Positive-codimension thickening reduces closed sources to the open case

The formal section and fibre-preserving homotopy in [F1] show that $r_F$ is a homotopy equivalence, not merely a map with contractible fibres. Indeed $r_F{S}_F=\mathrm{i}\mathrm{d}$, and fibre contraction followed by interpolation of the tangential component of the vertical map gives $S_F{r}_F\simeq \mathrm{i}\mathrm{d}$. All maps are jointly continuous in the weak smooth topology because fibre scaling, the fixed horizontal splitting, and smooth orthogonal projection are continuous in every compact jet seminorm.

For enriched genuine data $(f,\alpha )$ let $V:E\to f^{\ast }TN$ be the orthogonal lift of $\alpha$ relative to $df(TM)$. The germ $e_{f,\alpha }(x,v)={\mathrm{\ell }}_{f(x)}(V_{x}v)$ is an immersion near the zero section: its derivative there is $(df,V):TM\oplus E\to f^{\ast }TN$, an isomorphism. For a compact family of these data, compactness of the parameter space times $M$ gives a single positive radius on which every germ is defined and immersive. Define ${\mathrm{\Psi }}_{\delta }(x,v)=(x,\frac{\delta v}{\sqrt{{\delta }^2+\mathrm{\mid }v{\mathrm{\mid }}^2}})\mathrm{.}$ Its fibre image has radius less than $\delta$, its derivative at zero is the identity, and its radial and tangential eigenvalues away from zero are respectively ${\delta }^3\mathrm{/}({\delta }^2+\mathrm{\mid }v{\mathrm{\mid }}^2{)}^{3\mathrm{/}2}$ and $\delta \mathrm{/}({\delta }^2+\mathrm{\mid }v{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$, both positive. Thus for sufficiently small common $\delta >0$, $e_{f,\alpha }^{\delta }=e_{f,\alpha }\circ {\mathrm{\Psi }}_{\delta }$ is an immersion of the entire fixed $B$ with $r_I(e_{f,\alpha }^{\delta })=(f,\alpha )$.

Let $g_p:B\to N$ be a compact continuous family of immersions and write $f_p=g_p{\mathrm{\mid }}_M$, $B_p=d_v{g}_p{\mathrm{\mid }}_M$, and ${\alpha }_p=[B_p]$. Since $\dim B=\dim N$, $(d{f}_p,B_p)$ is an isomorphism. For $t>0$ and $v$ near zero put $H_{p,t}(x,v)={\mathrm{\ell }}_{f_p(x)}(t^{-1}{\log }_{f_p(x)}{g}_p(x,tv)),H_{p,0}(x,v)={\mathrm{\ell }}_{f_p(x)}(B_{p,x}v)\mathrm{.}$ In bundle charts Taylor's integral formula writes the argument at $t>0$ as ${\int }_0^1{d}_v({\log }_{f_p(x)}{g}_p)(x,utv)vdu$. This extends continuously in all spatial derivatives to $t=0$, smoothly when the parameter family is smooth. Every $H_{p,t}$ has derivative $(d{f}_p,B_p)$ along the zero section, independent of $t$. Compactness of parameters times $[0,1]\times M$ and openness of invertibility consequently give a common small radius on which all $H_{p,t}$ are defined and immersive.

Let $Z_F\subset \mathrm{FImm}(M,N)$ consist of data with normal bundle isomorphic to $E$, and let $Z_I$ be the corresponding immersion locus. Fix $z_0=(f_0,F_0)$ and an identification ${\alpha }_0:E\to {\nu }_{z_0}$. For $z=(f,F)$ close to $z_0$, [F2] identifies $f_0^{\ast }TN$ with $f^{\ast }TN$ by the invertible vertical derivative of ${\mathrm{\ell }}_{f_0(x)}$ at ${\log }_{f_0(x)}f(x)$. Project this identification from $F_0(TM{)}^{\perp }$ onto $F(TM{)}^{\perp }$. At $z=z_0$ it is the identity, hence it remains invertible on a weak-smooth neighbourhood, by compactness of $M$. Taking quotients gives a continuously varying isomorphism ${\theta }_z:{\nu }_{z_0}\to {\nu }_z$. Therefore $(z,a)⟼(z,{\theta }_z{\alpha }_{0}a),a\in \mathrm{Aut}(E),$ is a local product chart for the formal forgetful map. The same chart restricted to $F=df$ gives the genuine forgetful chart. In particular $Z_F$ and $Z_I$ are open loci, and the derivative map induces the identity on each common gauge fibre.

