Let be a map with . Associated to this, one can form a CW complex ; that is, we attach a -cell to via the map . This CW complex has one cell in dimension and one cell in dimension (and one cell in dimension ). The map determines a generator of and the map determines a generator of ; there are no other elements in cohomology other than the unit. Consequently, we have

Definition 1The number as above such that is theHopf invariantof .

The homotopy type of determines only on the homotopy class of , so the Hopf invariant is a homotopy invariant.

**Example 1** The Hopf fibration is, by definition, the map such that the mapping cone is ; it follows that the Hopf fibration has Hopf invariant one.

The Hopf invariant is clearly identically zero for odd, but when is even the Hopf invariant is never identically zero; in fact, it defines a homomorphism

which for even has image containing the even integers. (This is where the exceptional summand in the homotopy groups of spheres comes from.)

A classical problem in topology was the following:

Question:For which does there exist a map of Hopf invariant one? (more…)