# Hopf invariant

In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.

## Motivation

In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map

$\eta \colon S^{3}\to S^{2}$ ,

and proved that $\eta$  is essential, i.e., not homotopic to the constant map, by using the fact that the linking number of the circles

$\eta ^{-1}(x),\eta ^{-1}(y)\subset S^{3}$

is equal to 1, for any $x\neq y\in S^{2}$ .

It was later shown that the homotopy group $\pi _{3}(S^{2})$  is the infinite cyclic group generated by $\eta$ . In 1951, Jean-Pierre Serre proved that the rational homotopy groups

$\pi _{i}(S^{n})\otimes \mathbb {Q}$

for an odd-dimensional sphere ($n$  odd) are zero unless $i$  is equal to 0 or n. However, for an even-dimensional sphere (n even), there is one more bit of infinite cyclic homotopy in degree $2n-1$ .

## Definition

Let $\phi \colon S^{2n-1}\to S^{n}$  be a continuous map (assume $n>1$ ). Then we can form the cell complex

$C_{\phi }=S^{n}\cup _{\phi }D^{2n},$

where $D^{2n}$  is a $2n$ -dimensional disc attached to $S^{n}$  via $\phi$ . The cellular chain groups $C_{\mathrm {cell} }^{*}(C_{\phi })$  are just freely generated on the $i$ -cells in degree $i$ , so they are $\mathbb {Z}$  in degree 0, $n$  and $2n$  and zero everywhere else. Cellular (co-)homology is the (co-)homology of this chain complex, and since all boundary homomorphisms must be zero (recall that $n>1$ ), the cohomology is

$H_{\mathrm {cell} }^{i}(C_{\phi })={\begin{cases}\mathbb {Z} &i=0,n,2n,\\0&{\mbox{otherwise}}.\end{cases}}$

Denote the generators of the cohomology groups by

$H^{n}(C_{\phi })=\langle \alpha \rangle$  and $H^{2n}(C_{\phi })=\langle \beta \rangle .$

For dimensional reasons, all cup-products between those classes must be trivial apart from $\alpha \smile \alpha$ . Thus, as a ring, the cohomology is

$H^{*}(C_{\phi })=\mathbb {Z} [\alpha ,\beta ]/\langle \beta \smile \beta =\alpha \smile \beta =0,\alpha \smile \alpha =h(\phi )\beta \rangle .$

The integer $h(\phi )$  is the Hopf invariant of the map $\phi$ .

## Properties

Theorem: The map $h\colon \pi _{2n-1}(S^{n})\to \mathbb {Z}$  is a homomorphism. Moreover, if $n$  is even, $h$  maps onto $2\mathbb {Z}$ .

The Hopf invariant is $1$  for the Hopf maps, where $n=1,2,4,8$ , corresponding to the real division algebras $\mathbb {A} =\mathbb {R} ,\mathbb {C} ,\mathbb {H} ,\mathbb {O}$ , respectively, and to the fibration $S(\mathbb {A} ^{2})\to \mathbb {PA} ^{1}$  sending a direction on the sphere to the subspace it spans. It is a theorem, proved first by Frank Adams, and subsequently by Adams and Michael Atiyah with methods of topological K-theory, that these are the only maps with Hopf invariant 1.

## Generalisations for stable maps

A very general notion of the Hopf invariant can be defined, but it requires a certain amount of homotopy theoretic groundwork:

Let $V$  denote a vector space and $V^{\infty }$  its one-point compactification, i.e. $V\cong \mathbb {R} ^{k}$  and

$V^{\infty }\cong S^{k}$  for some $k$ .

If $(X,x_{0})$  is any pointed space (as it is implicitly in the previous section), and if we take the point at infinity to be the basepoint of $V^{\infty }$ , then we can form the wedge products

$V^{\infty }\wedge X$ .

Now let

$F\colon V^{\infty }\wedge X\to V^{\infty }\wedge Y$

be a stable map, i.e. stable under the reduced suspension functor. The (stable) geometric Hopf invariant of $F$  is

$h(F)\in \{X,Y\wedge Y\}_{\mathbb {Z} _{2}}$ ,

an element of the stable $\mathbb {Z} _{2}$ -equivariant homotopy group of maps from $X$  to $Y\wedge Y$ . Here "stable" means "stable under suspension", i.e. the direct limit over $V$  (or $k$ , if you will) of the ordinary, equivariant homotopy groups; and the $\mathbb {Z} _{2}$ -action is the trivial action on $X$  and the flipping of the two factors on $Y\wedge Y$ . If we let

$\Delta _{X}\colon X\to X\wedge X$

denote the canonical diagonal map and $I$  the identity, then the Hopf invariant is defined by the following:

$h(F):=(F\wedge F)(I\wedge \Delta _{X})-(I\wedge \Delta _{Y})(I\wedge F).$

This map is initially a map from

$V^{\infty }\wedge V^{\infty }\wedge X$  to $V^{\infty }\wedge V^{\infty }\wedge Y\wedge Y$ ,

but under the direct limit it becomes the advertised element of the stable homotopy $\mathbb {Z} _{2}$ -equivariant group of maps. There exists also an unstable version of the Hopf invariant $h_{V}(F)$ , for which one must keep track of the vector space $V$ .