and proved that is essential, i.e. not homotopic to the constant map, by using the linking number (=1) of the circles
- for any .
for an odd-dimensional sphere ( odd) are zero unless i = 0 or n. However, for an even-dimensional sphere (n even), there is one more bit of infinite cyclic homotopy in degree . There is an interesting way of seeing this:
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
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 .
Theorem: 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 double cover sending a direction on the sphere to the subspace it spans). It is a theorem, proved first by Frank Adams and subsequently by Michael Atiyah with methods of topological K-theory, that these are the only maps with Hopf invariant 1.
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
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 .
- Adams, J.F. (1960), "On the non-existence of elements of Hopf invariant one", Ann. Math. (The Annals of Mathematics, Vol. 72, No. 1) 72 (1): 20–104, doi:10.2307/1970147, JSTOR 1970147
- Adams, J.F.; Atiyah, M.F. (1966), "K-Theory and the Hopf Invariant", The Quarterly Journal of Mathematics 17 (1): 31–38, doi:10.1093/qmath/17.1.31
- Crabb, M.; Ranicki, A. (2006), The geometric Hopf invariant
- 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), "Hopf invariant", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4