---
tag: XZG6MTSWCN
kind: concept
name: Homotopy equivalence
msc: [55P10]
specializes: [0VGFHTM75Y]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy
---

## Summary

A homotopy equivalence is a continuous map $f \colon X \to Y$ that has an inverse up to homotopy.
This is a map $g \colon Y \to X$ with $g \circ f$ [[homotopic|F2F143KQ2M]] to $\id_X$ and $f \circ g$ homotopic to $\id_Y$.
It is a weaker notion than a [[homeomorphism|0QT00PB5G2]].
Homotopy invariants, such as the fundamental group and homology, cannot distinguish spaces that are related by a homotopy equivalence.

## Explanations

### L1 human

A homotopy equivalence says that two spaces have the same shape once you allow stretching, shrinking and squashing, but no tearing.
A thick object can be squashed to a thin one: a solid cylinder squashes to a disk, and further to a point.
For example, the plane without the origin is homotopy equivalent to the circle: push each point radially onto the unit circle.
The two spaces are not homeomorphic, but they have the same holes.