Fundamental class

This is the standard setup.

Given topological spaces $N,M,A\subset N,B\subset M,$ and a map $f:(N,A)\rightarrow(M,B)$ between them, let

• $H_{k}(N,A),$ $H_{k}(M,B),$ $k=0,1,2...,$ be the homology groups of $M,N$ over $\mathbf{Z}$,
• $H^{k}(N,A),$ $H^{k}(M,B),$ $k=0,1,2...,$ be the cohomology groups of $M,N$ over $\mathbf{Z}$,
• $f_{\ast} :H_{k}(N,A)\rightarrow H_{k}(M,B)$ the homology operator, and
• $f^{\ast}:H^{k}(M,B)\rightarrow H^{k}(N,A)$ the cohomology operator of $f$.

Also keep in mind $H_{k}(M)=H_{k}(M,\varnothing).$

Suppose $M$ is a compact, oriented, $n$-dimensional manifold, then the $n$-th integral homology has dimension $1$: $$H_n(M)={\bf Z}.$$ Therefore there are two choices for a generator of this group. Once it's chosen it's called the fundamental class of $M$. It can be interpreted as a choice of "orientation" of $M$.

If $M$ is the realization of a cell complex, the fundamental class is the homology class represented by the sum of all of its $2$-cells, compatibly oriented. See especially orientable surface.

More generally one considers manifolds with boundary.

If $M$ is a compact orientable $n$-dimensional manifold with boundary $\partial M$ then $$H_{n}(M,\partial M)=H^{n}(M,\partial M)=\mathbf{Z},$$ generated by $O_{M},$ the fundamental class of $M$, and its dual $\overline{O}_{M},$ respectively.

Further, $$H_{0}(M)=H^{0}(M)=\mathbf{Z}.$$ The generators of these groups are denoted by $1.$

The Poincare duality isomorphism of a manifold is given by the cap product with the fundamental class $O_M$ of $M$: $$D(a)=a \frown O_M.$$ In particular it shows that $$H_{0}(M)\cong H^{n}(M)=\mathbf{Z}.$$