Fibración de Hopf

Es posible visualizar la fibración de Hopf utilizando una proyección estereográfica de S³ a R³ y luego comprimir R³ en una bola. Esta imagen muestra puntos en S² y sus correspondientes fibras con el mismo color.

Here keyrings mimic part of the Hopf fibration by showing some of the circles of the Hopf fibration which lie on a common torus.

En el ámbito de la rama de la matemáticas denominado topologia, la fibración de Hopf (también denominada el Hopf bundle o mapa de Hopf) describe una 3-esfera (una hiperesfera en el espacio de cuatro dimensiones) mediante círculos y una esfera ordinaria. Descubierta en 1931 por Heinz Hopf, es un ejemplo inicial importante de un haz de fibras. Tecnicamente, Hopf descubrió una función continua (o "mapa") de varios a uno de la 3-esfera en la 2-esfera tal que cada puntoen particular de la 2-esfera proviene de un círculo específico de la 3-esfera (Hopf, 1931). Por lo tanto la 3-esfera is composed of fibras, donde cada fubra es un círculo — uno para cada punto de la 2-esfera.

This fiber bundle structure is denoted

$S^{1}\hookrightarrow S^{3}{\xrightarrow {\ p\,}}S^{2},$

meaning that the fiber space S¹ (a circle) is embedded in the total space S³ (la 3-esfera), y p: S³→S² (Mapa de Hopf) proyecta S³ en el espacio base S² (la 2-sphere ordinaria). La fibración de Hopf, al igual que todo fiber bundle, has the important property that it is locally a product space. However it is not a trivial fiber bundle, i.e., S³ is not globally a product de S² y S¹ although locally it is indistinguishable from it.

This has many implications: for example the existence of this bundle shows that the higher homotopy groups of spheres are not trivial in general. It also provides a basic example of a principal bundle, by identifying the fiber with the circle group.

Stereographic projection of the Hopf fibration induces a remarkable structure on R³, in which space is filled with nested tori made of linking Villarceau circles. Here each fiber projects to a circle in space (one of which is a line, thought of as a "circle through infinity"). Each torus is the stereographic projection of the inverse image of a circle of latitude of the 2-sphere. (Topologically, a torus is the product of two circles.) These tori are illustrated in the images at right. When R³ is compressed to a ball, some geometric structure is lost although the topological structure is retained (see Topology and Geometry). The loops are homeomorphic to circles, although they are not geometric circles.

There are numerous generalizations of the Hopf fibration. The unit sphere in Cⁿ⁺¹ fibers naturally over CPⁿ 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:

$S^{0}\hookrightarrow S^{1}\rightarrow S^{1},\,\!$

$S^{1}\hookrightarrow S^{3}\rightarrow S^{2},\,\!$

$S^{3}\hookrightarrow S^{7}\rightarrow S^{4},\,\!$

$S^{7}\hookrightarrow S^{15}\rightarrow S^{8}.\,\!$

By Adams' theorem such fibrations can occur only in these dimensions.

La fibración de Hopf es importante en el ámbito de la twistor theory.

Referencias

• Cayley, Arthur (1845), «On certain results relating to quaternions», Philosophical Magazine 26: 141-145 .; reprinted as article 20 in Cayley, Arthur (1889), The collected mathematical papers of Arthur Cayley, I, (1841–1853), Cambridge University Press, pp. 123-126 .
• Hopf, Heinz (1931), «Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche», Mathematische Annalen (Berlin: Springer) 104 (1): 637-665, ISSN 0025-5831, doi:10.1007/BF01457962 .
• Hopf, Heinz (1935), «Über die Abbildungen von Sphären auf Sphären niedrigerer Dimension», Fundamenta Mathematicae (Warsaw: Polish Acad. Sci.) 25: 427-440, ISSN 0016-2736 .
• Lyons, David W. (April de 2003), «An Elementary Introduction to the Hopf Fibration» (PDF), Mathematics Magazine 76 (2): 87-98, ISSN 0025-570X, JSTOR 3219300, doi:10.2307/3219300  .
• Mosseri, R.; Dandoloff, R. (2001), «Geometry of entangled states, Bloch spheres and Hopf fibrations», J. Phys. A: Math. Gen. 34 (47): 10243-10252, arXiv:quant-ph/0108137, doi:10.1088/0305-4470/34/47/324 ..
• Steenrod, Norman (1951), The Topology of Fibre Bundles, PMS 14, Princeton University Press (publicado el 1999), ISBN 978-0-691-00548-5 .
• Urbantke, H.K. (2003), «The Hopf fibration-seven times in physics», Journal of Geometry and Physics 46 (2): 125-150, doi:10.1016/S0393-0440(02)00121-3 ..

Enlaces externos

• Dimensions Math Chapters 7 and 8 illustrate the Hopf fibration with animated computer graphics.
• YouTube animation of the construction of the 120-cell By Gian Marco Todesco shows the Hopf fibration of the 120-cell.