Smale–Hirsch for open source manifolds

If $M$ is empty, both spaces are singleton spaces. An open zero-manifold with no compact component is empty, since every singleton component is compact. Treat a nonempty connected positive-dimensional source first. Choose the compact filtration $K_j\subset \mathrm{int}{K}_{j+1}$ in [F1]. Each component of $K_0$ has nonempty boundary: a boundaryless codimension-zero component would be open and closed in the ambient connected noncompact manifold. Dual elimination gives a finite presentation with indices $k<m\le n$. Every later band has a finite presentation with the same strict inequality $k<m$. Hence all its handles satisfy the integration hypothesis $k<n$, also when $m=n$.

Specify the stage choices to avoid replacing countable choice by dependent choice. Fix beforehand countable coordinate and framed covers, target embedding/retraction, bump functions and mollifier translates for the source, and finite covers for the compact parameter manifold. Countable choice suffices for this countable fixed geometric data. On a finite stage, finite rational linear combinations of the fixed smooth bump translates form a countable dense family of Euclidean smooth corrections in the needed first-jet norm. Indeed first mollify a compactly supported correction; its differentiated convolution converges by [F4]. Approximate that integral and its first derivatives by a finite Riemann sum of bump translates, then approximate the finitely many centres and coefficients by rationals. Uniform continuity of the bump derivatives gives simultaneous first-jet approximation. Apply this on the finitely many coordinate pieces and paste with the fixed partition. The same argument approximates bundle-column corrections in zeroth order.

We extend genuine-family smoothing to noncompact $M$ before using it. Let $a(p,x)=e(g_p(x))$ for a continuous compact-parameter genuine family, smooth on a parameter neighbourhood $W$ of its relative set. The permitted ambient first jets are $(z,L)$ with $z$ in the tubular domain and $d{r}_{z}L$ injective. For each precompact source chart, compactness of its closure times $P$ gives a positive jet-error bound. A locally finite subordinate partition averages smaller such bounds to a positive continuous tolerance $\eta (x)$ valid at every $p$: at $x$ the average is at most the largest active bound, which is valid at that point. Fix a countable locally finite smooth partition $({\lambda }_j)$ with compact supports $C_j$. Mollify $a$ only in the parameter variables, using the tubular projection on $P_0$ and interval clamps as in the compact smoothing proof. Choose a constant radius ${\delta }_j$ for each support so that its value and first source-derivative errors are less than $2^{-j-4}{\min }_{C_j}\eta \mathrm{/}(1+{\sup }_{C_j}\mathrm{\mid }d{\lambda }_j\mathrm{\mid })$. These are independent countably many choices; least sufficiently small dyadic radii suffice. Put $b={\sum }_j{\lambda }_j{a}_{{\delta }_j}$. Since ${\sum }_{j}d{\lambda }_j=0$, $d_{x}b-d_{x}a={\sum }_j{\lambda }_j(d_x{a}_{{\delta }_j}-d_{x}a)+{\sum }_{j}d{\lambda }_j(a_{{\delta }_j}-a)\mathrm{.}$ Both this error and $b-a$ are smaller than $\eta (x)\mathrm{/}4$, by the displayed bounds and the convergent geometric sum. Thus the straight ambient homotopy from $a$ to $b$, followed by $r$, remains immersive everywhere. For the relative version replace $b$ by $\chi (p)a+(1-\chi (p))b$, with $\chi =1$ near the relative set and supported in $W$; there is no source derivative of $\chi$. This endpoint is jointly smooth, agrees with the original family near the relative set, and the homotopy is weakly continuous in every source derivative by local finiteness. Empty supports are omitted. The same estimates allow source cutoffs on compact stage constructions: their derivatives contribute bounded multiples of the value error. No uniform global rank margin or global convolution radius is required.

Start with the original global formal family, smoothed relative to a fixed parameter neighbourhood of $Q$ by [F3]. Apply [F2] over the finite initial presentation to make it genuine on $K_0$ and a small outward source collar. Choose once a compact collar enlargement $V_0$ in that genuine region. Inductively integrate the finite presentation of the next band relative to the genuine neighbourhood of $V_j$, retaining a smaller collar around its incoming face, and end genuine on a collar enlargement $V_{j+1}$ of $K_{j+1}$. Adding or shortening the incoming collar changes neither source dimension nor the handle indices. The relative construction fixes $V_j$ pointwise, and we retain it as a common fixed neighbourhood at every subsequent stage. The fixed parameter neighbourhood of $Q$ is also retained at every stage.

