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see also differential topology, algebraic topology, functional analysis and topological homotopy theory
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closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
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see also algebraic topology
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cobordism theory, Introduction
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The complex Hopf fibration [Hopf (1931)] is a canonical nontrivial circle principal bundle over the 2-sphere whose total space is the 3-sphere.
Its canonically associated complex line bundle is the basic line bundle on the 2-sphere.
This we discuss below in
More generally, there are four Hopf fibrations, on the 1-sphere, the 3-sphere, the 7-sphere and the 15-sphere, respectively. This we discuss in
The Eilenberg-MacLane space is the classifying space for circle group principal bundles. By its very nature, it has a single nontrivial homotopy group, the second, and this is isomorphic to the group of integers
This means that there is, up to homotopy, a canonical (up to sign), continuous map from the 2-sphere
such that .
As any map into this classifies a circle group principal bundle over its domain. This is the Hopf fibration, fitting into the long fiber sequence
In other words, the Hopf fibration is the -bundle with unit first Chern class on .
An explicit topological space presenting the Hopf fibration may be obtained as follows. Identify
and
Then the continuous function defined by
gives the Hopf fibration. (Thus, the Hopf fibration is the circle bundle naturally associated with the canonical line bundle.)
Alternatively, if we identify
and identify this presentation of the 2-sphere with the complex projective line via stereographic projection, then the Hopf fibration is identified with the map
Alternatively, we may regard as the unit sphere in the quaternions and as the unit sphere in the imaginary quaternions. Under this identification, the complex Hopf fibration is equivalently represented by
where is any unit imaginary quaternion.
Regard as equipped with its circle group structure. This makes in particular an H-space. The Hopf fibration is the Hopf construction applied to this H-space.
Consider
the Spin(3)-action on the 2-sphere which is induced by the defining action on under the identification ;
the Spin(3)-action on the 3-sphere which is induced under the exceptional isomorphism by the canonical left action of on via .
Then the complex Hopf fibration is equivariant with respect to these actions.
A way to make the -equivariance of the complex Hopf fibration fully explicit is to observe that it is equivalently the following map of coset spaces:
For each of the normed division algebras over , the real numbers, complex numbers, quaternions, octonions
there is a corresponding Hopf fibration of Hopf invariant one.
The total space of the fibration is the space of pairs of unit norm: . This gives spheres of dimension 1, 3, 7, and 15 respectively. The base space of the fibration is projective 1-space , giving spheres of dimension 1, 2, 4, and 8, respectively. In each case, the Hopf fibration is the map
() which sends the pair to .
When is a sphere that is an -space, namely, one of the groups the trivial group, the group of order 2, the 3-sphere special unitary group ; or the 7-sphere with its Moufang loop structure, then the Hopf construction produces the above four Hopf fibrations:
Let
,
be the homotopy classes of the complex Hopf fibration, the quaternionic Hopf fibration and the octonionic Hopf fibration, respectively. Then their suspensions are the generators of the corresponding stable homotopy groups of spheres:
see also at
and this MO comment
When line bundles are regarded as models for the topological structure underlying the electromagnetic field the Hopf fibration is often called “the magnetic monopole”. We may think of the homotopically as being the 3-dimensional Cartesian space with origin removed and think of this as being 3-dimensional physical space with a unit point magnetic charge at the origin removed. The corresponding electromagnetic field away from the origin is given by a connection on the corresponding Hopf fibration bundle.
In complex K-theory, the Hopf fibration represents a class which generates the cohomology ring , and satisfying the relation , or . (So in particular has an inverse , see at Bott generator.)
A succinct formulation of Bott periodicity for complex K-theory is that for a space whose homotopy type is that of a CW-complex, we have
(It would be interesting to see whether this can be proved by internalizing the (classically easy) calculation for to the topos of sheaves over .)
The Hopf fibrations over other normed division algebras also figure in the more complicated case of real K-theory : they can be used to provide generators for the non-zero homotopy groups for the classifying space of the stable orthogonal group, which are periodic of period 8 (not coincidentally, 8 is the dimension of the largest normed division algebra ). [To be followed up on.]
Original articles:
Heinz Hopf, Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, Mathematische Annalen 104 (1931) 637–665 [doi:10.1007/BF01457962]
Heinz Hopf, Über die Abbildungen von Sphären auf Sphäre niedrigerer Dimension, Fundamenta Mathematicae 25 1 (1935) 427-440 [eudml:212801]
Exposition:
Review:
Herman Gluck, Frank Warner, Chung Tao Yang, Section 6 of: Division algebras, fibrations of spheres by great spheres and the topological determination of space by the gross behavior of its geodesics, Duke Math. J. Volume 50, Number 4 (1983), 1041-1076 (euclid:dmj/1077303489)
Herman Gluck, Frank Warner, Wolfgang Ziller, The geometry of the Hopf fibrations, L’Enseignement Mathématique, 32 (1986), 173-198 [ResearchGate, pdf]
and via the Pontrjagin theorem:
Formulation in homotopy type theory:
Relation to skyrmions:
Discussion of supersymmetric Hopf fibrations:
A. P. Balachandran, G. Marmo, B.-S. Skagerstam and A. Stern, section 9.3 of Gauge Symmetries and Fibre Bundles, Lect. Notes in Physics 188, Springer-Verlag, Berlin, 1983 (arXiv:1702.08910)
Simon Davis, section 3 of Supersymmetry and the Hopf fibration (doi:10.4995/agt.2012.1623)
Last revised on October 17, 2024 at 18:51:05. See the history of this page for a list of all contributions to it.