---
tag: NJQMWVK3QK
kind: concept
name: Reduced homology
msc: [55N10]
sources:
  - url: https://en.wikipedia.org/wiki/Reduced_homology
same_as:
  wikidata: Q7306331
  wikipedia: Reduced homology
---

## Summary

Reduced homology is a small change of [[singular homology|M0Q5CFJ7ZH]] that makes the homology of a point vanish in every degree.
It differs from ordinary homology only in degree $0$, where one copy of $\Z$ is removed.
Many statements, such as the homology of spheres, the suspension isomorphism and Alexander duality, take their cleanest form in reduced homology.

## Explanations

### L1 human

Ordinary homology always sees at least one copy of $\Z$ in degree $0$ for a nonempty space, because the space has at least one piece.
Reduced homology throws this copy away, so that it only records how far the space is from being a single point.
For a space made of three separate points, $H_0$ is $\Z^3$, while the reduced group $\tilde H_0$ is $\Z^2$.
For the sphere $S^n$, the reduced homology is $\Z$ in degree $n$ and zero in all other degrees, with no special case for $n = 0$.