---
tag: EASHTX8EG5
kind: concept
name: H-space
msc: [55P45]
sources:
  - url: https://en.wikipedia.org/wiki/H-space
same_as:
  wikidata: Q1562471
  wikipedia: H-space
---

## Summary

An H-space is a pointed space with a continuous multiplication for which the basepoint is a two-sided unit up to homotopy.
Topological groups and [[loop spaces|2JFF5F3EGF]] are the main examples.
The multiplication gives the cohomology of an H-space extra structure, such as a [[Hopf algebra structure|WJXK0V9NE9]].
This strongly restricts which spaces can be H-spaces.
By a theorem of Adams, the only spheres that are H-spaces are $S^0$, $S^1$, $S^3$ and $S^7$.

## Explanations

### L1 human

A topological group has a multiplication with an exact identity element.
An H-space keeps only part of this: a multiplication with an identity that works up to deformation, and no inverses or associativity are required.
The circle $S^1$, seen as the unit complex numbers, is an H-space.
The sphere $S^7$, seen as the unit octonions, is an H-space although the octonions are not associative, so $S^7$ is not a group.