Enumerate candidate relative formal homotopies from those corrections, retaining the current input exactly near initial time and on the frozen source and parameter neighbourhoods. Use a fixed cutoff zero on the relative region, with its transition supported where an existing smooth witness is already unchanged. Precompose every candidate with a fixed smooth temporal reparametrization constant on both endpoint collars. Near final time replace the candidate bundle columns by the differential of its candidate final base map only on a source neighbourhood of the compact stage being holonomized. Use a temporal cutoff equal to one near final time, multiplied by a source cutoff $\zeta$ equal to one near that stage and supported in the final genuine collar of a successful witness. Outside this collar retain the candidate formal columns; in particular the original possibly nonholonomic data outside the compact support are unchanged. The finite presentation may include a fixed small outgoing collar, so such $\zeta$ can be included in the enumerated stage masks. Project Euclidean columns to the tangent bundle of the retracted candidate base map. At initial time this is the current formal datum; on the fixed regions it is unchanged because those data are genuine; at final time it is holonomic on the prescribed compact stage and its smaller outgoing collar by construction. A successful relative stage witness exists by [F2]. Smooth its parameter and time variables using [F3], relative to the frozen source and parameter neighbourhoods, then impose final holonomicity by the same endpoint derivative replacement. Compact first-jet margins make these operations rank-preserving. Make that witness constant in temporal endpoint collars. The candidate endpoint replacement is supported within its constant final-time collar, so candidate final derivatives and candidate bundle columns lie over the same base map. The density from step 1.2 approximates it closely enough that every candidate column map remains injective on the compact parameter-source-time set, the endpoint remains immersive, and the maps stay in the tubular domain. Source cutoffs contribute only their bounded derivatives times the small zeroth-order errors. Thus at least one candidate satisfies the finite-stage conditions. Select the least such candidate in the fixed natural-number enumeration. Likewise choose the least dyadic collar width and least integer subdivision satisfying the explicit compact bounds in the core lifting formulas. This is a uniquely defined selection rule depending on the current input, so ordinary recursion defines the entire stage sequence. There is no invocation of dependent choice to select unspecified successive deformation witnesses.

These are global formal homotopies with compact source support. The handle construction of [F2] is defined on an ambient neighbourhood of its compact handle: its cocore compression and local-addition formulas extend to slightly larger cocore disks, while its attaching region is already the given reference germ. Since $k<m$, the nonattaching faces have a cocore collar available; the finitely many formulas therefore extend into an outgoing buffer contained in a later $K_i$. Use the parameter-time smoothing of step 2.2 before this construction; source-dependent time substitution then remains source-smooth. On that buffer cut off the homotopy time by a source function $\chi$ equal to one near the processed compact stage and zero outside the buffer: use $(f_{t\chi (x)},F_{t\chi (x)})$. This is still a formal immersion because the same time is used in the base map and its fibre monomorphism; a formal bundle map need not equal the derivative of its base map on the transition region. Where $\chi =1$ the new family is genuine, and outside the buffer it is unchanged. At an incoming frozen neighbourhood no cutoff modification is made. Thus each stage has a global endpoint and all stages concatenate exactly. No arbitrary all-boundary-jet extension or dependent limit of local section margins is used.

Put the stage homotopies on $[t_j,t_{j+1}]$, where $t_j=1-2^{-j}$, and use a smooth reparametrization constant near both endpoints. Define the family at $t=1$ by its eventual value at each source point. Every compact source set $C$ lies inside some $V_j$ and all later homotopies are fixed on a common open neighbourhood of $C$. Hence each spatial derivative on $P\times C$ is stationary for all sufficiently late times; in particular it is jointly continuous at $t=1$. By [F3] the full homotopy is weakly continuous, its endpoint is smooth in the source and genuine everywhere, and it is relative to $Q$. In the smooth version, every source point has a neighbourhood on which the evaluation map is independent of $t$ near $1$, so the evaluation map is smooth there as well as at the finitely many earlier joins. This proves the stated smooth relative form too.

For surjectivity, represent a based formal homotopy class by a sphere family and collapse a small parameter neighbourhood of the marked point to that point. This precomposition is homotopic to the identity fixing the marked point and makes the family holonomic and smooth there; step 4.1 then gives a genuine representative with that basepoint fixed. For injectivity in a based test, first perform the same collapse near the marked boundary point and extend its formal derivative homotopy over the disk by [F5]. The resulting genuine boundary family is constant near that point. An arbitrary continuous genuine boundary family of a formal disk can then be smoothed through genuine families by step 1.3, relative to a constant neighbourhood of its marked point when a based test requires one. Extend its formal derivative homotopy to the disk by the finite CW boundary HEP of [F5]. Reparametrize the resulting disk radially to be constant in the inward collar coordinate near its boundary; its formal data there are the derivatives of the now smooth genuine boundary family. Step 4.1 applies relative to that collar and produces a genuine filling of the smoothed boundary. Attach the reverse preliminary genuine boundary homotopy as an annulus to recover a filling of the original boundary family. These are continuous genuine families on the entire noncompact source, so they prove injectivity in every degree, including degree one. The interval test gives the component comparison in the same way. All based corrections fix the marked genuine basepoint. This is weak equivalence, without requiring a compact parameter map to factor through a finite source stage.

A second-countable manifold has at most countably many components, because each open component contains a distinct member of a fixed countable basis. Mapping spaces split as products over them in the weak topology: a compact source set meets only finitely many components. Apply the deterministic construction componentwise using the fixed countable geometric data; coordinatewise products of the resulting maps and homotopies are therefore weakly continuous. Homotopy groups of products are computed by coordinate maps and homotopies, and the countable choice premise supplies the componentwise component representatives where needed. This proves both the weak-equivalence and relative-family assertions on all of $M$. Only the fixed countable geometric selections and the explicitly countable-choice suppliers use ${\mathrm{A}\mathrm{C}}_{\omega }$; the recursive deformation choices use the least-candidate rule.