Hopf invariant

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

Motivation[edit]

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 linking number (=1) of the circles

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 i = 0 or n. However, for an even-dimensional sphere (n even), there is one more bit of infinite cyclic homotopy in degree .

Definition[edit]

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 .

Properties[edit]

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 fibration 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[edit]

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[edit]

  • 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