---
tag: TEGTSY7XMX
kind: concept
name: Hopf invariant
msc: [55Q25]
sources:
  - url: https://en.wikipedia.org/wiki/Hopf_invariant
same_as:
  wikidata: Q5900509
  wikipedia: Hopf invariant
---

## Summary

The Hopf invariant is an integer $H(f)$ attached to a map $f \colon S^{2n-1} \to S^n$.
It is read off from the [[cup product|JX47MS1R1H]] in the cohomology of the space obtained by attaching a $2n$-cell to $S^n$ along $f$.
It depends only on the homotopy class of $f$ and gives a homomorphism $\pi_{2n-1}(S^n) \to \Z$.
It showed that $\pi_3(S^2)$ is infinite, and the question of which $n$ admit maps of Hopf invariant one is tied to division algebras.

## Explanations

### L1 human

The Hopf invariant measures how much a map from a big sphere to a smaller sphere is twisted.
For the Hopf map $S^3 \to S^2$, the preimages of two different points are two circles that are linked once, and the Hopf invariant is the linking number, $\pm 1$.
For a constant map nothing is linked, and the invariant is $0$.