Hopf invariant

Short description: Homotopy invariant of maps between n-spheres

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

[math]\displaystyle{ \eta\colon S^3 \to S^2 }[/math],

and proved that [math]\displaystyle{ \eta }[/math] is essential, i.e., not homotopic to the constant map, by using the fact that the linking number of the circles

[math]\displaystyle{ \eta^{-1}(x),\eta^{-1}(y) \subset S^3 }[/math]

is equal to 1, for any [math]\displaystyle{ x \neq y \in S^2 }[/math].

It was later shown that the homotopy group [math]\displaystyle{ \pi_3(S^2) }[/math] is the infinite cyclic group generated by [math]\displaystyle{ \eta }[/math]. In 1951, Jean-Pierre Serre proved that the rational homotopy groups

[math]\displaystyle{ \pi_i(S^n) \otimes \mathbb{Q} }[/math]

for an odd-dimensional sphere ([math]\displaystyle{ n }[/math] odd) are zero unless [math]\displaystyle{ i }[/math] 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 [math]\displaystyle{ 2n-1 }[/math].

Definition

Let [math]\displaystyle{ \phi \colon S^{2n-1} \to S^n }[/math] be a continuous map (assume [math]\displaystyle{ n\gt 1 }[/math]). Then we can form the cell complex

[math]\displaystyle{ C_\phi = S^n \cup_\phi D^{2n}, }[/math]

where [math]\displaystyle{ D^{2n} }[/math] is a [math]\displaystyle{ 2n }[/math]-dimensional disc attached to [math]\displaystyle{ S^n }[/math] via [math]\displaystyle{ \phi }[/math]. The cellular chain groups [math]\displaystyle{ C^*_\mathrm{cell}(C_\phi) }[/math] are just freely generated on the [math]\displaystyle{ i }[/math]-cells in degree [math]\displaystyle{ i }[/math], so they are [math]\displaystyle{ \mathbb{Z} }[/math] in degree 0, [math]\displaystyle{ n }[/math] and [math]\displaystyle{ 2n }[/math] 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 [math]\displaystyle{ n\gt 1 }[/math]), the cohomology is

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

Denote the generators of the cohomology groups by

[math]\displaystyle{ H^n(C_\phi) = \langle\alpha\rangle }[/math] and [math]\displaystyle{ H^{2n}(C_\phi) = \langle\beta\rangle. }[/math]

For dimensional reasons, all cup-products between those classes must be trivial apart from [math]\displaystyle{ \alpha \smile \alpha }[/math]. Thus, as a ring, the cohomology is

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

The integer [math]\displaystyle{ h(\phi) }[/math] is the Hopf invariant of the map [math]\displaystyle{ \phi }[/math].

Properties

Theorem: The map [math]\displaystyle{ h\colon\pi_{2n-1}(S^n)\to\mathbb{Z} }[/math] is a homomorphism. If [math]\displaystyle{ n }[/math] is odd, [math]\displaystyle{ h }[/math] is trivial (since [math]\displaystyle{ \pi_{2n-1}(S^n) }[/math] is torsion). If [math]\displaystyle{ n }[/math] is even, the image of [math]\displaystyle{ h }[/math] contains [math]\displaystyle{ 2\mathbb{Z} }[/math]. Moreover, the image of the Whitehead product of identity maps equals 2, i. e. [math]\displaystyle{ h([i_n, i_n])=2 }[/math], where [math]\displaystyle{ i_n \colon S^n \to S^n }[/math] is the identity map and [math]\displaystyle{ [\,\cdot\,,\,\cdot\,] }[/math] is the Whitehead product.