Precompose $g_p$ with the embeddings of $B$ obtained by interpolating fibrewise between $v$ and ${\mathrm{\Psi }}_{\delta }(v)$. Both radial derivative and tangential eigenvalue of the interpolation are positive, its norm is at most $\mathrm{\mid }v\mathrm{\mid }$, and its derivative at zero is the identity. This is a homotopy through immersions from $g_p$ to $g_p\circ {\mathrm{\Psi }}_{\delta }$, with $r_I$ fixed. Next decrease $t$ from one to zero in $H_{p,t}\circ {\mathrm{\Psi }}_{\delta }$. Finally decompose $B_p=V_p+d{f}_p{A}_p$, where $V_p$ is the orthogonal lift of ${\alpha }_p$, and interpolate $B_p$ to $V_p$ by $B_{p,s}=V_p+(1-s)d{f}_p{A}_p$. The maps $(d{f}_p,B_{p,s})$ remain isomorphisms because their quotient on $E$ is the fixed isomorphism ${\alpha }_p$. Shrinking the same common $\delta$ if necessary makes ${\mathrm{\ell }}_{f_p(x)}(B_{p,s,x}{\mathrm{\Psi }}_{\delta }(v))$ immersive for all $p,s$. This gives a compact-family homotopy, entirely over the unchanged enriched data, from $g_p$ to $e_{f_p,{\alpha }_p}^{\delta }$.

The two forgetful maps are Serre fibrations by a direct compact-cube lifting construction. For a continuous homotopy $z_{p,t}$ on a compact parameter cube, apply the preceding projection construction to pairs of nearby data, choosing the tangent identification from [F2]. It gives normal transport $T(z,z^{\mathrm{\prime }}):{\nu }_z\to {\nu }_{z^{\mathrm{\prime }}}$ defined on a neighbourhood of the diagonal and satisfying $T(z,z)=\mathrm{i}\mathrm{d}$. Compactness of the homotopy image permits a common finite subdivision $0=t_0<\cdots <t_r=1$ such that $T(z_{p,t_j},z_{p,t})$ is defined for every $p$ and $t\in [t_j,t_{j+1}]$. Given an initial identification, define recursively ${\alpha }_{p,t}=T(z_{p,t_j},z_{p,t}){\alpha }_{p,t_j}$ on each interval. The formulas agree at subdivision times and are continuous in all source jet seminorms. This lifts the homotopy with its prescribed initial family and proves cube HLP. It also shows that normal-isomorphism loci are path-saturated. No choice of one lift for every path, or assertion of contractible gauge fibres, is needed.

These constructions also give relative compact tests, rather than just fibrewise contraction. Given an enriched family on a compact parameter manifold $P$ and an existing genuine extension on an open neighbourhood $U$ of a closed parameter subset $Q$, choose $Q\subset U_0\subset \overline{U_0}\subset U$ and a parameter cutoff equal to zero near $Q$ and one near $P\setminus U_0$. Choose one $\delta$ small enough for the canonical family on $P$ and the step-2.1 deformation of the given extensions on $\overline{U_0}$. On $U_0$ run that deformation for the cutoff amount of time; outside $U_0$ use the canonical model. Near the boundary of $U_0$ both formulas equal the same model, so they paste continuously, smoothly for smooth families. The resulting extension has exactly the prescribed enriched restriction and equals the original extension near $Q$. Different admissible radii can be compared by decreasing both to a common smaller radius; the radial models and their interpolations remain in the common germ neighbourhood. For a disk test with a given genuine lift on its boundary, first precompose the enriched disk family with a radial parameter map homotopic to the identity relative to the boundary and constant in the inward collar coordinate near the boundary. Thus the enriched data on this collar equal their boundary values. Glue the step-2.1 homotopy of the genuine boundary lift to its canonical model along that parameter collar, and use the canonical lift of the reparametrized enriched family on the remaining inner disk. The resulting genuine family extends the boundary lift and its enriched restriction is homotopic, relative to the boundary, to the original disk family. This proves the relative lifting-up-to-homotopy tests for $r_I$, so it induces isomorphisms on all homotopy groups and a bijection on components.

If the enriched derivative map ${\mathcal{I}}_E\to {\mathcal{F}}_E$ is a weak homotopy equivalence, the commuting forgetful-fibration square descends it to $Z_I\to Z_F$. Indeed the fibre map is the identity on $\mathrm{Aut}(E)$ by step 1.4 and both vertical maps are Serre fibrations by step 2.2. The exact sequences in [F3] give the isomorphisms on base homotopy groups by the usual exact diagram chase (including the nonabelian degree-one groups). For components, the fibre-component action identifies ${\pi }_0$ of each total space over a fixed base component with the orbits of the same ${\pi }_0\mathrm{Aut}(E)$; surjectivity on enriched components supplies every formal base component, and injectivity follows by lifting a formal base path, correcting its endpoint in the common gauge fibre, and using the enriched equivalence. Thus the base component map is a bijection as well. Each ordinary datum belongs to the locus obtained by taking $E$ to be its own normal bundle, and each path or based compact homotopy stays in that locus by step 2.2. Hence this proves descent on the full ordinary immersion and formal-immersion spaces.

Every component of $\mathrm{int}B$ is noncompact because $E$ has positive rank; its dimension is $n$. Assuming the open-source derivative theorem [given], [F3] makes the derivative map on $B$ a weak homotopy equivalence. Its commuting restriction square has horizontal maps $r_I,r_F$ that are weak homotopy equivalences by steps 1.1 and 3.1, so two-of-three gives the enriched derivative equivalence. Step 3.2 then gives the ordinary closed-source derivative equivalence. This proves the reduction, with the open-source theorem as its explicit prerequisite. For empty $M$ all spaces are singleton spaces and the conclusions hold directly.