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Concept
Hopf invariant
Summary
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.
TOC
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 .
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 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 .
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 .
Theorem: The map is a homomorphism.
If is odd, is trivial (since is torsion).
If is even, the image of contains . Moreover, the image of the Whitehead product of identity maps equals 2, i. e. , where is the identity map and is the Whitehead product.
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 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.
Official source
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