- Hopf fibration
In the mathematical field of

topology , the**Hopf fibration**(also known as the**Hopf bundle**or**Hopf map**) describes a3-sphere (ahypersphere infour-dimensional space ) in terms ofcircles and an ordinarysphere . Discovered byHeinz Hopf in1931 , it is an influential early example of afiber bundle . Technically, Hopf found a many-to-onecontinuous function (or "map") from the 3-sphere onto the 2-sphere such that each distinct "point" of the 2-sphere comes from a distinct "circle" of the 3-sphere harv|Hopf|1931. Thus the 3-sphere is composed of fibers, where each fiber is a circle — one for each point of the 2-sphere.This fiber bundle structure is denoted:$S^1\; hookrightarrow\; S^3\; xrightarrow\{\; p\; ,\; \}\; S^2,$where "S"

^{3}(the 3-sphere) is the total space, "S"^{2}(the ordinary 2-sphere) the base space, "S"^{1}(a circle) the fiber space, and "p": "S"^{3}→"S"^{2}(Hopf's map) the bundle projection. The Hopf fibration, like any fiber bundle, has the important property that it is locally aproduct space . However it is "not" a trivial fiber bundle, i.e., "S"^{3}is not (globally) a product of "S"^{2}and "S"^{1}. This has many implications: for example the existence of this bundle shows that the higherhomotopy groups of spheres are not trivial in general. It also provides a basic example of aprincipal bundle , by identifying the fiber with thecircle group .Stereographic projection of the Hopf fibration induces a remarkable structure on**R**^{3}, in which space is filled with nested tori made of linkingVillarceau circles . Here each fiber projects to a circle in space (one of which is a "circle through infinity" — a line). Each torus is the stereographic projection of theinverse image of a circle of latitude of the 2-sphere. (Topologically, a torus is the product of two circles.) One of these toruses is illustrated by the image of linking keyrings on the right.There are numerous generalizations of the Hopf fibration. The unit sphere in

**C**^{"n"+1}fibers naturally over**CP**^{"n"}with circles as fibers, and there are also real, quaternionic, and octonionic versions of these fibrations. In particular, the Hopf fibration belongs to a family of four fiber bundles in which the total space, base space, and fiber space are all spheres::$S^0hookrightarrow\; S^1\; ightarrow\; S^1,\; ,!$:$S^1hookrightarrow\; S^3\; ightarrow\; S^2,\; ,!$:$S^3hookrightarrow\; S^7\; ightarrow\; S^4\; ,!$:$S^7hookrightarrow\; S^\{15\}\; ightarrow\; S^8.\; ,!$In fact these are the only such fibrations between spheres.The Hopf fibration is important in

twistor theory .**Definition and construction**For any

natural number "n", an "n"-dimensional sphere, orn-sphere , can be defined as the set of points in an ("n"+1)-dimensional space which are a fixed distance from a central point. For concreteness, the central point can be taken to be the origin, and the distance of the points on the sphere from this origin can be assumed to be a unit length. With this convention, the "n"-sphere, "S"^{"n"}, consists of the points ("x"_{1}, "x"_{2}, …, "x"_{"n"+1}) in**R**^{"n"+1}with "x"_{1}^{2}+ "x"_{2}^{2}+ ⋯+ "x"_{"n"+1}^{2}= 1. For example, the 3-sphere consists of the points ("x"_{1}, "x"_{2}, "x"_{3}, "x"_{4}) in**R**^{4}with "x"_{1}^{2}+ "x"_{2}^{2}+ "x"_{3}^{2}+ "x"_{4}^{2}= 1.The Hopf fibration "p": "S"

^{3}→ "S"^{2}of the 3-sphere over the 2-sphere can be defined in several ways.**Direct construction**Identify

**R**^{4}with**C**^{2}and**R**^{3}with**C**×**R**(where**C**denotes thecomplex number s) by writing::("x"_{1}, "x"_{2}, "x"_{3}, "x"_{4}) as ("z"_{0}= "x"_{1}+ i"x"_{2}, "z"_{1}= "x"_{3}+ i"x"_{4}); and:("x"_{1}, "x"_{2}, "x"_{3}) as ("z" = "x"_{1}+ i"x"_{2}, "x" = "x"_{3}).Thus "S"^{3}is identified with thesubset of all ("z"_{0}, "z"_{1}) in**C**^{2}such that |"z"_{0}|^{2}+ |"z"_{1}|^{2}= 1, and "S"^{2}is identified with the subset of all ("z", "x") in**C**×**R**such that |"z"|^{2}+ "x"^{2}= 1. (Here, for a complex number "z" = "x" + i"y", |"z"|^{2}= "z" "z"^{∗}= "x"^{2}+ "y"^{2}, where the star denotes thecomplex conjugate .) Then the Hopf fibration "p" is defined by:"p"("z"_{0}, "z"_{1}) = (2"z"_{0}"z"_{1}^{∗}, |"z"_{0}|^{2}− |"z"_{1}|^{2}).The first component is a complex number, whereas the second component is real. Any point on the 3-sphere must have the property that |"z"

