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