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.

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 linking number (=1) of the circles

$\eta ^{{-1}}(x),\eta ^{{-1}}(y)\subset S^{3}$ 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 = 0 or n. However, for an even-dimensional sphere (n even), there is one more bit of infinite cyclic homotopy in degree $2n-1$ . There is an interesting way of seeing this:

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 $D^{{2n}}$ is a $2n$ -dimensional disc attached to $S^{n}$ via $\phi$ . 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 $2n$ 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

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

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

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 topological K-theory, that these are the only maps with Hopf invariant 1.

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.F. (1960), "On the non-existence of elements of Hopf invariant one", Ann. Math. (The Annals of Mathematics, Vol. 72, No. 1) 72 (1): 20–104, doi:10.2307/1970147, JSTOR 1970147
Adams, J.F.; Atiyah, M.F. (1966), "K-Theory and the Hopf Invariant", The Quarterly Journal of Mathematics 17 (1): 31–38, doi:10.1093/qmath/17.1.31

Crabb, M.; Ranicki, A. (2006), The geometric Hopf invariant
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), "Hopf invariant", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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