I'm trying to prove this using purely topological arguments, no differential geometry as I haven't been exposed to it.
I've been playing around with definitions a bit and here's what I have so far.
Let $M$ be an orientable manifold. Let $N$ be its covering space. Then we have an orientation function $\mu : M \rightarrow \{\pm 1\}$ that satisfies:
$\forall x \in M, \exists U \cong D^n$ an open neighborhood of $x$ and $\mu_u$ a generator of $H_n(M, M-U) \cong \mathbb{Z}$ such that $\forall y \in U$ $H_n(M, M – {y}) \longleftarrow H_n(M, M – U) \longrightarrow H_n(M, M-{x})$ where each map is an isomorphism.
Now since $N$ is the covering space of $M$ we know that there is a function $p: N \rightarrow M$ surjective and continuous s.t. $\forall x \in U$ open, $p^-1(U)$ is a disjoint union of open sets $v_i \in N$ s.t. $p(v_i)$ is homeomorphic to $U$.
My conjecture is that $\mu \circ p$ is an orientation function on $N$ satisfying the compatibility conditions i.e. $\forall x \in N, \exists U \cong D^n$ an open neighborhood of $x$ and $\mu_u$ a generator of $H_n(N, N-U) \cong \mathbb{Z}$ such that $\forall y \in U$ $H_n(N, N – {y}) \longleftarrow H_n(N, N – U) \longrightarrow H_n(N, N-{x})$ where each map is an isomorphism.
I need some help moving forward from here. Is it perhaps true because for $x \in N$ we can choose a neighborhood $V$ s.t. $V$ is one of the open sets which maps homeomorphically onto some neighborhood $U$ of $x' \in M$ and we know that the compatibility conditions are true for any $y' \in U$ so it must be true for any $y \in V$?
Best Answer
You can use orientation coverings or orientation reversing curves (together with the observation, that every curve in the covering comes from a curve in your manifold) to prove your statement in one line. If you want to prove it by definition, as you are conjecturing correctly, $\mu$ gives rise to $\hat \mu$ on $\hat M$ by post-composing (here $\hat M \to M$ is the covering. (note that in nyour definition you forgot to mention the important fact that you need preferred generators on local homology i.e. you need that the maps, induced by inclusions map $\mu_U \mapsto \mu_x$).
To finish your proof consider the diagram: $$ \begin{array}{c} H_n(\hat M, \hat M-U) &\to& H_n(\hat M, \hat M - \{x\})\\ \downarrow && \downarrow\\ H_n(M,M-\pi(U)) &\to & H_n(M,M-\{\pi(x)\}), \end{array} $$ where $U$ is chosen such that $\pi|U$ is a homeomorphism and the lower horizontal map is as desired and sends preferred generator to preferred generators. Since such diagrams commute, you certainly obtain your $\hat \mu_x =\mu_{\pi(x)}$.