---
tag: YHK51GHK2T
kind: concept
name: Eilenberg–Steenrod axioms
msc: [55N40]
sources:
  - url: https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_axioms
same_as:
  wikidata: Q5349493
  wikipedia: Eilenberg–Steenrod axioms
---

## Summary

The Eilenberg–Steenrod axioms list the properties that a homology theory on pairs of spaces should have.
They are functoriality, homotopy invariance, a long exact sequence for each pair, excision, and a dimension axiom fixing the homology of a point.
[[Singular homology|M0Q5CFJ7ZH]] satisfies them, and on finite CW pairs any two theories that satisfy them with the same coefficient group agree.
Dropping the dimension axiom gives generalized homology and cohomology theories such as K-theory and cobordism.

## Explanations

### L1 human

There are many ways to build homology: from simplices, from cells, from singular chains.
The axioms say which properties really matter, so that one can prove things about all these constructions at once.
They act like a contract: any construction that keeps the contract gives the same answer on reasonable spaces.
For example, from the axioms alone one can compute that the $n$-sphere has homology only in degrees $0$ and $n$.