---
tag: SMEV5BSQ5S
kind: concept
name: Null-homotopic map
msc: [55Q05, 55P10]
specializes: [0VGFHTM75Y]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy
---

## Summary

A continuous map is null-homotopic when it is [[homotopic|F2F143KQ2M]] to a constant map.
Null-homotopic maps are the trivial maps of homotopy theory.
For example, a map $S^n \to Y$ is null-homotopic exactly when it extends over the disk $D^{n+1}$.

## Explanations

### L1 human

A map is null-homotopic when its image can be shrunk continuously to a single point inside the target.
Every map into a disk is null-homotopic.
Every loop in the plane is null-homotopic, but in the plane without the origin a loop that goes once around the origin is not.
Null-homotopic maps carry no information about the holes of the target.