nLab
fundamental class

Definition

For manifolds

For $X$ a connected oriented closed manifold, its integral homology group in degree the dimension of $X$ is isomorphic to the integers

$$H_{\dim X}(X,\mathbb{Z})\simeq \mathbb{Z}.$$H_{dim X}(X, \mathbb{Z}) \simeq \mathbb{Z} \,.

The generator of this corresponding to the choice of orientation is called the fundamental class of $X$.

Virtual fundamental class

(…)

• virtual fundamental class.