Fundamental class

In mathematics, the fundamental class is a homology class [M] associated to an oriented manifold M of dimension n, which corresponds to the generator of the homology group ${\displaystyle H_{n}(M;\mathbf {Z} )\cong \mathbf {Z} }$ . The fundamental class can be thought of as the orientation of the top-dimensional simplices of a suitable triangulation of the manifold.

When M is a connected orientable closed manifold of dimension n, the top homology group is infinite cyclic: ${\displaystyle H_{n}(M,\mathbf {Z} )\cong \mathbf {Z} }$, and an orientation is a choice of generator, a choice of isomorphism ${\displaystyle \mathbf {Z} \to H_{n}(M,\mathbf {Z} )}$. The generator is called the fundamental class.

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