Smooth manifolds have CW homotopy type

Given any sequence $(X_n{)}_{n\in \mathbb{N}}$ of nonempty sets, apply [A1] to its range $\{X_n:n\in \mathbb{N}\}$ and compose the resulting choice function with $n\mapsto X_n$. Thus the assumed AC implies [F1], which is the exact choice strength required by the published suppliers below.

Each point of $M$ has a chart into an open subset of ${\mathbb{R}}^n$ or a relatively open subset of the closed half-space by [F14] or [F15]. If $n\ge 1$, around its coordinate image choose a small open ball (intersected with the half-space when needed) whose closed ball lies in the chart image; this closed, bounded subset is compact by [F25]. Its chart preimage is compact because the chart is a homeomorphism (pull back any open cover), and it contains an open neighbourhood of the original point. Hence $M$ is locally compact; it is Hausdorff by the given hypothesis. If $n=0$, the local model is a point, which is itself a compact neighbourhood.

If $f:K\to M$ is continuous with $K$ compact Hausdorff, then $f(K)$ is compact: pull any open cover of $f(K)$ back to a cover of $K$ and take a finite subcover. Since $M$ is Hausdorff, [F12] makes $f(K)$ closed. By [F13], $M$ is weak Hausdorff.

For any second-countable locally compact Hausdorff space $X$, [F8] gives the closure-shrinking property, so [F10] makes $X$ regular; [F9] makes it Lindelöf under [F1], and [F11] then makes it paracompact. Applying this implication to $M$ proves its paracompactness.

Let $A\subseteq M$ be k-closed in the sense of [F13], and fix $x\mathrm{\notin }A$. By [F7], choose a compact neighbourhood $K$ of $x$ and an open $U$ with $x\in U\subseteq K$. The inclusion $K\hookrightarrow M$ is a compact Hausdorff test, so $A\cap K$ is closed in $K$. There is therefore an open $W\subseteq M$ with $W\cap K=K\setminus A$. Then $U\cap W$ is an open neighbourhood of $x$ disjoint from $A$. Thus every k-closed subset of $M$ is closed; the reverse implication follows by continuity of every test map, so $kM=M$. Hence $M$ is compactly generated and, with step 1.3, CGWH.

First suppose $M$ has no boundary. By [F2] and step 1.1 it admits a proper smooth embedding $i:M\hookrightarrow {\mathbb{R}}^N$, where $N=2n+1$.

Now suppose $\mathrm{\partial }M\mathrm{\ne }\mathrm{\varnothing }$. By [F4] take a collar $c:\mathrm{\partial }M\times [0,1)\to V\subseteq M$ with open image. By [F24], $\mathrm{int}M$ is open. Apply [F5] to the cover $\{V,\mathrm{int}M\}$ and let $\lambda$ be the partition function assigned to $V$. Its support lies in $V$, and $\lambda =1$ on $\mathrm{\partial }M$ because the other cover member misses the boundary. Put $\rho (u)=0$ for $u\le 0$ and $\rho (u)=e^{-1\mathrm{/}u}$ for $u>0$, and define $\chi (t)=\rho (\frac{1}{2}-t)\mathrm{/}(\rho (\frac{1}{2}-t)+\rho (t-\frac{1}{4}))$. Then $\chi =1$ near $0$ and $\chi =0$ for $t\ge \frac{1}{2}$. On the collar set $H_s(c(p,t))=c(p,t+\frac{s}{4}\lambda (c(p,t))\chi (t))$ and set $H_s(x)=x$ outside $V$. The new collar coordinate stays below $1$; because $\lambda$ has support contained in $V$, this formula glues continuously to the identity. For $s>0$ every boundary point moves into the interior, and every interior point stays there. Thus $r:=H_1$ maps $M$ to $\mathrm{int}M$, while $H_s$ gives both $ir\simeq {\mathrm{id}}_M$ and, on the interior, $ri\simeq {\mathrm{id}}_{\mathrm{int}M}$ for the inclusion $i:\mathrm{int}M\hookrightarrow M$.