_{0}|^{2}+ |"z"_{1}|^{2}= 1. If that is so, then "p"("z"_{0}, "z"_{1}) lies on the unit 2-sphere in**C**×**R**, as may be shown by squaring the complex and real components of "p":$2\; z\_\{0\}\; z\_\{1\}^\{ast\}\; cdot\; 2\; z\_\{0\}^\{ast\}\; z\_\{1\}\; +\; left(\; left|\; z\_\{0\}\; ight|^\{2\}\; -\; left|\; z\_\{1\}\; ight|^\{2\}\; ight)^\{2\}\; =\; 4\; left|\; z\_\{0\}\; ight|^\{2\}\; left|\; z\_\{1\}\; ight|^\{2\}\; +\; left|\; z\_\{0\}\; ight|^\{4\}\; -\; 2\; left|\; z\_\{0\}\; ight|^\{2\}\; left|\; z\_\{1\}\; ight|^\{2\}\; +\; left|\; z\_\{0\}\; ight|^\{4\}\; =\; left(\; left|\; z\_\{0\}\; ight|^\{2\}\; +\; left|\; z\_\{1\}\; ight|^\{2\}\; ight)^\{2\}\; =\; 1$

Furthermore, if two points on the 3-sphere map to the same point on the 2-sphere, i.e., if "p"("z"

_{0}, "z"_{1}) = "p"("w"_{0}, "w"_{1}), then ("w"_{0}, "w"_{1}) must equal ("λ" "z"_{0}, "λ" "z"_{1}) for some complex number "λ" with |"λ"|^{2}= 1. The converse is also true; any two points on the 3-sphere that differ by a common complex factor "λ" map to the same point on the 2-sphere. These conclusions follow, because the complex factor "λ" cancels with its complex conjugate "λ"^{∗}in both parts of "p": in the complex 2"z"_{0}"z"_{1}^{∗}component and in the real component |"z"_{0}|^{2}− |"z"_{1}|^{2}.Since the set of complex numbers "λ" with |"λ"|

^{2}= 1 form the unit circle in the complex plane, it follows that for each point "m" in "S"^{2}, theinverse image "p"^{−1}("m") is a circle, i.e., "p"^{−1}"m" ≅ "S"^{1}. Thus the 3-sphere is realized as adisjoint union of these circular fibers.**Geometric interpretation using the complex projective line**A geometric interpretation of the fibration may be obtained using the

complex projective line ,**CP**^{1}, which is defined to be the set of all complex one dimensional subspaces of**C**^{2}. Equivalently,**CP**^{1}is the quotient of**C**^{2}{0} by theequivalence relation which identifies ("z"_{0}, "z"_{1}) with ("λ" "z"_{0}, "λ" "z"_{1}) for any nonzero complex number "λ". On any complex line in**C**^{2}there is a circle of unit norm, and so the restriction of thequotient map to the points of unit norm is a fibration of "S"^{3}over**CP**^{1}.**CP**^{1}is diffeomorphic to a 2-sphere: indeed it can be identified with theRiemann sphere **C**_{∞}=**C**∪ {∞}, which is theone point compactification of**C**(obtained by adding apoint at infinity ). The formula given for "p" above defines an explicit diffeomorphism between the complex projective line and the ordinary 2-sphere in 3-dimensional space. Alternatively, the point ("z"_{0}, "z"_{1}) can be mapped to the ratio "z"^{1}/"z"^{0}in the Riemann sphere**C**_{∞}.**Fiber bundle structure**The Hopf fibration defines a

fiber bundle , with bundle projection "p". This means that it has a "local product structure", in the sense that every point of the 2-sphere has some neighborhood "U" whose inverse image in the 3-sphere can be identified with the product of "U" and a circle: "p"^{−1}("U") ≅ "U"×"S"^{1}. Such a fibration is said to belocally trivial .For the Hopf fibration, it is enough to remove a single point "m" from "S"

