{{Short description|Homotopy invariant of maps between n-spheres}} In [[mathematics]], in particular in [[algebraic topology]], the '''Hopf invariant''' is a [[homotopy]] invariant of certain maps between [[n-sphere|''n''-spheres]]. __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 fact that the [[linking number]] of the circles :<math>\eta^{-1}(x),\eta^{-1}(y) \subset S^3</math> is equal to 1, 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 <ref>{{cite journal |last1=Serre |first1=Jean-Pierre |title=Groupes D'Homotopie Et Classes De Groupes Abeliens |journal=The Annals of Mathematics |date=September 1953 |volume=58 |issue=2 |pages=258–294 |doi=10.2307/1969789|jstor=1969789 }}</ref> :<math>\pi_i(S^n) \otimes \mathbb{Q}</math>
for an odd-dimensional sphere (<math>n</math> odd) are zero unless <math>i</math> 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 <math>2n-1</math>.
== Definition == Let <math>\varphi \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_\varphi = S^n \cup_\varphi D^{2n},</math>
where <math>D^{2n}</math> is a <math>2n</math>-dimensional disc attached to <math>S^n</math> via <math>\varphi</math>. The cellular chain groups <math>C^*_\mathrm{cell}(C_\varphi)</math> are just freely generated on the <math>i</math>-cells in degree <math>i</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_\varphi) = \begin{cases} \mathbb{Z} & i=0,n,2n, \\ 0 & \text{otherwise}. \end{cases}</math>
Denote the generators of the cohomology groups by
: <math>H^n(C_\varphi) = \langle\alpha\rangle</math> and <math>H^{2n}(C_\varphi) = \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_\varphi) = \mathbb{Z}[\alpha,\beta]/\langle \beta\smile\beta = \alpha\smile\beta = 0, \alpha\smile\alpha=h(\varphi)\beta\rangle.</math>
The integer <math>h(\varphi)</math> is the '''Hopf invariant''' of the map <math>\varphi</math>.
== Properties == '''Theorem''': The map <math>h\colon\pi_{2n-1}(S^n)\to\mathbb{Z}</math> is a homomorphism. If <math>n</math> is odd, <math>h</math> is trivial (since <math>\pi_{2n-1}(S^n)</math> is torsion). If <math>n</math> is even, the image of <math>h</math> contains <math>2\mathbb{Z}</math>. Moreover, the image of the [[Whitehead product]] of identity maps equals 2, i. e. <math>h([i_n, i_n])=2</math>, where <math>i_n \colon S^n \to S^n </math> is the identity map and <math>[\,\cdot\,,\,\cdot\,]</math> is the [[Whitehead product]].
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 fibration <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 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.<ref>{{cite journal |last1=Whitehead |first1=J. H. C. |title=An Expression of Hopf's Invariant as an Integral |journal=Proceedings of the National Academy of Sciences |date=1 May 1947 |volume=33 |issue=5 |pages=117–123 |doi=10.1073/pnas.33.5.117|pmid=16578254 |doi-access=free |pmc=1079004 |bibcode=1947PNAS...33..117W }}</ref><ref>{{cite book |last1=Bott |first1=Raoul |last2=Tu |first2=Loring W |title=Differential forms in algebraic topology |date=1982 |location=New York |isbn=9780387906133}}</ref>{{rp|prop. 17.22}} Given a map <math>\varphi \colon S^{2n-1} \to S^n</math>, one considers a [[volume form]] <math>\omega_n</math> on <math>S^n</math> such that <math>\int_{S^n}\omega_n = 1</math>. Since <math>d\omega_n = 0</math>, the [[Pullback (differential geometry)|pullback]] <math>\varphi^* \omega_n</math> is a [[closed differential form]]: <math>d(\varphi^* \omega_n) = \varphi^* (d\omega_n) = \varphi^* 0 = 0</math>. By [[Poincaré's lemma]] it is an [[exact differential form]]: there exists an <math>(n - 1)</math>-form <math>\eta</math> on <math>S^{2n - 1}</math> such that <math>d\eta = \varphi^* \omega_n</math>. The Hopf invariant is then given by :<math> \int_{S^{2n - 1}} \eta \wedge d \eta. </math>
== 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== {{Reflist}}
* {{citation | first = J. Frank|last= Adams|authorlink=Frank Adams | year = 1960 | title = On the non-existence of elements of Hopf invariant one | journal = [[Annals of Mathematics]] | volume = 72 | pages = 20–104 | doi = 10.2307/1970147 | issue = 1 | jstor = 1970147 | mr=0141119 |citeseerx= 10.1.1.299.4490}} * {{citation | first1 = J. Frank|last1= Adams| author1-link=Frank Adams | first2 = Michael F.|last2=Atiyah| author2-link=Michael Atiyah | year = 1966 | title = K-Theory and the Hopf Invariant | journal = [[Quarterly Journal of Mathematics]] | volume = 17 | issue = 1 | pages = 31–38 | doi = 10.1093/qmath/17.1.31 | mr=0198460 }} * {{cite web | first1 = Michael|last1= Crabb | first2=Andrew |last2= Ranicki|author2-link=Andrew 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]]