Let $E\to M$ be a finite-rank real or complex smooth vector bundle. In the real case [F18] gives a linear trivializing cover. For a complex bundle presented as a real smooth bundle with smooth fiberwise $J^2=-I$, fix any point $p$ and a complex basis in $E_p$; extend its vectors to smooth local sections in a real trivialization. Those sections and their $J$-images remain real-linearly independent after shrinking, since their coordinate determinant is nonzero at $p$ and varies continuously. They form local complex frames; on overlaps the transition maps commute with $J$ and have smooth real matrix entries, so are smooth complex-linear trivializations. The pointwise construction selects no global family. If $M$ has boundary, [F5] supplies a locally finite smooth partition subordinate to either cover; otherwise [F6] does. Forgetting smoothness gives a topological linear trivializing cover, and [F20] makes the cover with its partition a numeration. This includes rank zero and the empty manifold, where the empty cover and empty partition satisfy the definition.

In the boundaryless branch of step 2.3, the image $i(M)$ is closed. Indeed, for $y\mathrm{\notin }i(M)$ take an open Euclidean ball $V$ about $y$ contained in a compact closed ball $K$, compact by [F25]. Properness means compact sets have compact preimages, so $i^{-1}(K)$ is compact; its image is compact and closed by [F12]. The open set $V\setminus i(i^{-1}(K))$ contains $y$ and misses $i(M)$, proving closedness. Now [F3] gives an open tubular neighbourhood $U$ of $i(M)$, diffeomorphic to a disk neighbourhood in its normal bundle. The homotopy $E(p,v)\mapsto E(p,(1-s)v)$, $0\le s\le 1$, is defined inside that disk neighbourhood and retracts $U$ onto $i(M)$. Thus in this branch $M\simeq U$.

In the boundaryless branch, the open set $U\subseteq {\mathbb{R}}^N$ from step 3.1 is second-countable, locally compact, and Hausdorff, so the implication proved in step 2.1 makes it paracompact. Let $\mathcal{U}$ be the set of all Euclidean open balls whose closed balls lie in $U$. This is an open cover; each nonempty finite intersection is convex and hence contractible by straight-line contraction to any point in that intersection. Since $U$ is itself a smooth manifold, [F6] supplies a subordinate partition of unity. Hatcher, Algebraic Topology, §4G, Proposition 4G.2 and Corollary 4G.3, then give $U\simeq \mathrm{\mid }N\mathcal{U}\mathrm{\mid }$: the proposition uses the subordinate partition, and the corollary uses the paracompactness and contractible finite intersections just verified.

In the boundaryless branch, the nerve $N\mathcal{U}$ is the abstract simplicial complex whose vertices are the balls and whose finite simplices are the subfamilies with nonempty intersection. In its realization, distinct points differ at some vertex coordinate; that coordinate is continuous by the weak topology, so disjoint intervals separate the points. The open simplices are cells, each closed simplex meets only its finitely many faces, and the weak topology in [F22] is exactly the closed-cell topology in [F23]. Attaching the simplices in increasing dimension therefore gives a CW structure on $\mathrm{\mid }N\mathcal{U}\mathrm{\mid }$. From steps 3.1 and 4.1, the boundaryless $M$ has the homotopy type of this CW complex.

The interior is a finite-dimensional Hausdorff second-countable smooth manifold without boundary by [F14] and restriction of the charts and smooth structure in [F16] and [F17]. Applying the boundaryless argument of steps 2.3, 3.1, 4.1, and 5.1 to $\mathrm{int}M$ gives it CW homotopy type; step 2.4 makes its inclusion into $M$ a homotopy equivalence. Therefore $M$ also has CW homotopy type.

Step 2.1 establishes paracompactness; steps 1.3 and 2.2 establish weak Hausdorffness and compact generation, while Hausdorffness is assumed. Steps 5.1 and 6.1 establish CW homotopy type in the boundaryless and boundary cases, and step 2.5 establishes numerability. Together with step 1.1, all AC-dependent supplier hypotheses are met, proving the statement.