Hopf Invariant - Generalisations For Stable Maps

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 .