In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.
Motivation
editIn 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 [1]
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 .
Definition
editLet 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
editTheorem: 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.
Whitehead integral formula
editJ. H. C. Whitehead has proposed the following integral formula for the Hopf invariant.[2][3]: prop. 17.22 Given a map , one considers a volume form on such that . Since , the pullback is a closed differential form: . By Poincaré's lemma it is an exact differential form: there exists an -form on such that . The Hopf invariant is then given by
Generalisations for stable maps
editA 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- ^ Serre, Jean-Pierre (September 1953). "Groupes D'Homotopie Et Classes De Groupes Abeliens". The Annals of Mathematics. 58 (2): 258–294. doi:10.2307/1969789. JSTOR 1969789.
- ^ Whitehead, J. H. C. (1 May 1947). "An Expression of Hopf's Invariant as an Integral". Proceedings of the National Academy of Sciences. 33 (5): 117–123. Bibcode:1947PNAS...33..117W. doi:10.1073/pnas.33.5.117. PMC 1079004. PMID 16578254.
- ^ Bott, Raoul; Tu, Loring W (1982). Differential forms in algebraic topology. New York. ISBN 9780387906133.
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- Adams, J. Frank (1960), "On the non-existence of elements of Hopf invariant one", Annals of Mathematics, 72 (1): 20–104, CiteSeerX 10.1.1.299.4490, doi:10.2307/1970147, JSTOR 1970147, MR 0141119
- Adams, J. Frank; Atiyah, Michael F. (1966), "K-Theory and the Hopf Invariant", Quarterly Journal of Mathematics, 17 (1): 31–38, doi:10.1093/qmath/17.1.31, MR 0198460
- Crabb, Michael; Ranicki, Andrew (2006). "The geometric Hopf invariant" (PDF).
- 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) [1994], "Hopf invariant", Encyclopedia of Mathematics, EMS Press