Let \(M\) be a closed connected \(n\)-manifold with an orientation \(x\mapsto\mu_x\in H_n(M,M\setminus\ts{x};\ZZ)\). A fundamental class of \(M\) is a class \([M]\in H_n(M;\ZZ)\) whose image under \(H_n(M;\ZZ)\to H_n(M,M\setminus\ts{x};\ZZ)\) is \(\mu_x\) for every \(x\in M\).
Let \(M\) be a closed connected \(n\)-manifold with an orientation \(x\mapsto\mu_x\). Then \(H_n(M;\ZZ)\cong\ZZ\), the fundamental class \([M]\) exists and is unique, and it generates \(H_n(M;\ZZ)\) [@Hat02].