^{2}and the corresponding circle "p"^{-1}("m") from "S"^{3}; thus one can take "U" = "S"^{2}{"m"}, and any point in "S"^{2}has a neighborhood of this form.**Geometric interpretation using rotations**Another geometric interpretation of the Hopf fibration can be obtained by considering rotations of the 2-sphere in ordinary 3-dimensional space. The

rotation group SO(3) has adouble cover , thespin group Spin(3),diffeomorphic to the 3-sphere. The spin group acts transitively on "S"^{2}by rotations. The stabilizer of a point is isomorphic to thecircle group . It follows easily that the 3-sphere is aprincipal circle bundle over the 2-sphere, and this is the Hopf fibration.To make this more explicit, there are two approaches: the group Spin(3) can either be identified with the group Sp(1) of unit

quaternion s, or with thespecial unitary group SU(2) .In the first approach, a vector ("x"

_{1}, "x"_{2}, "x"_{3}, "x"_{4}) in**R**^{4}is interpreted as a quaternion "q" ∈**H**by writing:$q\; =\; x\_1+old\{i\}x\_2+old\{j\}x\_3+old\{k\}x\_4.,!$ The 3-sphere is then identified with the quaternions of unit norm, i.e., those "q" ∈**H**for which |"q"|^{2}= 1, where |"q"|^{2}= "q q"^{∗}, which is equal to "x"_{1}^{2}+ "x"_{2}^{2}+ "x"_{3}^{2}+ "x"_{4}^{2}for "q" as above.On the other hand, a vector ("y"

_{1}, "y"_{2}, "y"_{3}) in**R**^{3}can be interpreted as an imaginary quaternion:$p\; =\; old\{i\}y\_1+old\{j\}y\_2+old\{k\}y\_3.\; ,!$Then, as is well-known since Harvtxt|Cayley|1845, the mapping:$p\; mapsto\; q\; p\; q^*\; ,!$is a rotation in**R**^{3}: indeed it is clearly anisometry , since |"q p q"^{∗}|^{2}= "q p q"^{∗}"q p"^{∗}"q"^{∗}= "q p p"^{∗}"q"^{∗}= |"p"|^{2}, and it is not hard to check that it preserves orientation.In fact, this identifies the group of unit quaternions with the group of rotations of

**R**^{3}, modulo the fact that the unit quaternions "q" and −"q" determine the same rotation. As noted above, the rotations act transitively on "S"^{2}, and the set of unit quaternions "q" which fix a given unit imaginary quaternion "p" have the form "q" = "u" + "v" "p", where "u" and "v" are real numbers with "u"^{2}+ "v"^{2}= 1. This is a circle subgroup. For concreteness, one can take "p" =**k**, and then the Hopf fibration can be defined as the map sending a unit quaternion "q" to "q"**k**"q"^{∗}.This approach is related to the direct construction by identifying a quaternion "q" = "x"

_{1}+**i**"x"_{2}+**j**"x"_{3}+**k**"x"_{4}with the 2×2 matrix::$egin\{bmatrix\}\; x\_1+old\; i\; x\_2\; x\_3+old\; i\; x\_4\; \backslash \backslash \; -x\_3+old\; i\; x\_4\; x\_1-old\; i\; x\_2\; end\{bmatrix\}.,!$This identifies the group of unit quaternions with SU(2), and the imaginary quaternions with the skew-hermitian 2×2 matrices (isomorphic to**C**×**R**).**Explicit formulae**The rotation induced by a unit quaternion "q" = "w" +

**i**"x" +**j**"y" +**k**"z" is given explicitly by the orthogonal matrix:$egin\{bmatrix\}1-2(y^2+z^2)\; 2(xy\; -\; wz)\; 2(xz+wy)\backslash \backslash 2(xy\; +\; wz)\; 1-2(x^2+z^2)\; 2(yz-wx)\backslash \backslash 2(xz-wy)\; 2(yz+wx)\; 1-2(x^2+y^2)end\{bmatrix\}\; .$Here we find an explicit real formula for the bundle projection. For, the fixed unit vector along the "z" axis, (0,0,1), rotates to another unit vector,:$Big(2(xz+wy)\; ,\; 2(yz-wx)\; ,\; 1-2(x^2+y^2)Big)\; ,\; ,!$which is a continuous function of ("w","x","y","z"). That is, the image of "q" is where it aims the "z" axis. The fiber for a given point on "S"^{2}consists of all those unit quaternions that aim there.To write an explicit formula for the fiber over a point ("a","b","c") in "S"

