Hopf invariant explained

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

Motivation

In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map

η \colon S ^{3} \to S ^{2,}

and proved that

η is essential, i.e., not homotopic to the constant map, by using the fact that the linking number of the circles η ^{-1} (x), η ^{-1} (y) \subset S ^{3} is equal to 1, for any x ≠ y \in S ^{2} .\pi ^{2 )} _{3 (S} is the infinite cyclic group generated by η . In 1951, Jean-Pierre Serre proved that the rational homotopy groups ^[1] \pi ^{n)} _{i(S} ⊗ Q

for an odd-dimensional sphere (

n odd) are zero unless i 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 2 n -1 .

Definition

Let

\varphi \colon S ^{2n-1} \to S ^{n} be a continuous map (assume n> 1 ). Then we can form the cell complexC _{\varphi} = S ^{n} \cup _{\varphi} D ^{2n} ,

where

D ^{2n} is a 2 n -dimensional disc attached to S ^{n} via \varphi .The cellular chain groups C ^{*} _{cell (C} _{\varphi)} are just freely generated on the i -cells in degree i , so they are Z in degree 0, n and 2 n 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 isH ^{i} _{cell (C} _{\varphi)} = \begin{cases} Z & i =0 ,n, 2 n, \ 0 & otherwise . \end{cases}

Denote the generators of the cohomology groups by

H ^{n(C} _{\varphi)} = \langle\alpha\rangle and H ^{2n} (C _{\varphi)} = \langle\beta\rangle.

For dimensional reasons, all cup-products between those classes must be trivial apart from

\alpha \smile \alpha . Thus, as a ring, the cohomology isH ^{*(C} _{\varphi)} = Z [\alpha,\beta]/\langle \beta\smile\beta = \alpha\smile\beta = 0 , \alpha\smile\alpha = h(\varphi)\beta\rangle.

The integer

h(\varphi) is the Hopf invariant of the map \varphi .

Properties

Theorem: The map

h\colon\pi _{2n-1} (S ^{n)\toZ} is a homomorphism. If n is odd, h is trivial (since \pi _{2n-1} (S ^{n)} is torsion).If n is even, the image of h contains 2 Z . Moreover, the image of the Whitehead product of identity maps equals 2, i. e. h([i _{n,} i _{n])=2} , where i _{n} \colon S ^{n} \to S ^{n} is the identity map and [ ⋅ , ⋅ ] is the Whitehead product.

The Hopf invariant is

1 for the Hopf maps, where n =1 , 2 , 4 , 8 , corresponding to the real division algebras A = R , C , H , O , respectively, and to the fibration S( A ^{2)\toPA} ^{1} 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

J. H. C. Whitehead has proposed the following integral formula for the Hopf invariant.^{[2] [3]} Given a map

\varphi \colon S ^{2n-1} \to S ^{n} , one considers a volume form \omega _{n} on S ^{n} such that \int _{S ^{n}} \omega _{n} = 1 .Since d\omega _{n} = 0 , the pullback \varphi ^{*} \omega _{n} is a closed differential form: d(\varphi ^{*} \omega _{n)} = \varphi ^{*} (d\omega _{n)} = \varphi ^{*} 0 = 0 .By Poincaré's lemma it is an exact differential form: there exists an (n - 1 ) -form η on S ^{2n} such that d η = \varphi ^{*} \omega _{n} . The Hopf invariant is then given by \int _{S ^{2n}} η \wedge d η .

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 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 the point 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 \to 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\} _{Z _{2}} ,

an element of the stable

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 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 \to 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

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

• • • Web site: Michael. Crabb . Andrew . Ranicki. Andrew Ranicki . 2006 . The geometric Hopf invariant .

Notes and References

1. Serre . Jean-Pierre . Groupes D'Homotopie Et Classes De Groupes Abeliens . The Annals of Mathematics . September 1953 . 58 . 2 . 258–294 . 10.2307/1969789. 1969789 .
2. Whitehead . J. H. C. . An Expression of Hopf's Invariant as an Integral . Proceedings of the National Academy of Sciences . 1 May 1947 . 33 . 5 . 117–123 . 10.1073/pnas.33.5.117. 16578254 . free . 1079004 . 1947PNAS...33..117W .
3. Book: Bott . Raoul . Tu . Loring W . Differential forms in algebraic topology . 1982 . New York . 9780387906133.