotopy of families. Empty source or parameter spaces need no smoothing. This proves the relative assertion.

For an arbitrary continuous path $\gamma$, choose a smooth $\eta :[0,1]\to [0,1]$ equal to zero near zero and one near one. The path $t\mapsto \gamma (\eta (t))$ is homotopic to $\gamma$ relative to endpoints by precomposition with $(1-s)t+s\eta (t)$. Its adjoint is smooth near the endpoints, where it is independent of $t$ and equals the prescribed smooth immersion. Apply steps 1.1–4.1 with $P=[0,1]$, $Q=\{0,1\}$ to produce a smooth path with exactly those endpoints. A smooth path of immersions is a regular homotopy by Regular homotopy of immersions, and every regular homotopy is weakly continuous by [F1]. Hence the component and regular-homotopy classifications agree.