The Hopf invariant is [math]\displaystyle{ 1 }[/math] for the Hopf maps, where [math]\displaystyle{ n=1,2,4,8 }[/math], corresponding to the real division algebras [math]\displaystyle{ \mathbb{A}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O} }[/math], respectively, and to the fibration [math]\displaystyle{ S(\mathbb{A}^2)\to\mathbb{PA}^1 }[/math] 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 [math]\displaystyle{ \phi \colon S^{2n-1} \to S^n }[/math], one considers a volume form [math]\displaystyle{ \omega_n }[/math] on [math]\displaystyle{ S^n }[/math] such that [math]\displaystyle{ \int_{S^n}\omega_n = 1 }[/math]. Since [math]\displaystyle{ d\omega_n = 0 }[/math], the pullback [math]\displaystyle{ \varphi^* \omega_n }[/math] is a Closed differential form: [math]\displaystyle{ d(\varphi^* \omega_n) = \varphi^* (d\omega_n) = \varphi^* 0 = 0 }[/math]. By Poincaré's lemma it is an exact differential form: there exists an [math]\displaystyle{ (n - 1) }[/math]-form [math]\displaystyle{ \eta }[/math] on [math]\displaystyle{ S^{2n - 1} }[/math] such that [math]\displaystyle{ d\eta = \varphi^* \omega_n }[/math]. The Hopf invariant is then given by

[math]\displaystyle{ \int_{S^{2n - 1}} \eta \wedge d \eta. }[/math]

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 [math]\displaystyle{ V }[/math] denote a vector space and [math]\displaystyle{ V^\infty }[/math] its one-point compactification, i.e. [math]\displaystyle{ V \cong \mathbb{R}^k }[/math] and

[math]\displaystyle{ V^\infty \cong S^k }[/math] for some [math]\displaystyle{ k }[/math].

If [math]\displaystyle{ (X,x_0) }[/math] 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 [math]\displaystyle{ V^\infty }[/math], then we can form the wedge products

[math]\displaystyle{ V^\infty \wedge X }[/math].

Now let

[math]\displaystyle{ F \colon V^\infty \wedge X \to V^\infty \wedge Y }[/math]

be a stable map, i.e. stable under the reduced suspension functor. The (stable) geometric Hopf invariant of [math]\displaystyle{ F }[/math] is

[math]\displaystyle{ h(F) \in \{X, Y \wedge Y\}_{\mathbb{Z}_2} }[/math],

an element of the stable [math]\displaystyle{ \mathbb{Z}_2 }[/math]-equivariant homotopy group of maps from [math]\displaystyle{ X }[/math] to [math]\displaystyle{ Y \wedge Y }[/math]. Here "stable" means "stable under suspension", i.e. the direct limit over [math]\displaystyle{ V }[/math] (or [math]\displaystyle{ k }[/math], if you will) of the ordinary, equivariant homotopy groups; and the [math]\displaystyle{ \mathbb{Z}_2 }[/math]-action is the trivial action on [math]\displaystyle{ X }[/math] and the flipping of the two factors on [math]\displaystyle{ Y \wedge Y }[/math]. If we let

[math]\displaystyle{ \Delta_X \colon X \to X \wedge X }[/math]

denote the canonical diagonal map and [math]\displaystyle{ I }[/math] the identity, then the Hopf invariant is defined by the following:

[math]\displaystyle{ h(F) := (F \wedge F) (I \wedge \Delta_X) - (I \wedge \Delta_Y) (I \wedge F). }[/math]

This map is initially a map from

[math]\displaystyle{ V^\infty \wedge V^\infty \wedge X }[/math] to [math]\displaystyle{ V^\infty \wedge V^\infty \wedge Y \wedge Y }[/math],

but under the direct limit it becomes the advertised element of the stable homotopy [math]\displaystyle{ \mathbb{Z}_2 }[/math]-equivariant group of maps. There exists also an unstable version of the Hopf invariant [math]\displaystyle{ h_V(F) }[/math], for which one must keep track of the vector space [math]\displaystyle{ V }[/math].

References

• "On the non-existence of elements of Hopf invariant one", Annals of Mathematics 72 (1): 20–104, 1960, doi:10.2307/1970147 
• "K-Theory and the Hopf Invariant", Quarterly Journal of Mathematics 17 (1): 31–38, 1966, doi:10.1093/qmath/17.1.31 
• Crabb, Michael (2006). "The geometric Hopf invariant". http://www.maths.ed.ac.uk/~aar/slides/hopfbeam.pdf. 
• Hopf, Heinz (1931), "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche", Mathematische Annalen 104: 637–665, doi:10.1007/BF01457962, ISSN 0025-5831 
• Hazewinkel, Michiel, ed. (2001), "Hopf invariant", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=h/h048000