---
tag: AXT498MW87
kind: definition
name: Hopf invariant
msc: [55Q25]
defines: TEGTSY7XMX
uses: [P976WYEFMB, 6N7BC70CX0, JX47MS1R1H, 5W95QFHXPR]
sources:
  - url: https://en.wikipedia.org/wiki/Hopf_invariant
---

## Definition

Let $n \ge 2$, and let $f \colon S^{2n-1} \to S^n$ be a continuous map.
Let $C_f = S^n \cup_f D^{2n}$ be the [[CW complex|5W95QFHXPR]] obtained by attaching a $2n$-cell to $S^n$ along $f$.
Its [[singular cohomology|6N7BC70CX0]] with integer coefficients is $H^k(C_f; \Z) \cong \Z$ for $k = 0, n, 2n$ and is zero otherwise.
Let $\alpha \in H^n(C_f; \Z)$ and $\beta \in H^{2n}(C_f; \Z)$ be the generators given by fixed orientations of $S^n$ and $D^{2n}$.
The *Hopf invariant* of $f$ is the integer $H(f)$ such that
$$\alpha \smile \alpha = H(f) \, \beta,$$
where $\smile$ is the [[cup product|JX47MS1R1H]].

## Remarks

The integer $H(f)$ depends only on the homotopy class of $f$, and $H$ is a homomorphism $\pi_{2n-1}(S^n) \to \Z$ on [[homotopy groups|P976WYEFMB]].
A change of the generator $\beta$ changes only the sign of $H(f)$.

Example: for the Hopf map $h \colon S^3 \to S^2$, the space $C_h$ is $\C P^2$.
The square of a generator of $H^2(\C P^2; \Z)$ generates $H^4(\C P^2; \Z)$, so $H(h) = \pm 1$.

Non-example: for $n = 1$ the construction breaks down.
A map $S^1 \to S^1$ of degree $d$ gives $H^2(C_f; \Z) \cong \Z/d$, which is not $\Z$ when $d \ne 0$, so no generator $\beta$ as above exists.

If $n$ is odd, then $\alpha \smile \alpha = -\alpha \smile \alpha$ by graded commutativity, and since $H^{2n}(C_f; \Z) \cong \Z$ has no torsion, $H(f) = 0$.
If $n$ is even, the Whitehead product $[\iota_n, \iota_n]$ of the identity of $S^n$ with itself has Hopf invariant $\pm 2$.