Hopf invariant

In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.__toc__

Motivation

In 1931 Heinz Hopf used Clifford parallels to construct the "Hopf map" $etacolon\; S^3\; o\; S^2$, 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 $x\; eq\; y\; in\; S^2$. 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 $phi\; colon\; S^\{2n-1\}\; o\; S^n$ 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

:

Denote the generators of the cohomology groups by

: $H^n(C\_phi)\; =\; langlealpha\; angle$ and

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

:

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

Properties

Theorem: $hcolonpi\_\{2n-1\}(S^n)\; omathbb\{Z\}$ is a homomorphism. Moreover, if $n$ is even, $h$ maps onto $2mathbb\{Z\}$.

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 $S(mathbb\{A\}^2)\; omathbb\{PA\}^1$ 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 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 $F\; colon\; V^infty\; wedge\; X\; o\; V^infty\; wedge\; Y$ 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 $Delta\_X\; colon\; X\; o\; X\; wedge\; X$ 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

* citation
first = J.F.|last= Adams
year = 1960
title = On the non-existence of elements of Hopf invariant one
journal = Ann. Math.
volume = 72
pages = 20-104
* citation
first = J.F.|last= Adams
first2 = M.F. |last2Atiyah
year = 1966
title = K-Theory and the Hopf Invariant
journal = The Quarterly Journal of Mathematics
volume = 17
issue = 1
pages = 31-38

* citation
first = M.|last= Crabb
first2=A. |last2= Ranicki
year = 2006
title = The geometric Hopf invariant
url = http://www.maths.ed.ac.uk/~aar/slides/hopfbeam.pdf
*
*springer|first=A.V. |last=Shokurov|title=Hopf invariant|id=h/h048000