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

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

.

It was later shown that the homotopy group is the infinite cyclic group. In 1951, Jean-Pierre Serre proved that the rational homotopy groups

is equal to 0 or n. However, for an even-dimensional sphere (n.

Definition

be a continuous map). Then we can form the cell complex

and zero everywhere else. Cellular (co-)homology is the (co-)homology of this chain complex), the cohomology is

Denote the generators of the cohomology groups by

. Thus, as a ring, the cohomology is

is the Hopf invariant.

Properties

Theorem. Moreover, the image of the Whitehead product is the Whitehead product.

for the Hopf maps 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.

Whitehead integral formula

J. H. C. Whitehead has proposed the following integral formula for the Hopf invariant., one considers a volume form, the pullback is a closed differential form. By Poincaré's lemma it is an exact differential form. The Hopf invariant is then given by

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:

its one-point compactification and

1. .

is any pointed space (as it is implicitly in the previous section), and if we take the point at infinity, 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 is

. If we let

the identity, then the Hopf invariant is defined by the following:

This map is initially a map from

.

References