Hopf invariant

Motivation

In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map :\eta\colon S^3 \to S^2,

and proved that \eta is essential, i.e., not homotopic to the constant map, by using the fact that the linking number of the circles :\eta^{-1}(x),\eta^{-1}(y) \subset S^3 is equal to 1, for any x \neq y \in S^2.

It was later shown that the homotopy group \pi_3(S^2) is the infinite cyclic group generated by \eta. In 1951, Jean-Pierre Serre proved that the rational homotopy groups :\pi_i(S^n) \otimes \mathbb{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 2n-1.

Definition

Let \varphi \colon S^{2n-1} \to S^n be a continuous map (assume n1). Then we can form the cell complex

: C_\varphi = S^n \cup_\varphi D^{2n},

where D^{2n} is a 2n-dimensional disc attached to S^n via \varphi. The cellular chain groups C^*\mathrm{cell}(C\varphi) are just freely generated on the i-cells in degree i, 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 n1), the cohomology is

: H^i_\mathrm{cell}(C_\varphi) = \begin{cases} \mathbb{Z} & i=0,n,2n, \ 0 & \text{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 is

: H^*(C_\varphi) = \mathbb{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)\to\mathbb{Z} 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\mathbb{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 [,\cdot,,,\cdot,] 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 \mathbb{A}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}, respectively, and to the fibration S(\mathbb{A}^2)\to\mathbb{PA}^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. 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 \eta on S^{2n - 1} such that d\eta = \varphi^* \omega_n. The Hopf invariant is then given by : \int_{S^{2n - 1}} \eta \wedge d \eta.

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 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 is :h(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 \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 \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

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References

1. (September 1953). "Groupes D'Homotopie Et Classes De Groupes Abeliens". The Annals of Mathematics.
2. (1 May 1947). "An Expression of Hopf's Invariant as an Integral". Proceedings of the National Academy of Sciences.
3. (1982). "Differential forms in algebraic topology".