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 ( odd) are zero unless   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 .

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, 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: The map  is a homomorphism. 
If  is odd,  is trivial (since  is torsion).
If  is even, the image of  contains . Moreover, the image of the Whitehead product of identity maps equals 2, i. e. , where  is the identity map and  is the Whitehead product.

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 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.
Given a map , one considers a volume form  on  such that .
Since , the pullback  is a Closed differential form: .
By Poincaré's lemma it is an exact differential form: there exists an -form  on  such that . 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:

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 .

References

 
 
 
 

Homotopy theory