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

${\displaystyle \eta \colon S^{3}\to S^{2},}$

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

${\displaystyle \eta ^{-1}(x),\eta ^{-1}(y)\subset S^{3}}$

is equal to 1, for any ${\displaystyle x\neq y\in S^{2}}$.

It was later shown that the homotopy group ${\displaystyle \pi _{3}(S^{2})}$ is the infinite cyclic group generated by ${\displaystyle \eta }$. In 1951, Jean-Pierre Serre proved that the rational homotopy groups ^[1]

${\displaystyle \pi _{i}(S^{n})\otimes \mathbb {Q} }$

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

Definition

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

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

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

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

Denote the generators of the cohomology groups by

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

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

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

The integer ${\displaystyle h(\varphi )}$ is the Hopf invariant of the map ${\displaystyle \varphi }$.

Properties

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

The Hopf invariant is ${\displaystyle 1}$ for the Hopf maps, where ${\displaystyle n=1,2,4,8}$, corresponding to the real division algebras ${\displaystyle \mathbb {A} =\mathbb {R} ,\mathbb {C} ,\mathbb {H} ,\mathbb {O} }$, respectively, and to the fibration ${\displaystyle S(\mathbb {A} ^{2})\to \mathbb {PA} ^{1}}$ 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.^{[2][3]: prop. 17.22} Given a map ${\displaystyle \varphi \colon S^{2n-1}\to S^{n}}$, one considers a volume form ${\displaystyle \omega _{n}}$ on ${\displaystyle S^{n}}$ such that ${\displaystyle \int _{S^{n}}\omega _{n}=1}$. Since ${\displaystyle d\omega _{n}=0}$, the pullback ${\displaystyle \varphi ^{*}\omega _{n}}$ is a closed differential form: ${\displaystyle d(\varphi ^{*}\omega _{n})=\varphi ^{*}(d\omega _{n})=\varphi ^{*}0=0}$. By Poincaré's lemma it is an exact differential form: there exists an ${\displaystyle (n-1)}$-form ${\displaystyle \eta }$ on ${\displaystyle S^{2n-1}}$ such that ${\displaystyle d\eta =\varphi ^{*}\omega _{n}}$. The Hopf invariant is then given by

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

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

${\displaystyle V^{\infty }\cong S^{k}}$ for some ${\displaystyle k}$.

If ${\displaystyle (X,x_{0})}$ 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 ${\displaystyle V^{\infty }}$, then we can form the wedge products

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

Now let

${\displaystyle F\colon V^{\infty }\wedge X\to V^{\infty }\wedge Y}$

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

${\displaystyle h(F)\in \{X,Y\wedge Y\}_{\mathbb {Z} _{2}},}$

an element of the stable ${\displaystyle \mathbb {Z} _{2}}$-equivariant homotopy group of maps from ${\displaystyle X}$ to ${\displaystyle Y\wedge Y}$. Here "stable" means "stable under suspension", i.e. the direct limit over ${\displaystyle V}$ (or ${\displaystyle k}$, if you will) of the ordinary, equivariant homotopy groups; and the ${\displaystyle \mathbb {Z} _{2}}$-action is the trivial action on ${\displaystyle X}$ and the flipping of the two factors on ${\displaystyle Y\wedge Y}$. If we let

${\displaystyle \Delta _{X}\colon X\to X\wedge X}$

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

${\displaystyle h(F):=(F\wedge F)(I\wedge \Delta _{X})-(I\wedge \Delta _{Y})(I\wedge F).}$

This map is initially a map from

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

but under the direct limit it becomes the advertised element of the stable homotopy ${\displaystyle \mathbb {Z} _{2}}$-equivariant group of maps. There exists also an unstable version of the Hopf invariant ${\displaystyle h_{V}(F)}$, for which one must keep track of the vector space ${\displaystyle V}$.