---
tag: ZEA0PEM3B2
kind: concept
name: Fundamental class
msc: [57N65, 55N10]
sources:
  - url: https://en.wikipedia.org/wiki/Fundamental_class
same_as:
  wikidata: Q4493388
  wikipedia: Fundamental class
---

## Summary

The fundamental class $[M]$ of a closed oriented $n$-manifold $M$ is a class in $H_n(M; \Z)$ that restricts to the chosen local [[orientation|913WEQERBE]] at every point.
It represents the manifold itself as a single homology cycle.
Capping with it gives Poincaré duality, and pairing top-degree cohomology classes with it gives numbers such as degrees of maps and characteristic numbers.

## Explanations

### L1 human

Cut a closed oriented surface into triangles and orient them all the same way.
The sum of all these triangles is a cycle, because each edge appears twice with opposite signs and cancels.
This cycle is the fundamental class: the whole surface counted once.
For a connected closed oriented manifold it generates the top homology group, which is $\Z$.