MHB Is Orientability Preserved by Local Diffeomorphisms?

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Here is this week's POTW:

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Suppose $f : M \to N$ is a local diffeomorphism between two smooth manifolds. Show that orientability of $N$ implies orientability of $M$.

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No one answered this week's problem. You can read my solution below.

Spoiler

Assume $N$ is orientable. Then $N$ can be expressed as the union of open sets of the form $x_\alpha(U_\alpha)$, where $x_\alpha : U_\alpha \to N$ is a local parametrization in $N$ such that whenever $W_{\alpha\beta}:= x_\alpha(U_\alpha) \cap x_\beta(U_\beta) \neq \emptyset$, the change of coordinates map $x_\beta^{-1}\circ x_\alpha : x_{\alpha}^{-1}(W_{\alpha\beta}) \to x_{\beta}^{-1}(W_{\alpha\beta})$ has positive Jacobian. Let $m\in M$, and $x_m : U_m \to N$ be a local parametrization with $x_m(U_m)\ni f(m)$. Since $f$ is a local diffeomorphism, there exists an open neighborhood $V_m$ of $m$ such that $f(V_m)$ is open in $N$. Consider the open set $Z_m:= x_m^{-1}(x_m(U_m) \cap f(V_m))$. Then $f^{-1}\circ x_m : Z_m \to f^{-1}(x_m(U_m) \cap f(V_m))\subset M$ is a parametrization in $M$.

The collection $\{(Z_m, f^{-1}\circ x_m)\}_{m\in M}$ of parametrizations make an orientable cover of $M$. For if $W_{mn} := Z_m \cap Z_n \neq \emptyset$, the change of variables map $(f^{-1}\circ x_m)^{-1}\circ (f^{-1}\circ x_n) : x_n^{-1}(f^{-1}(W_{mn})) \to x_m^{-1}(f^{-1}(W_{mn}))$ is the map $x_m^{-1}\circ f \circ f^{-1}\circ x_n = x_m^{-1}\circ x_n$, which has positive Jacobian by assumption. Therefore, $M$ is orientable.