Fundamental class

Remarks

For a closed oriented $n$-manifold a fundamental class exists and is unique. If $M$ is also connected, then $H_n(M; \Z) \cong \Z$, generated by $[M]$. With $\Z/2$ coefficients every closed manifold, orientable or not, has a unique fundamental class in $H_n(M; \Z/2)$. The groups $H_n(M, M \setminus \set{x})$ are relative homology groups.

Example: for $M = S^1$ with its counterclockwise orientation, the loop $\sigma \colon [0, 1] \to S^1$, $\sigma(t) = e^{2\pi i t}$, is a singular $1$-cycle. Its class generates $H_1(S^1) \cong \Z$ and restricts to the counterclockwise local generator at each point, so $[S^1] = [\sigma]$.

Non-example: the compactness hypothesis is needed, since $H_n(\R^n) = 0$ and so $\R^n$ has no fundamental class. The orientation hypothesis is needed with $\Z$ coefficients, since $H_2(\R P^2; \Z) = 0$ for the real projective plane.