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 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.
It was later shown that the homotopy group is the infinite cyclic group generated by. In 1951, Jean-Pierre Serre proved that the rational homotopy groups
for an odd-dimensional sphere are zero unless is equal to 0 or n. However, for an even-dimensional sphere, there is one more bit of infinite cyclic homotopy in degree.

Definition

Let be a continuous map. Then we can form the cell complex
where is a -dimensional disc attached to via.
The cellular chain groups are just freely generated on the -cells in degree, so they are in degree 0, and and zero everywhere else. Cellular homology is the homology of this chain complex, and since all boundary homomorphisms must be zero, 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. Thus, as a ring, the cohomology is
The integer is the Hopf invariant of the map.

Properties

Theorem: The map is a homomorphism. Moreover, if is even, maps onto.
The Hopf invariant is for the Hopf maps, where, corresponding to the real division algebras, respectively, and to the fibration sending a direction on the sphere to the subspace it spans. It is a theorem, proved first by Frank Adams and subsequently by 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 denote a vector space and its one-point compactification, i.e. and
If is any pointed space, and if we take the point at infinity to be the basepoint of, then we can form the wedge products
Now let
be a stable map, i.e. stable under the reduced suspension functor. The geometric Hopf invariant of is
an element of the stable -equivariant homotopy group of maps from to. Here "stable" means "stable under suspension", i.e. the direct limit over of the ordinary, equivariant homotopy groups; and the -action is the trivial action on and the flipping of the two factors on. If we let
denote the canonical diagonal map and 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 -equivariant group of maps.
There exists also an unstable version of the Hopf invariant, for which one must keep track of the vector space.