Fundamental class

Summary

The fundamental class $[M]$ of a closed oriented $n$-manifold $M$ is a class in $H_n(M; \Z)$ that restricts to the chosen local orientation at every point. It represents the manifold itself as a single homology cycle. Capping with it gives Poincaré duality, and pairing top-degree cohomology classes with it gives numbers such as degrees of maps and characteristic numbers.