Hopf invariant

In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.

MotivationEdit

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  .

DefinitionEdit

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  .

PropertiesEdit

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 Adams and Michael Atiyah with methods of topological K-theory, that these are the only maps with Hopf invariant 1.

Generalisations for stable mapsEdit

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  .

ReferencesEdit

  • Adams, J. Frank (1960), "On the non-existence of elements of Hopf invariant one", Annals of Mathematics, 72 (1): 20–104, CiteSeerX 10.1.1.299.4490, doi:10.2307/1970147, JSTOR 1970147, MR 0141119 CS1 maint: discouraged parameter (link)
  • Adams, J. Frank; Atiyah, Michael F. (1966), "K-Theory and the Hopf Invariant", Quarterly Journal of Mathematics, 17 (1): 31–38, doi:10.1093/qmath/17.1.31, MR 0198460
  • Crabb, Michael; Ranicki, Andrew (2006). "The geometric Hopf invariant" (PDF).
  • Hopf, Heinz (1931), "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche", Mathematische Annalen, 104: 637–665, doi:10.1007/BF01457962, ISSN 0025-5831
  • Shokurov, A.V. (2001) [1994], "Hopf invariant", Encyclopedia of Mathematics, EMS Press