---
tag: SK0ZJGWXDP
kind: concept
name: Homotopy type
msc: [55P10, 55P15]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy
---

## Summary

Two topological spaces have the same homotopy type when there is a [[homotopy equivalence|XZG6MTSWCN]] between them.
The homotopy type of a space is its class under this equivalence relation.
Most invariants of algebraic topology, such as homotopy groups, homology and cohomology, depend only on the homotopy type.
Classifying spaces up to homotopy type is a central aim of homotopy theory.

## Explanations

### L1 human

The homotopy type of a space is what is left of it after you allow any continuous squashing that can be undone up to deformation.
A solid ball, a line and a point all have the same homotopy type.
A circle, an annulus and a Möbius band also share one homotopy type, different from that of a point.
The homotopy type remembers the holes of a space but forgets its size and dimension.