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

$\eta :S^3\to S^2,$

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 $x\ne y\in S^2$. 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

${\pi }_i(S^n)\otimes \mathbb{\mathbb{Q}}$

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

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

where $D^{2n}$ is a $2n$-dimensional disc attached to $S^n$ via $\phi$. The cellular chain groups $C_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^{*}(C_{\phi })$ are just freely generated on the $i$-cells in degree $i$, so they are $\mathbb{\mathbb{Z}}$ 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

$H_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}^i(C_{\phi })=\{\begin{array}{ll}\mathbb{\mathbb{Z}}& i=0,n,2n,\\ 0& \text{otherwise}.\end{array}$

Denote the generators of the cohomology groups by

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

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

$H^{*}(C_{\phi })=\mathbb{\mathbb{Z}}[\alpha ,\beta ]/⟨\beta ⌣\beta =\alpha ⌣\beta =0,\alpha ⌣\alpha =h(\phi )\beta ⟩.$

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

Properties

Theorem: The map $h:{\pi }_{2n-1}(S^n)\to \mathbb{\mathbb{Z}}$ 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 $2\mathbb{\mathbb{Z}}$. Moreover, the image of the Whitehead product of identity maps equals 2, i. e. $h([i_n,i_n])=2$, where $i_n:S^n\to S^n$ 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 $S({\mathbb{𝔸}}^2)\to {\mathbb{\mathbb{P}}\mathbb{𝔸}}^1$ 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 $\phi :S^{2n-1}\to S^n$, one considers a volume form ${\omega }_n$ on $S^n$ such that $\underset{S^n}{\int }{\omega }_n=1$. Since $d{\omega }_n=0$, the pullback ${\phi }^{*}{\omega }_n$ is a closed differential form: $d({\phi }^{*}{\omega }_n)={\phi }^{*}(d{\omega }_n)={\phi }^{*}0=0$. By Poincaré's lemma it is an exact differential form: there exists an $(n-1)$-form $\eta$ on $S^{2n-1}$ such that $d\eta ={\phi }^{*}{\omega }_n$. The Hopf invariant is then given by

$\underset{S^{2n-1}}{\int }\eta \wedge d\eta .$

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 $V^{\mathrm{\infty }}$ its one-point compactification, i.e. $V\cong {\mathbb{ℝ}}^k$ and

$V^{\mathrm{\infty }}\cong S^k$ 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 $V^{\mathrm{\infty }}$, then we can form the wedge products

$V^{\mathrm{\infty }}\wedge X.$

Now let

$F:V^{\mathrm{\infty }}\wedge X\to V^{\mathrm{\infty }}\wedge Y$

be a stable map, i.e. stable under the reduced suspension functor. The (stable) geometric Hopf invariant of $F$ is

$h(F)\in \{X,Y\wedge Y{\}}_{{\mathbb{\mathbb{Z}}}_2},$

an element of the stable ${\mathbb{\mathbb{Z}}}_2$-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 ${\mathbb{\mathbb{Z}}}_2$-action is the trivial action on $X$ and the flipping of the two factors on $Y\wedge Y$. If we let

${\mathrm{\Delta }}_X:X\to X\wedge X$

denote the canonical diagonal map and $I$ the identity, then the Hopf invariant is defined by the following:

$h(F):=(F\wedge F)(I\wedge {\mathrm{\Delta }}_X)-(I\wedge {\mathrm{\Delta }}_Y)(I\wedge F).$

This map is initially a map from

$V^{\mathrm{\infty }}\wedge V^{\mathrm{\infty }}\wedge X$ to $V^{\mathrm{\infty }}\wedge V^{\mathrm{\infty }}\wedge Y\wedge Y,$

but under the direct limit it becomes the advertised element of the stable homotopy ${\mathbb{\mathbb{Z}}}_2$-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

• Adams, J. Frank (1960), "On the non-existence of elements of Hopf invariant one", Annals of Mathematics, 72 (1): 20–104, CiteSeerX 10.1.1.299.4490, doi:10.2307/1970147, JSTOR 1970147, MR 0141119
• Adams, J. Frank; Atiyah, Michael F. (1966), "K-Theory and the Hopf Invariant", Quarterly Journal of Mathematics, 17 (1): 31–38, doi:10.1093/qmath/17.1.31, MR 0198460
• Crabb, Michael; Ranicki, Andrew (2006). "The geometric Hopf invariant" (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
• Shokurov, A.V. (2001) [1994], "Hopf invariant", Encyclopedia of Mathematics, EMS Press