HOPF INVARIANT

(Redirected from Hopf invariant one)
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.
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Contents

Motivation

Definition

Properties

Generalisations for stable maps

References

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. It was later shown that $eta$ generates the homotopy group $pi\_3(S^2)$. 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
:
angle and
angle.
For dimensional reasons, all cup-products between those classes must be trivial apart from . Thus, as a ''ring'', the cohomology is
: $H^$
★ (C_phi) = mathbb{Z}[lpha,eta]/langle etasmileeta = lphasmileeta = 0, lphasmilelpha=h(phi)eta
angle.
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

★ Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, H. Hopf, , , Math. Ann., 1931

★ On the non-existence of elements of Hopf invariant one, J.F. Adams, , , Ann. Math., 1960

★ K-Theory and the Hopf Invariant, J.F. Adams, , , The Quarterly Journal of Mathematics, 1966

★ The geometric Hopf invariant, M. Crabb, , , http://www.maths.ed.ac.uk/~aar/slides/hopfbeam.pdf, 2006