---
tag: K99YMX66XD
kind: definition
name: Fundamental class
msc: [57N65, 55N10]
defines: ZEA0PEM3B2
uses: [913WEQERBE, QQQ9B1DK4P, M0Q5CFJ7ZH, 4BVWH0BN28, 0TVJZYB6SH, 1DDYB6VVSR]
sources:
  - url: https://en.wikipedia.org/wiki/Fundamental_class
---

## Definition

Let $M$ be a closed [[topological manifold|QQQ9B1DK4P]] of dimension $n$, that is, a [[compact|0TVJZYB6SH]] manifold without boundary, and let $x \mapsto \mu_x$ be an [[orientation|913WEQERBE]] of $M$.
A *fundamental class* of $M$ is a class $[M] \in H_n(M; \Z)$ in [[singular homology|M0Q5CFJ7ZH]] with the following property.
For every $x \in M$, the map $H_n(M) \to H_n(M, M \setminus \set{x})$ induced by inclusion sends $[M]$ to $\mu_x$.

## 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|4BVWH0BN28]] 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|1DDYB6VVSR]].