Let \(M\) be an \(n\)-manifold, and for \(A\subseteq M\) write \(H_n(M\mid A)\coloneqq H_n(M, M\sm A;\ZZ)\). For \(x\in M\), excision gives \(H_n(M\mid x)\cong H_n(\RR^n, \RR^n\sm\ts{0};\ZZ)\cong\ZZ\).
A local orientation of \(M\) at \(x\) is a generator \(\mu_x\) of \(H_n(M\mid x)\).
An orientation of \(M\) is a function \(x\mapsto\mu_x\) assigning to each \(x\in M\) a local orientation at \(x\), such that each \(x\in M\) has an open neighborhood \(U\subseteq M\) with a homeomorphism \(U\cong\RR^n\) and an open ball \(B\subseteq U\) of finite radius containing \(x\), together with a generator \(\mu_B\) of \(H_n(M\mid B)\cong\ZZ\) whose image under the map \(H_n(M\mid B)\to H_n(M\mid y)\) induced by inclusion is \(\mu_y\) for every \(y\in B\).
\(M\) is orientable if an orientation of \(M\) exists.
Each \(H_n(M\mid x)\) has exactly two generators, so \(M\) has exactly two local orientations at each point. The local orientations form the orientation double cover \(\tilde M = \ts{\mu_x \st x\in M,\ \mu_x \text{ a local orientation of } M \text{ at } x}\), with \(\mu_x\mapsto x\). The manifold \(\tilde M\) is orientable, and a connected manifold \(M\) is orientable if and only if \(\tilde M\) has two components [@Hat02].
Let \(M\) be a connected \(n\)-manifold with orientation double cover \(p\colon\tilde M\to M\), and let \(x\in M\). The following are equivalent:
\(M\) is orientable.
\(\tilde M\) is disconnected.
\(\tilde M\) has two components, each mapped homeomorphically onto \(M\) by \(p\).
\(p\) has a continuous section \(s\colon M\to\tilde M\), \(p\circ s = \id_M\).
The homomorphism \(\pi_1(M, x)\to\ZZ/2\) given by the action of \(\pi_1(M,x)\) on the two-point fiber \(\inverseof{p}(x)\) is trivial.
[@Hat02].