Compact smooth manifolds have finite CW models under countable choice

Empty manifolds have the empty CW model. A compact zero-dimensional manifold is finite, since its singleton open cover has a finite subcover, and has a finite discrete CW model. Hence assume $\dim W>0$. Use [L1] to collar the boundary. For the absolute model take $(W;\mathrm{\varnothing },\mathrm{\partial }W)$ and choose $f_0:W\to [0,1]$ equal to $1-t$ near the outgoing face, with all other values in $(0,1)$; cut off this collar formula to the constant $1\mathrm{/}2$ in the interior. When the boundary is empty use $f_0=1\mathrm{/}2$. For the relative model instead take $(W;\mathrm{\partial }W,\mathrm{\varnothing })$ and use $f_0=t$ near its incoming face. These functions have no critical point on a fixed boundary strip and the required boundary level sets.

Let $K$ be the compact complement of a smaller boundary strip, contained in the interior. Take finitely many coordinate charts with compact cores covering $K$ and bumps ${\rho }_i$ equal to one near those cores, supported in the interior, by [L2]. The smooth functions ${\rho }_i{x}_i^j$, extended by zero, have differentials spanning $T_p^{\ast }W$ on an open neighbourhood $O$ of $K$.

Put $f_t=f_0+\sum t_{ij}{\rho }_i{x}_i^j$ with parameter space ${\mathbb{R}}^N$. The section $(p,t)\mapsto d{f}_t(p)$ over $O\times {\mathbb{R}}^N$ is transverse to the zero section because its parameter derivatives span the fibre. Its zero set $Z$ is therefore a smooth $N$-manifold by [L3]. At a zero its tangent equation in local coordinates is $H_{p}v+B_p\tau =0$, where $H_p$ is the Hessian and $B_p:{\mathbb{R}}^N\to T_p^{\ast }W$ is onto. Thus the projection $Z\to {\mathbb{R}}^N$ is regular at $(p,t)$ exactly when $H_p$ is onto, equivalently invertible.

Apply Euclidean Sard in countably many charts of $Z$. Under [A1] their critical-value sets have null union (choose covers with budgets $ϵ2^{-i}$), so that union contains no open parameter ball. Take a regular parameter $t$ arbitrarily near zero. On the remaining compact boundary strip $d{f}_0$ is bounded away from zero in a metric built by a finite chart partition; a sufficiently small $t$ preserves this property. Its support misses a neighbourhood of the boundary, and a small perturbation keeps the interior values strictly between zero and one. Hence $f_t$ is adapted and Morse throughout $W$.

The critical set is finite by [L4]. Choose disjoint small critical-point neighbourhoods and bumps constant one near each critical point. Adding sufficiently small independent constants times those bumps preserves the critical points and their Hessians; on the compact transition annuli the differential was bounded away from zero, so it remains nonzero. Choose the constants to make the finitely many critical values distinct, retaining the boundary formulas and range. This gives an excellent adapted function $f$.

By [L7], choose Morse charts at the finitely many critical points and their prescribed negative Euclidean gradient fields. Away from those charts choose local fields with $df(X)<0$, and on the boundary collars take the descending collar direction. A partition as in [L1], equal to one near the critical points, glues these fields: strict negativity is preserved by convex combination. Append exterior collars, extend the collar fields, and cut off outside a compact neighbourhood of $W$. The resulting ambient field is complete by [L7] and has the adapted local models and boundary signs. Thus $(f,X)$ meets the pair hypotheses of [L5].

Apply [L5] to obtain a finite handle decomposition relative to the chosen incoming face. In the absolute construction this face is empty, and [L6] gives a finite CW model homotopy equivalent to $W$. In the relative construction the incoming face is the supplied finite CW boundary; [L6] gives a finite relative CW pair and an equivalence fixing that face, with its initial collar compressed onto it.

These models prove both assertions. Only finite selections and the countable chart, null-cover, collar, partition and completeness suppliers used ${\mathrm{A}\mathrm{C}}_{\omega }$.