Hopf invariant

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

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 $$n>1$$). 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 $$n>1$$), 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 $$V$$.