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

$H_{\mathrm {cell} }^{i}(C_{\varphi })={\begin{cases}\mathbb {Z} &i=0,n,2n,\\0&{\text{otherwise}}.\end{cases}}$

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

• 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