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In [[mathematics]], in particular in [[algebraic topology]], the Hopf invariant is a [[homotopy]] invariant of certain maps between [[sphere]]s.
__TOC__
== Motivation ==
In 1931 [[Heinz Hopf]] used [[Clifford parallel]]s to construct the ''[[Hopf map]]''
:<math>\eta\colon S^3 \to S^2</math>,

and proved that <math>\eta</math> is essential, i.e. not [[homotopic]] to the constant map, by using the linking number (=1) of the circles
:<math>\eta^{-1}(x),\eta^{-1}(y) \subset S^3</math> for any <math>x \neq y \in S^2</math>.

It was later shown that the [[homotopy group]] <math>\pi_3(S^2)</math> is the infinite [[cyclic group]] generated by <math>\eta</math>. In 1951, [[Jean-Pierre Serre]] proved that the [[rational homotopy]] groups
:<math>\pi_i(S^n) \otimes \mathbb{Q}</math>

for an odd-dimensional sphere (<math>n</math> 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 <math>2n-1</math>. There is an interesting way of seeing this:

== Definition ==
Let <math>\phi \colon S^{2n-1} \to S^n</math> be a [[continuous map]] (assume <math>n>1</math>). Then we can form the [[cell complex]]

: <math>C_\phi = S^n \cup_\phi D^{2n},</math>

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

: <math>H^i_\mathrm{cell}(C_\phi) = \begin{cases} \mathbb{Z} & i=0,n,2n, \\ 0 & \mbox{otherwise}. \end{cases}</math>

Denote the generators of the cohomology groups by

: <math>H^n(C_\phi) = \langle\alpha\rangle</math> and <math>H^{2n}(C_\phi) = \langle\beta\rangle.</math>

For dimensional reasons, all cup-products between those classes must be trivial apart from <math>\alpha \smile \alpha</math>. Thus, as a ''ring'', the cohomology is

: <math>H^*(C_\phi) = \mathbb{Z}[\alpha,\beta]/\langle \beta\smile\beta = \alpha\smile\beta = 0, \alpha\smile\alpha=h(\phi)\beta\rangle.</math>

The integer <math>h(\phi)</math> is the '''Hopf invariant''' of the map <math>\phi</math>.

== Properties ==
'''Theorem''': <math>h\colon\pi_{2n-1}(S^n)\to\mathbb{Z}</math> is a homomorphism. Moreover, if <math>n</math> is even, <math>h</math> maps onto <math>2\mathbb{Z}</math>.

The Hopf invariant is <math>1</math> for the ''Hopf maps'' (where <math>n=1,2,4,8</math>, corresponding to the real division algebras <math>\mathbb{A}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}</math>, respectively, and to the double cover <math>S(\mathbb{A}^2)\to\mathbb{PA}^1</math> 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 <math>V</math> denote a vector space and <math>V^\infty</math> its [[one-point compactification]], i.e. <math>V \cong \mathbb{R}^k</math> and
:<math>V^\infty \cong S^k</math> for some <math>k</math>.

If <math>(X,x_0)</math> 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 <math>V^\infty</math>, then we can form the wedge products
:<math>V^\infty \wedge X</math>.

Now let
:<math>F \colon V^\infty \wedge X \to V^\infty \wedge Y</math>

be a stable map, i.e. stable under the [[reduced suspension]] functor. The ''(stable) geometric Hopf invariant'' of <math>F</math> is
:<math>h(F) \in \{X, Y \wedge Y\}_{\mathbb{Z}_2}</math>,

an element of the stable <math>\mathbb{Z}_2</math>-equivariant homotopy group of maps from <math>X</math> to <math>Y \wedge Y</math>. Here "stable" means "stable under suspension", i.e. the direct limit over <math>V</math> (or <math>k</math>, if you will) of the ordinary, equivariant homotopy groups; and the <math>\mathbb{Z}_2</math>-action is the trivial action on <math>X</math> and the flipping of the two factors on <math>Y \wedge Y</math>. If we let
:<math>\Delta_X \colon X \to X \wedge X</math>

denote the canonical diagonal map and <math>I</math> the identity, then the Hopf invariant is defined by the following:
:<math>h(F) := (F \wedge F) (I \wedge \Delta_X) - (I \wedge \Delta_Y) (I \wedge F).</math>

This map is initially a map from
:<math>V^\infty \wedge V^\infty \wedge X</math> to <math>V^\infty \wedge V^\infty \wedge Y \wedge Y</math>,

but under the direct limit it becomes the advertised element of the stable homotopy <math>\mathbb{Z}_2</math>-equivariant group of maps.
There exists also an unstable version of the Hopf invariant <math>h_V(F)</math>, for which one must keep track of the vector space <math>V</math>.

==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
| doi = 10.2307/1970147
| issue = 1
| publisher = The Annals of Mathematics, Vol. 72, No. 1
| jstor = 1970147
}}
* {{citation
| first = J.F.|last= Adams
| first2 = M.F.
| year = 1966
| title = K-Theory and the Hopf Invariant
| journal = The Quarterly Journal of Mathematics
| volume = 17
| issue = 1
| pages = 31–38|last2=Atiyah
| doi = 10.1093/qmath/17.1.31
}}

* {{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

}}
* {{Citation | last1=Hopf | first1=Heinz | title=Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche | year=1931 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=104 | pages=637–665 | doi=10.1007/BF01457962}}
*{{springer|first=A.V. |last=Shokurov|title=Hopf invariant|id=h/h048000}}

[[Category:Homotopy theory]]

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