Fundamental class of a manifold

Definition

Let $M$ be a compact connected orientable manifold of dimension $n$ and $R$ a commutative ring. A fundamental class for $M$ is an element of $H_{n}(M;R)$ whose image in $H_{n}(M,M\setminus x;R)$ is a generator (note that $H_{n}(M,M\setminus x;R)$ is clearly a free $R$-module of rank $1$).

A fundamental class corresponds to a choice of orientation.

Note that if $M$ is not compact or if $M$ is not orientable, no fundamental class exists. If $M$ is a union of connected components each of which is compact and orientable, we can mimic the above definition by defining a fundamental class of $M$ as a sum of fundamental classes of each of its connected components; however, the term fundamental class is typically reserved for connected manifolds.