In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.
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
is the infinite cyclic group generated by
In 1951, Jean-Pierre Serre proved that the rational homotopy groups [1] for an odd-dimensional sphere (
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
be a continuous map (assume
Then we can form the cell complex where
The cellular chain groups
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 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
Moreover, the image of the Whitehead product of identity maps equals 2, i. e.
, corresponding to the real division algebras
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.
J. H. C. Whitehead has proposed the following integral formula for the Hopf invariant.[2][3]: prop.
By Poincaré's lemma it is an exact differential form: there exists an
The Hopf invariant is then given by 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
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
-equivariant homotopy group of maps from
, if you will) of the ordinary, equivariant homotopy groups; and the
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 but under the direct limit it becomes the advertised element of the stable homotopy
There exists also an unstable version of the Hopf invariant
, for which one must keep track of the vector space