---
tag: 6TKGN8K8KT
kind: concept
name: Homology manifold
msc: [57P05, 55N10]
specializes: [0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Homology_manifold
same_as:
  wikidata: Q5891460
  wikipedia: Homology manifold
---

## Summary

A homology $n$-manifold is a space whose local homology at every point is that of a point in $\R^n$.
Every [[topological manifold|QQQ9B1DK4P]] is a homology manifold, but there are homology manifolds that are not manifolds.
Poincaré duality holds for compact oriented homology manifolds, so they are the natural setting for many results first proved for manifolds.

## Explanations

### L1 human

Near any point of an $n$-dimensional manifold, the space looks like $\R^n$.
Homology cannot see the whole shape of a neighbourhood, only some of its features, such as the holes left when the point is removed.
A homology manifold is a space where, near every point, these features are the same as in $\R^n$.
For example, take the cone over a space with the homology of a sphere but a nontrivial fundamental group.
It passes this test at its tip, though it is not a manifold there.