Fundamental class

In mathematics, the fundamental class is a homology class [M] associated to a connected orientable compact manifold of dimension n, which corresponds to the generator of the homology group $H_{n}(M,\partial M;\mathbf {Z} )\cong \mathbf {Z}$ . 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 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, $H_{n}(M;\mathbf {Z} )\ncong \mathbf {Z}$ , and so one cannot define a fundamental class M living inside the integers. 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 so the notion of the fundamental class can be extended to the manifold with boundary case.

Poincaré duality

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

[M]\frown ~:H^{q}(M;G)\rightarrow H_{n-q}(M;G)

.

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

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

.

Using the notion of fundamental class for manifolds with boundary, we can extend Poincaré duality to that case too (see Lefschetz duality). In fact, the cap product with a fundamental class gives a stronger duality result saying that we have isomorphisms $H^{q}(M,A;R)\cong H_{n-q}(M,B;R)$ , assuming we have that $A,B$ are $(n-1)$ -dimensional manifolds with $\partial A=\partial B=A\cap B$ and $\partial M=A\cup B$ .^[1]