Fundamental class

For the fundamental class in class field theory, see class formation.

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

Definition

Closed, orientable

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

If M is disconnected (but still orientable), a fundamental class is the direct sum of the fundamental classes for each connected component (corresponding to an orientation for each component).

In relation with de Rham cohomology It represents a integration over M; namely for M a smooth manifold, an n-form ω can be paired with the fundamental class as

$\langle\omega, [M]\rangle = \int_M \omega\ ,$

which is the integral of ω over M, and depends only on the cohomology class of ω.

Stiefel-Whitney class

If M is not orientable, the homology group is not infinite cyclic : $H_n(M,\mathbf{Z}) \ncong \mathbf{Z}$ , one cannot define a orientation of M, Indeed, one cannot integrate differential n-forms over non-orientable manifolds.

However, every closed manifold is $\mathbf{Z}_2$-orientable, and $H_n(M;\mathbf{Z}_2)=\mathbf{Z}_2$ (for M connected). Thus every closed manifold is $\mathbf{Z}_2$-oriented (not just orientable: there is no ambiguity in choice of orientation), and has a $\mathbf{Z}_2$-fundamental class.

This $\mathbf{Z}_2$-fundamental class is used in defining Stiefel–Whitney class.

With boundary

If M is a compact orientable manifold with boundary, then the top relative homology group is again infinite cyclic $H_n(M,\partial M)\cong \mathbf{Z}$, and the notion of the fundamental class is extended to the relative case.

Poincaré duality

Main article: Poincaré duality

For any abelian group $G$ and non negative integer $q \ge 0$ one can obtain an isomorphism

$[M]\cap:H^q(M;G) \rightarrow H_{n-q}(M;G)$ .

using the cap product of the fundamental class and the $q$ -homology group . This isomorphism gives Poincaré duality:

$H^* (M; G) \cong H_{n-*}(M; G)$ .

Poincaré duality is extended to the relative case .