---
tag: WQWEY93BKB
kind: definition
name: Null-homotopic map
msc: [55Q05, 55P10]
defines: SMEV5BSQ5S
uses: [F2F143KQ2M]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy
---

## Definition

Let $X$ and $Y$ be topological spaces.
A continuous map $f \colon X \to Y$ is *null-homotopic* if it is [[homotopic|F2F143KQ2M]] to a constant map $X \to Y$.

## Remarks

Example: every continuous map $f \colon X \to \R^n$ is null-homotopic, via $H(x, t) = (1 - t) f(x)$.
Non-example: the identity of the circle $S^1$ is not null-homotopic, since a constant map induces the trivial map on the [[fundamental group|S1XK3HXHFX]] while the identity induces the identity of $\Z$.
A map $f \colon S^n \to Y$ is null-homotopic if and only if it extends to a continuous map on the closed ball $D^{n+1}$.
A space $X$ is [[contractible|R5WP8Z69Q8]] if and only if it is nonempty and $\id_X$ is null-homotopic.