Let \(M\) be an \(n\)-manifold and \(x\in M\). Then \(H_n(M, M\sm\ts{x};\ZZ)\cong H_n(\RR^n, \RR^n\sm\ts{0};\ZZ)\cong\ZZ\).
Let \(M\) be an \(n\)-manifold and \(x\in M\). A local orientation of \(M\) at \(x\) is a choice of generator \(\mu_x\) of the infinite cyclic group \(H_n(M, M\sm\ts{x};\ZZ)\).
See [@Hat02].