---
tag: 8NAXJ0JH71
kind: concept
name: Homotopy group with coefficients
msc: [55Q05, 55Q70]
sources:
  - url: https://en.wikipedia.org/wiki/Homotopy_group_with_coefficients
same_as:
  wikidata: Q16928261
  wikipedia: Homotopy group with coefficients
---

## Summary

A homotopy group with coefficients in $\Z/k$ is the set of homotopy classes of maps from a mod $k$ [[Moore space|68MA0RM5P6]] into a space.
It plays the role for [[homotopy groups|P976WYEFMB]] that homology with coefficients plays for homology.
These groups sit in a short exact sequence with the ordinary homotopy groups and are used to study $p$-primary phenomena in homotopy theory.

## Explanations

### L1 human

An ordinary homotopy group records maps from a sphere into a space.
A homotopy group with coefficients replaces the sphere by a space that is a sphere with a disk glued on by a degree $k$ map.
Maps from that space see the parts of homotopy that are killed by $k$ and the classes modulo $k$.
For example, for the sphere $S^n$ with $n \ge 3$, the group $\pi_n(S^n; \Z/k)$ is cyclic of order $k$.