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

${\displaystyle \eta \colon S^{3}\to S^{2}}$ ,

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

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

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

It was later shown that the homotopy group ${\displaystyle \pi _{3}(S^{2})}$  is the infinite cyclic group generated by ${\displaystyle \eta }$ . In 1951, Jean-Pierre Serre proved that the rational homotopy groups ^[1]

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

for an odd-dimensional sphere (${\displaystyle n}$  odd) are zero unless ${\displaystyle i}$  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 ${\displaystyle 2n-1}$ .

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

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

where ${\displaystyle D^{2n}}$  is a ${\displaystyle 2n}$ -dimensional disc attached to ${\displaystyle S^{n}}$  via ${\displaystyle \phi }$ . The cellular chain groups ${\displaystyle C_{\mathrm {cell} }^{*}(C_{\phi })}$  are just freely generated on the ${\displaystyle i}$ -cells in degree ${\displaystyle i}$ , so they are ${\displaystyle \mathbb {Z} }$  in degree 0, ${\displaystyle n}$  and ${\displaystyle 2n}$  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 ${\displaystyle n>1}$ ), the cohomology is

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

Denote the generators of the cohomology groups by

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

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

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

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

Theorem: The map ${\displaystyle h\colon \pi _{2n-1}(S^{n})\to \mathbb {Z} }$  is a homomorphism. If ${\displaystyle n}$  is odd, ${\displaystyle h}$  is trivial (since ${\displaystyle \pi _{2n-1}(S^{n})}$  is torsion). If ${\displaystyle n}$  is even, the image of ${\displaystyle h}$  contains ${\displaystyle 2\mathbb {Z} }$ . Moreover, the image of the Whitehead product of identity maps equals 2, i. e. ${\displaystyle h([i_{n},i_{n}])=2}$ , where ${\displaystyle i_{n}\colon S^{n}\to S^{n}}$  is the identity map and ${\displaystyle [\,\cdot \,,\,\cdot \,]}$  is the Whitehead product.

The Hopf invariant is ${\displaystyle 1}$  for the Hopf maps, where ${\displaystyle n=1,2,4,8}$ , corresponding to the real division algebras ${\displaystyle \mathbb {A} =\mathbb {R} ,\mathbb {C} ,\mathbb {H} ,\mathbb {O} }$ , respectively, and to the fibration ${\displaystyle S(\mathbb {A} ^{2})\to \mathbb {PA} ^{1}}$  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.

J. H. C. Whitehead has proposed the following integral formula for the Hopf invariant.^{[2][3]: prop. 17.22} Given a map ${\displaystyle \phi \colon S^{2n-1}\to S^{n}}$ , one considers a volume form ${\displaystyle \omega _{n}}$  on ${\displaystyle S^{n}}$  such that ${\displaystyle \int _{S^{n}}\omega _{n}=1}$ . Since ${\displaystyle d\omega _{n}=0}$ , the pullback ${\displaystyle \varphi ^{*}\omega _{n}}$  is a Closed differential form: ${\displaystyle d(\varphi ^{*}\omega _{n})=\varphi ^{*}(d\omega _{n})=\varphi ^{*}0=0}$ . By Poincaré's lemma it is an exact differential form: there exists an ${\displaystyle (n-1)}$ -form ${\displaystyle \eta }$  on ${\displaystyle S^{2n-1}}$  such that ${\displaystyle d\eta =\varphi ^{*}\omega _{n}}$ . The Hopf invariant is then given by

${\displaystyle \int _{S^{2n-1}}\eta \wedge d\eta .}$ 

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

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

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

If ${\displaystyle (X,x_{0})}$  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 ${\displaystyle V^{\infty }}$ , then we can form the wedge products

${\displaystyle V^{\infty }\wedge X}$ .

Now let

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

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

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

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

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

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

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

This map is initially a map from

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

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

• Adams, J. Frank (1960), "On the non-existence of elements of Hopf invariant one", Annals of Mathematics, 72 (1): 20–104, CiteSeerX 10.1.1.299.4490, doi:10.2307/1970147, JSTOR 1970147, MR 0141119
• Adams, J. Frank; Atiyah, Michael F. (1966), "K-Theory and the Hopf Invariant", Quarterly Journal of Mathematics, 17 (1): 31–38, doi:10.1093/qmath/17.1.31, MR 0198460
• Crabb, Michael; Ranicki, Andrew (2006). "The geometric Hopf invariant" (PDF).
• Hopf, Heinz (1931), "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche", Mathematische Annalen, 104: 637–665, doi:10.1007/BF01457962, ISSN 0025-5831
• Shokurov, A.V. (2001) [1994], "Hopf invariant", Encyclopedia of Mathematics, EMS Press