---
tag: ADDS6HE5SV
kind: concept
name: Relative homotopy
msc: [55Q05, 55P10]
specializes: [9YA6S6HD9V]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy
---

## Summary

A homotopy relative to a subspace $A$ is a [[homotopy|9YA6S6HD9V]] that keeps every point of $A$ fixed during the whole deformation.
It lets one deform maps while keeping a boundary condition, such as the end points of a path.
Path homotopy, the fundamental group and deformation retractions are all defined with relative homotopies.

## Explanations

### L1 human

Imagine a rubber band pinned to a board at some points.
You may move the band, but the pins never move.
A homotopy relative to $A$ is such a movement, where $A$ is the set of pinned points.
For example, two paths in the plane with the same end points can be deformed into each other with both end points held fixed.