^{2}, we may proceed as follows. Multiplication of unit quaternions produces composition of rotations, and:$q\_\{\; heta\}\; =\; cos\; heta\; +\; old\{k\}\; sin\; heta\; ,!$is a rotation by 2θ around the "z" axis. As θ varies, this sweeps out agreat circle of "S"^{3}, our prototypical fiber. So long as the base point, ("a","b","c"), is not the antipode, (0,0,−1), the quaternion:$q\_\{(a,b,c)\}\; =\; frac\{1\}\{sqrt\{2(1+c)(1+c-old\{i\}b+old\{j\}a)$will aim there. Thus the fiber of ("a","b","c") is given by quaternions of the form "q"_{("a","b","c")}"q"_{θ}, which are the "S"^{3}points:$frac\{1\}\{sqrt\{2(1+c)\; Big((1+c)\; cos\; (\; heta\; ),\; a\; sin\; (\; heta\; )-b\; cos\; (\; heta\; ),\; a\; cos\; (\; heta\; )+b\; sin\; (\; heta\; ),\; (1+c)\; sin\; (\; heta\; )Big)\; .\; ,!$Since multiplication by "q"_{("a","b","c")}acts as a rotation of quaternion space, the fiber is not merely a topological circle, it is a geometric circle. The final fiber, for (0,0,−1), can be given by using "q"_{(0,0,−1)}=**i**, producing:$Big(0,cos\; (\; heta\; ),-sin\; (\; heta\; ),0Big)\; ,\; ,!$which completes the bundle.Thus, a simple way of visualizing the Hopf fibration is as follows. Any point on the 3-sphere is equivalent to a

quaternion , which in turn is equivalent to a particular rotation of a Cartesian coordinate frame in three dimensions. The set of all possible quaternions produces the set of all possible rotations, which moves the tip of one unit vector of such a coordinate frame (say, the**z**vector) to all possible points on a unit 2-sphere. However, fixing the tip of the**z**vector does not specify the rotation fully; a further rotation is possible about the**z**-axis. Thus, the 3-sphere is mapped onto the 2-sphere, plus a single rotation.**Generalizations**The Hopf construction, viewed as a fiber bundle "p": "S"

^{3}→**CP**^{"1"}, admits several generalizations, which are also often known as Hopf fibrations. First, one can replace the projective line by an "n"-dimensionalprojective space . Second, one can replace the complex numbers by any (real)division algebra , including (for "n" = 1) theoctonion s.**Real Hopf fibrations**A real version of the Hopf fibration is obtained by regarding "S"

^{1}as a subset of**R**^{2}in the usual way and factoring out by unit real multiplication to obtain and a fiber bundle "S"^{1}→**RP**^{1}over thereal projective line with fiber "S"^{0}= {1, -1}. Just as**CP**^{1}is diffeomorphic to a sphere,**RP**^{1}is diffeomorphic to a circle.More generally, the "n"-sphere "S"

^{"n"}fibers overreal projective space **RP**^{"n"}with fiber "S"^{0}.**Complex Hopf fibrations**The Hopf construction gives circle bundles "p" : "S"

^{2"n"+1}→**CP**^{"n"}overcomplex projective space . This is actually the restriction of thetautological line bundle over**CP**^{"n"}to the unit sphere in**C**^{"n"+1}.**Quaternionic Hopf fibrations**Similarly, one can regard "S"

^{4"n"−1}as lying in**H**^{"n"}(quaternionic "n"-space) and factor out by unit quaternion (= "S"^{3}) multiplication to get**HP**^{"n"}. In particular, since "S"^{4}=**HP**^{1}, there is a bundle "S"^{7}→ "S"^{4}with fiber "S"^{3}.**Octonionic Hopf fibrations**A similar construction with the

octonion s yields a bundle "S"^{15}→ "S"^{8}with fiber "S"^{7}. One can regard "S"^{8}as theoctonionic projective line **OP**^{1}.Although one can also define an octonionic projective plane,

**OP**^{2}, "S"^{31}does not fiber over it.**Fibrations between spheres**Sometimes the term "Hopf fibration" is restricted to the fibrations between spheres obtained above, which are

