Let \(M\) be an \(n\)-manifold. An orientation of \(M\) is a function \(x\mapsto\mu_x\) assigning to each \(x\in M\) a local orientation \(\mu_x\), a generator of \(H_n(M, M\sm\ts{x};\ZZ)\cong\ZZ\), such that every \(x\in M\) has an open ball neighborhood \(B\) of finite radius in a chart \(\RR^n\subseteq M\) and a class \(\mu_B\in H_n(M, M\sm B;\ZZ)\) whose image under the map \(H_n(M, M\sm B;\ZZ)\to H_n(M, M\sm\ts{y};\ZZ)\) induced by inclusion is \(\mu_y\) for every \(y\in B\). An oriented manifold is a pair consisting of a manifold and an orientation of it. The manifold \(M\) is orientable if an orientation of \(M\) exists.
A nonempty connected orientable \(n\)-manifold has exactly two orientations.
Let \(M\) be a closed connected oriented \(n\)-manifold with orientation \(\mu\). There is a unique class \([M]\in H_n(M;\ZZ)\) whose image in \(H_n(M, M\sm\ts{x};\ZZ)\) is \(\mu_x\) for every \(x\in M\), and \([M]\) generates \(H_n(M;\ZZ)\cong\ZZ\).
The class \([M]\) is the fundamental class of \(M\).
See [@Hat02].