Euler characteristic of a finite covering of a compact smooth manifold

By [F1], choose a finite triangulation of $B$. Write $c_k$ for the number of its $k$-simplices. The triangulation supplier and the choices of lifts below require only finite choices; AC is retained solely as the declared ambient hypothesis.

Let $\varphi :D^k\to B$ be the characteristic map of one triangulation cell, choose $x_0\in D^k$, and form its pullback covering $q:{\tilde{D}}^k\to D^k$. For each of the $d$ points $e_0$ over $\varphi (x_0)$, and each $x\in D^k$, lift the radial path $t\mapsto \varphi ((1-t)x_0+tx)$ from $e_0$ using [F5], and define $s_{e_0}(x)$ to be its endpoint. To verify continuity, fix $x_{\ast }$ and consider the parameter map $F(x,t)=\varphi ((1-t)x_0+tx)$. Cover the compact path $F(\{x_{\ast }\}\times I)$ by evenly covered open sets. By [F6], choose a subdivision $0=t_0<\cdots <t_m=1$ so each closed segment $F(\{x_{\ast }\}\times [t_j,t_{j+1}])$ lies in one such set $U_j$. Continuity of $F$ and compactness of each segment allow one to shrink a neighborhood $N$ of $x_{\ast }$ so that $F(N\times [t_j,t_{j+1}])\subset U_j$ for every $j$. Starting with $e_0$, successively use the inverse homeomorphism on the sheet over $U_j$ that contains the already constructed lift at $t_j$; after finitely shrinking $N$ so each preceding endpoint remains in that open sheet, these formulas give a continuous lift on $N\times I$. Uniqueness in [F5] identifies it with the radial path lifts, so $s_{e_0}$ is continuous near $x_{\ast }$. The $d$ sections are disjoint by uniqueness: if two meet, lifting the reversed radial path from the meeting point forces their distinct initial points over $\varphi (x_0)$ to agree. They exhaust every fiber because a point over $x$ lifts the reversed radial path to one of the $d$ points over $\varphi (x_0)$. On a connected evenly covered neighborhood in $D^k$, each continuous section lies in one sheet, so its image is open and the restricted projection is a homeomorphism. Thus the pullback consists of exactly $d$ copies of $D^k$. Applying this cell by cell, and inducting over the finite skeleta so each lifted boundary map lands in the lifted lower skeleton, gives a finite CW structure on $E$ with $d{c}_k$ cells in each dimension $k$.

The lifted CW structure has alternating cell sum ${\sum }_k(-1{)}^{k}d{c}_k=d{\sum }_k(-1{)}^k{c}_k$. By [F2], this equals the homological Euler characteristic of each finite CW complex, so $\chi (E)=d\chi (B)$. The finite CW structure is built from finitely many compact closed cell images, so $E$ is compact.