* "S"^{1}→ "S"^{1}with fiber "S"^{0}

* "S"^{3}→ "S"^{2}with fiber "S"^{1}

* "S"^{7}→ "S"^{4}with fiber "S"^{3}

* "S"^{15}→ "S"^{8}with fiber "S"^{7}As a consequence ofAdams' theorem , these are the only fiber bundles withsphere s as total space, base space, and fiber.**Geometry and applications**[

Villarceau circles in**R**^{3}.] The Hopf fibration has many implications, some purely attractive, others deeper. For example,stereographic projection of "S"^{3}to**R**^{3}induces a remarkable structure in**R**^{3}, which in turn illuminates the topology of the bundle Harv|Lyons|2003. Stereographic projection preserves circles and maps the Hopf fibers to geometrically perfect circles in**R**^{3}which fill space. Here there is one exception: the Hopf circle containing the projection point maps to a straight line in**R**^{3}— a "circle through infinity".The fibers over a circle of latitude on "S"

^{2}form atorus in "S"^{3}(topologically, a torus is the product of two circles) and these project to nestedtorus es in**R**^{3}which also fill space. The individual fibers map to linkingVillarceau circles on these tori, with the exception of the circle through the projection point and the one through its opposite point: the former maps to a straight line, the latter to a unit circle perpendicular to, and centered on, this line, which may be viewed as a degenerate torus whose radius has shrunken to zero. Every other fiber image encircles the line as well, and so, by symmetry, each circle is linked through "every" circle, both in "R"^{3}and in "S"^{3}. Two such linking circles form aHopf link in**R**^{3}Hopf proved that the Hopf map has

Hopf invariant 1, and therefore is notnull-homotopic . In fact it generates thehomotopy group π_{3}(S^{2}) and has infinite order.In

quantum mechanics , the Riemann sphere is known as theBloch sphere , and the Hopf fibration describes the topological structure of a quantum mechanicaltwo-level system orqubit . Similarly, the topology of a pair of entangled two-level systems is given by the Hopf fibration :$S^3\; hookrightarrow\; S^7\; o\; S^4\; ,!$.harv|Mosseri|Dandoloff|2001.**References*** citation

last=Cayley

first=Arthur

author-link=Arthur Cayley

title=On certain results relating to quaternions

journal=Philosophical Magazine

volume=26

pages=141–145

year=1845

url=http://www.archive.org/details/collmathpapers01caylrich ; reprinted as article 20 in citation

last=Cayley

first=Arthur

author-link=Arthur Cayley

year=1889

title=The collected mathematical papers of Arthur Cayley

publisher=Cambridge University Press

volume=I (1841–1853)

pages=123–126

isbn=

url=http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=ABS3153.0001.001;didno=ABS3153.0001.001;view=image;seq=00000140

* citation

last=Hopf

first=Heinz

author-link=Heinz Hopf

title=Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche

journal=Mathematische Annalen

publisher=Springer

location=Berlin

volume=104

pages=637–665

year=1931

issn=0025-5831

url=http://www.digizeitschriften.de/index.php?id=loader&tx_jkDigiTools_pi1%5BIDDOC%5D=363429&L=2

* citation

last=Hopf

first=Heinz

author-link=Heinz Hopf

title=Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension

journal=Fundamenta Mathematicae

publisher=Polish Acad. Sci.

location=Warsaw

volume=25

pages=427–440

year=1935

issn=0016-2736

* citation

last=Lyons

first=David W.

title=An Elementary Introduction to the Hopf Fibration

journal=Mathematics Magazine

volume=76

issue=2

pages=87–98

year=2003

month=April

url=http://csunix1.lvc.edu/~lyons/pubs/hopf_paper_preprint.pdf

format =PDF

issn=0025-570X

*citation

last1 = Mosseri | first1 = R.

last2 = Dandoloff | first2 = R.

title = Geometry of entangled states, Bloch spheres and Hopf fibrations

journal = J. Phys. A: Math. Gen.

volume = 34 | pages = 10243–10252 | year = 2001 | doi = 10.1088/0305-4470/34/47/324

id = arxiv | quant-ph/0108137.

* citation

last=Steenrod

first=Norman

title=The Topology of Fibre Bundles

publisher=Princeton University Press

year=1951

publication-date=1999

series=PMS 14

isbn=978-0-691-00548-5

url=http://books.google.com/books?id=m_wrjoweDTgC&printsec=frontcover&dq=%22The+Topology+of+Fibre+Bundles%22

*Wikimedia Foundation.
2010.*