Hopf invariant

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

1 Motivation
2 Definition
3 Properties
4 Generalisations for stable maps
5 References

[edit] Motivation

In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map $\eta\colon S^3 \to S^2$ , and proved that $η$ is essential, i.e. not homotopic to the constant map. It was later shown that $η$ generates the homotopy group $π 3 (S 2)$ . 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 = 0 or n. However, for an even-dimensional sphere (n even), there is one more bit of infinite cyclic homotopy in degree $2 n - 1$ . There is an interesting way of seeing this:

[edit] Definition

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

$C_\phi = S^n \cup_\phi D^{2n},$

where $D 2 n$ is a $2 n$ -dimensional disc attached to $S n$ via $φ$ . 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 $2 n$ 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 is

$H^i_\mathrm{cell}(C_\phi) = \begin{cases} \mathbb{Z} & i=0,n,2n, \\ 0 & \mbox{otherwise}. \end{cases}$

Denote the generators of the cohomology groups by

$H^n(C_\phi) = \langle\alpha\rangle$ and $H^{2n}(C_\phi) = \langle\beta\rangle.$

For dimensional reasons, all cup-products between those classes must be trivial apart from $\alpha \smile \alpha$ . Thus, as a ring, the cohomology is

$H^*(C_\phi) = \mathbb{Z}[\alpha,\beta]/\langle \beta\smile\beta = \alpha\smile\beta = 0, \alpha\smile\alpha=h(\phi)\beta\rangle.$

The integer $h (φ)$ is the Hopf invariant of the map $φ$ .

[edit] Properties

Theorem: $h\colon\pi_{2n-1}(S^n)\to\mathbb{Z}$ is a homomorphism. Moreover, if $n$ is even, $h$ maps onto $2\mathbb{Z}$ .

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 $S(\mathbb{A}^2)\to\mathbb{PA}^1$ sending a direction on the sphere to the subspace it spans). It is a theorem, proved first by Frank Adams and subsequently by Michael Atiyah with methods of K-theory, that these are the only maps with Hopf invariant 1.

[edit] 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 $V$ .

[edit] References

H. Hopf (1931). "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche". Math. Ann. 104: 637-665.
J.F. Adams (1960). "On the non-existence of elements of Hopf invariant one". Ann. Math. 72: 20-104.
J.F. Adams; M.F. Atiyah (1966). "K-Theory and the Hopf Invariant". The Quarterly Journal of Mathematics 17 (1): 31-38.