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" 
and proved that formula_2 is essential, i.e., not homotopic to the constant map, by using the fact that the linking number of the circles 
is equal to 1, for any formula_4.
It was later shown that the homotopy group formula_5 is the infinite cyclic group generated by formula_2. In 1951, Jean-Pierre Serre proved that the rational homotopy groups 
for an odd-dimensional sphere (formula_8 odd) are zero unless formula_9 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 formula_10. 
Definition.
Let formula_11 be a continuous map (assume formula_12). Then we can form the cell complex
where formula_14 is a formula_15-dimensional disc attached to formula_16 via formula_17.
The cellular chain groups formula_18 are just freely generated on the formula_9-cells in degree formula_9, so they are formula_21 in degree 0, formula_8 and formula_15 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 formula_12), the cohomology is
Denote the generators of the cohomology groups by
For dimensional reasons, all cup-products between those classes must be trivial apart from formula_28. Thus, as a "ring", the cohomology is
The integer formula_30 is the Hopf invariant of the map formula_17.
Properties.
Theorem: The map formula_32 is a homomorphism. Moreover, if formula_8 is even, formula_34 maps onto formula_35.
The Hopf invariant is formula_36 for the "Hopf maps", where formula_37, corresponding to the real division algebras formula_38, respectively, and to the fibration formula_39 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 formula_40 denote a vector space and formula_41 its one-point compactification, i.e. formula_42 and 
If formula_45 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 formula_41, then we can form the wedge products 
Now let 
be a stable map, i.e. stable under the reduced suspension functor. The "(stable) geometric Hopf invariant" of formula_49 is
an element of the stable formula_51-equivariant homotopy group of maps from formula_52 to formula_53. Here "stable" means "stable under suspension", i.e. the direct limit over formula_40 (or formula_44, if you will) of the ordinary, equivariant homotopy groups; and the formula_51-action is the trivial action on formula_52 and the flipping of the two factors on formula_53. If we let 
denote the canonical diagonal map and formula_60 the identity, then the Hopf invariant is defined by the following:
This map is initially a map from 
but under the direct limit it becomes the advertised element of the stable homotopy formula_51-equivariant group of maps.
There exists also an unstable version of the Hopf invariant formula_65, for which one must keep track of the vector space formula_40.