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 $H_{n}(M;\mathbf {Z} )\cong \mathbf {Z}$ . The fundamental class can be thought of as the orientation of the top-dimensional simplices of a suitable triangulation of the manifold.

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

For any abelian group $G$ and non negative integer $q\geq 0$ 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 .

Applications

In the Bruhat decomposition of the flag variety of a Lie group, the fundamental class corresponds to the top-dimension Schubert cell, or equivalently the longest element of a Coxeter group.

External links

Fundamental class at the Manifold Atlas.
The Encyclopedia of Mathematics article on the fundamental class.