- Hopf invariant
In
mathematics , in particular inalgebraic topology , the Hopf invariant is ahomotopy invariant of certain maps betweensphere s.__toc__Motivation
In 1931
Heinz Hopf used Clifford parallels to construct the "Hopf map " etacolon S^3 o S^2, and proved that eta is essential, i.e. nothomotopic to the constant map, by using the linking number (=1) of the circleseta^{-1}(x),eta^{-1}(y) subset S^3 for any x eq y in S^2. It was later shown that thehomotopy group pi_3(S^2) is the infinitecyclic group generated by eta. In 1951,Jean-Pierre Serre proved that therational homotopy groups pi_i(S^n) otimes mathbb{Q} for an odd-dimensional sphere (n 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 2n-1. There is an interesting way of seeing this:Definition
Let phi colon S^{2n-1} o S^n be a
continuous map (assume n>1). Then we can form thecell complex : C_phi = S^n cup_phi D^{2n},
where D^{2n} is a 2n-dimensional disc attached to S^n via phi.The cellular chain groups C^*_mathrm{cell}(C_phi) are just freely generated on the n-cells in degree n, so they are mathbb{Z} in degree 0, n and 2n 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 n>1), the cohomology is: H^i_mathrm{cell}(C_phi) = egin{cases} mathbb{Z} & i=0,n,2n, \ 0 & mbox{otherwise}. end{cases}
Denote the generators of the cohomology groups by
: H^n(C_phi) = langlealpha angle and H^{2n}(C_phi) = langleeta angle.
For dimensional reasons, all cup-products between those classes must be trivial apart from alpha smile alpha. Thus, as a "ring", the cohomology is
: H^*(C_phi) = mathbb{Z} [alpha,eta] /langle etasmileeta = alphasmileeta = 0, alphasmilealpha=h(phi)eta angle.
The integer h(phi) is the Hopf invariant of the map phi.
Properties
Theorem: hcolonpi_{2n-1}(S^n) omathbb{Z} is a homomorphism. Moreover, if n is even, h maps onto 2mathbb{Z}.
The Hopf invariant is 1 for the "Hopf maps" (where n=1,2,4,8, corresponding to the real division algebras mathbb{A}=mathbb{R},mathbb{C},mathbb{H},mathbb{O}, respectively, and to the double cover S(mathbb{A}^2) omathbb{PA}^1 sending a direction on the sphere to the subspace it spans). It is a theorem, proved first by
Frank Adams and subsequently byMichael Atiyah with methods ofK-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 V denote a vector space and V^infty its
one-point compactification , i.e. V cong mathbb{R}^k and V^infty cong S^k for some k. If X,x_0) is any pointed space (as it is implicitly in the previous section), and if we take thepoint at infinity to be the basepoint of V^infty, then we can form the wedge products V^infty wedge X.Now let F colon V^infty wedge X o V^infty wedge Y be a stable map, i.e. stable under the
reduced suspension functor. The "(stable) geometric Hopf invariant" of F ish(F) in {X, Y wedge Y}_{mathbb{Z}_2},
an element of the stable mathbb{Z}_2-equivariant homotopy group of maps from X to Y wedge Y. Here "stable" means "stable under suspension", i.e. the direct limit over V (or k, if you will) of the ordinary, equivariant homotopy groups; and the mathbb{Z}_2-action is the trivial action on X and the flipping of the two factors on Y wedge Y. If we let Delta_X colon X o X wedge X denote the canonical diagonal map and I the identity, then the Hopf invariant is defined by the following:
h(F) := (F wedge F) (I wedge Delta_X) - (I wedge Delta_Y) (I wedge F).
This map is initially a map from V^infty wedge V^infty wedge X to V^infty wedge V^infty wedge Y wedge Y, but under the direct limit it becomes the advertised element of the stable homotopy mathbb{Z}_2-equivariant group of maps.
There exists also an unstable version of the Hopf invariant h_V(F), for which one must keep track of the vector space V.
References
* citation
first = J.F.|last= Adams
year = 1960
title = On the non-existence of elements of Hopf invariant one
journal = Ann. Math.
volume = 72
pages = 20-104
* citation
first = J.F.|last= Adams
first2 = M.F. |last2Atiyah
year = 1966
title = K-Theory and the Hopf Invariant
journal = The Quarterly Journal of Mathematics
volume = 17
issue = 1
pages = 31-38* citation
first = M.|last= Crabb
first2=A. |last2= Ranicki
year = 2006
title = The geometric Hopf invariant
url = http://www.maths.ed.ac.uk/~aar/slides/hopfbeam.pdf
*
*springer|first=A.V. |last=Shokurov|title=Hopf invariant|id=h/h048000
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