---
tag: HK8NS02NNE
kind: concept
name: Homology sphere
msc: [57N65, 57K30]
sources:
  - url: https://en.wikipedia.org/wiki/Homology_sphere
same_as:
  wikidata: Q1502277
  wikipedia: Homology sphere
---

## Summary

A homology $n$-sphere is a closed $n$-manifold with the same integral homology groups as the [[$n$-sphere|T62369RFH6]].
The first example that is not a sphere is the Poincaré homology sphere, a $3$-manifold with a nontrivial fundamental group of order $120$.
Homology spheres show that homology alone does not determine a manifold, and they were central in the history of the Poincaré conjecture.

## Explanations

### L1 human

Homology counts holes of each dimension.
A homology sphere has exactly the holes of an ordinary sphere and no others.
Still, it can differ from the sphere: it may contain loops that cannot be shrunk to a point, though each such loop has trivial homology class.
The Poincaré homology sphere, obtained by gluing opposite faces of a solid dodecahedron with a twist, is the classical example.