In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in particular in algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

, the Hopf invariant is a homotopy

Homotopy

In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

 invariant of certain maps between sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

s.

Motivation

In 1931 Heinz Hopf

Heinz Hopf

Heinz Hopf was a German mathematician born in Gräbschen, Germany . He attended Dr. Karl Mittelhaus' higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age...

 used Clifford parallel

Clifford parallel

A Clifford parallel is a line which lies at a constant distance from some "base" line but, unlike an ordinary parallel line, does not lie in the same plane...

s to construct the Hopf map,

and proved that is essential, i.e. not homotopic to the constant map, by using the linking number (=1) of the circles for any .

It was later shown that the homotopy group

Homotopy group

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...

  is the infinite cyclic group

Cyclic group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

 generated by . In 1951, Jean-Pierre Serre

Jean-Pierre Serre

Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...

 proved that the rational homotopy groups

for an odd-dimensional sphere ( odd) are zero unless i = 0 or n. However, for an even-dimensional sphere (n even), there is one more bit of infinite cyclic homotopy in degree . There is an interesting way of seeing this:

Definition

Let be a continuous map (assume ). 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 (co-)homology is the (co-)homology of this chain complex

Chain complex

In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...

, and since all boundary homomorphisms must be zero (recall that ), the cohomology is

Denote the generators of the cohomology groups by

and

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: 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 double cover sending a direction on the sphere to the subspace it spans). It is a theorem, proved first by Frank Adams

Frank Adams

John Frank Adams FRS was a British mathematician, one of the founders of homotopy theory.-Life:He was born in Woolwich, a suburb in south-east London. He began research as a student of Abram Besicovitch, but soon switched to algebraic topology. He received his Ph.D. from the University of...

 and subsequently by Michael Atiyah

Michael Atiyah

Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...

 with methods of K-theory

K-theory

In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...

, 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 for some .

If 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 , 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 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 (or , if you will) 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 to ,

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 .