Let \(M\) be an \(n\)-manifold, and for \(A\subseteq M\) write \(H_n(M\mid A) \coloneqq H_n(M, M\sm A;\ZZ)\). The orientation double cover of \(M\) is the set \(\tilde M\) of all local orientations \(\mu_x\) of \(M\) at points \(x\in M\), with the map \(p\colon\tilde M\to M\), \(\mu_x\mapsto x\), and the topology with basis the sets \[ U(\mu_B) \coloneqq \ts{ \mu_x \st x\in B,\ \mu_x \text{ the image of } \mu_B \text{ under } H_n(M\mid B)\to H_n(M\mid x) }, \] where \(B\) ranges over the open balls of finite radius in charts \(\RR^n\subseteq M\) and \(\mu_B\) over the generators of \(H_n(M\mid B)\cong\ZZ\).
Let \(M\) be an \(n\)-manifold with orientation double cover \(p\colon\tilde M\to M\). Then \(p\) is a two-sheeted covering space, and \(\tilde M\) is an orientable \(n\)-manifold. If \(M\) is connected, then \(M\) is orientable if and only if \(\tilde M\) has two components.
[@Hat02].