Compact manifolds are retracts of finite polyhedra without choice

Choose finitely many chart domains $V_i$ whose smaller concentric coordinate balls cover $M$, using compactness. For boundary charts use the relatively open unit half-ball $B_{+}^n=\{x:\mathrm{\mid }x\mathrm{\mid }<1,x_n\ge 0\}$; for interior charts use the unit ball. Denote the chart by ${\phi }_i$ and arrange that the radius-$1\mathrm{/}3$ smaller balls cover. Choose the explicit radial bump ${\rho }_i$ equal to $1$ at radius at most $1\mathrm{/}3$ and zero at radius at least $2\mathrm{/}3$, interpolating linearly in between, and extend it by zero outside $V_i$. Its support is compact in $V_i$. Define $e(x)=({\rho }_i(x),{\rho }_i(x){\phi }_i(x){)}_{i=1}^s\in {\mathbb{R}}^{s(n+1)},$ with the vector entry zero outside $V_i$. Boundedness of the chart coordinates makes these extensions continuous. If $e(x)=e(y)$, choose an index with ${\rho }_i(x)>0$. The equal scalar and vector entries recover ${\phi }_i(x)={\phi }_i(y)$ and hence $x=y$. A continuous injection of a compact space into the Hausdorff Euclidean space is an embedding. From now on identify $M$ with its image. All choices here form one finite list.

Write an ambient point in the $i$th block as $(t_i,z_i)$. On the open ambient set $O_i=\{t_i>0,\mathrm{\mid }{z}_i\mathrm{/}{t}_i\mathrm{\mid }<1\}$ define $q_i(t,z)=e({\phi }_i^{-1}(p_i(z_i\mathrm{/}{t}_i))),$ where $p_i$ is the identity for an interior chart and $p_i(x_1,\dots ,x_n)=(x_1,\dots ,x_{n-1},\max (0,x_n))$ for a boundary chart. The latter decreases norm and retracts the unit ball onto its relative half-ball. Thus $q_i:O_i\to e(V_i)$ is continuous and $q_i(x)=x$ for every $x\in M\cap O_i$. The compact sets $K_i=e(\{{\rho }_i\ge 1\mathrm{/}2\})$ cover $M$ and lie in $O_i$. Choose a positive ${\delta }_i$ with the closed ${\delta }_i$ neighborhood of $K_i$ contained in $O_i$; such a number exists by compactness and the open cover by small ambient balls. Put ${\beta }_i(z)=\max (0,{\delta }_i-\mathrm{dist}(z,K_i))$. Its closed support $S_i$ is compact and contained in $O_i$, and ${\sum }_i{\beta }_i>0$ on $M$. Empty $K_i$ may simply be discarded. No nearest-point selection is made: distance means the infimum of the set of distances.

We construct successive continuous maps $r_j:D_j\to M$, where $D_j=U_j\cap \{{\beta }_1+\cdots +{\beta }_j>0\}$ and $U_j$ is an open neighborhood of $M$. In addition require the following controlled property even at points outside $D_j$: for every $x\in M$ and every neighborhood $V$ of $x$ in $M$, there is an ambient neighborhood $A$ of $x$ with $r_j(A\cap D_j)\subseteq V$. Begin with $U_0={\mathbb{R}}^N$ and the empty map. Suppose the property holds for $r_{j-1}$. There is an open neighborhood $U_j\subseteq U_{j-1}$ of $M$ such that $r_{j-1}(U_j\cap S_j\cap D_{j-1})\subseteq e(V_j)\mathrm{.}$ Indeed, at $x\in M\cap S_j$ the chart $e(V_j)$ is a neighborhood of $x$, so the controlled property supplies the required ambient neighborhood. At $x\in M\setminus S_j$ an ambient neighborhood avoids the closed set $S_j$. Take the union of all open subsets of $U_{j-1}$ satisfying the displayed containment. This union contains $M$ and itself satisfies that containment. Taking the union of all admissible neighborhoods avoids selecting one for every point.

On $D_j$ put $a={\beta }_1+\cdots +{\beta }_{j-1}$ and $b={\beta }_j$. Where $b=0$ use $r_j=r_{j-1}$; where $a=0$ use $r_j=q_j$. If $a,b>0$, both $r_{j-1}$ and $q_j$ lie in $e(V_j)$ by step 3.1, so define $r_j(z)=e{\phi }_j^{-1}(\frac{a}{a+b}{\phi }_j{e}^{-1}{r}_{j-1}(z)+\frac{b}{a+b}{\phi }_j{e}^{-1}{q}_j(z))\mathrm{.}$ The chart image is a bounded convex ball or half-ball, so the convex combination remains in it. This map is continuous where both weights are positive. At a point where $a=0,b>0$, the first coordinate vector is uniformly bounded in the unit ball and its coefficient tends to zero; the expression therefore tends to the $q_j$ value regardless of whether $r_{j-1}$ has a limit there. At $b=0,a>0$, either a neighborhood avoids $S_j$, or the point is in $S_j$ and the containment from step 3.1 puts $r_{j-1}$ in $e(V_j)$ there. Continuity of $r_{j-1},q_j$ and the bounded coordinate formula then gives the stated $r_{j-1}$ value. This proves continuity on all of $D_j$.

The controlled property also persists. At $x\mathrm{\notin }{S}_j$ the map locally is the old map. At $x\in M\cap S_j$, choose a sufficiently small convex coordinate neighborhood of ${\phi }_j(x)$ in the ball or half-ball whose inverse lies in a prescribed neighborhood of $x$. The old controlled property puts all defined old values in this neighborhood for ambient points sufficiently near $x$; continuity of $q_j$ and $q_j(x)=x$ puts the new values there too. Every convex blend, as well as each endpoint case, remains in that neighborhood. This proves the property at $x$. Whenever $r_j$ is defined at a point of $M$, all contributing maps have value that point, so $r_j{\mathrm{\mid }}_{M\cap D_j}$ is the identity. Finite induction ends with $M\subset D_s$, since the sum of the weights is positive there. The continuous map $r_s:D_s\to M$ is therefore the required neighborhood retraction. The induction and neighborhood unions require no infinite selection.

By compactness, some positive $\epsilon$ has the closed $\epsilon$ neighborhood of $M$ contained in the open set $D_s$: finitely many ambient balls with doubled balls inside $D_s$ give a uniform positive minimum radius. Choose a rectangular grid with cube side less than $\epsilon \mathrm{/}\sqrt{N}$. There are only finitely many grid cubes meeting the bounded compact set $M$, and their union $L$ contains $M$ and lies in $D_s$, because each such cube has diameter less than $\epsilon$. Include all cube faces. This is a finite CW complex by [F1], and the inclusion $M\hookrightarrow L$ followed by $r_s{\mathrm{\mid }}_L$ is literally the identity on $M$. Empty $M$ is the empty retract. A compact zero-manifold is finite, and the same disjoint-neighborhood construction, or a finite set of points as $L$, handles that case without dividing by $\sqrt{0}$. For positive dimension the construction above applies to interior and boundary charts alike. Every noncanonical datum chosen in the proof belongs to a finite chart or parameter list. Thus the asserted embedding, retraction and finite domination hold without a choice axiom.