In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space)...
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: 198 The Hopf fibration S 1 → S 3 → S 2 {\displaystyle S^{1}\to S^{3}\to S^{2}} is a non-trivial fiber bundle and, specifically, a Serre fibration. A mapping...
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600-cell (section Hopf spherical coordinates)
All the fiber bundles are aspects of the same discrete Hopf fibration, because the fibration is the various expressions of the same distinct left-right...
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time. At this time Hopf discovered the Hopf invariant of maps S 3 → S 2 {\displaystyle S^{3}\to S^{2}} and proved that the Hopf fibration has invariant 1...
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Serre spectral sequence (section Hopf fibration)
{\displaystyle d_{k}:E_{p,q}^{k}\to E_{p-k,q-1+k}^{k}} . Recall that the Hopf fibration is given by S 1 ↪ S 3 → S 2 {\displaystyle S^{1}\hookrightarrow S^{3}\to...
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Homotopy groups of spheres (section Hopf fibrations)
The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere S3 around the usual sphere S2 in a non-trivial...
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the intersection curve is real. The torus plays a central role in the Hopf fibration of the 3-sphere, S3, over the ordinary sphere, S2, which has circles...
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3-sphere (section Hopf coordinates)
d\xi _{1}\wedge d\xi _{2}.} To get the interlocking circles of the Hopf fibration, make a simple substitution in the equations above z 1 = e i ( ξ 1 +...
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Homotopy group (redirect from Exact sequence of a fibration)
Example: the Hopf fibration. Let B equal S 2 {\displaystyle S^{2}} and E equal S 3 . {\displaystyle S^{3}.} Let p be the Hopf fibration, which has fiber...
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a Hopf link. This was Hopf's motivation for studying the Hopf link: because each two fibers are linked, the Hopf fibration is a nontrivial fibration. This...
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four-group is the smallest noncyclic group,... Shokurov, A.V. (2002). "Hopf fibration". In Michiel Hazewinkel (ed.). Encyclopedia of Mathematics. Helsinki:...
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space C 2 {\displaystyle \mathbb {C} ^{2}} to the Bloch sphere is the Hopf fibration, with each ray of spinors mapping to one point on the Bloch sphere....
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in a Hopf fibration. Four cell-disjoint 6-cell rings comprise the same discrete fibration. The 24-cell has four such discrete hexagonal fibrations, and...
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singularities comes from considering the flow along the lines of a Hopf fibration. Let r {\textstyle r} be a constant radius of the inner coil. One set...
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Hopf fibration in the case r = i; other vectors give isomorphic, but not identical fibrations. In 2003 David W. Lyons wrote "the fibers of the Hopf map...
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Fiber bundle (redirect from Fibre bundle)
then the quotient map is a fiber bundle. One example of this is the Hopf fibration, S 3 → S 2 {\displaystyle S^{3}\to S^{2}} , which is a fiber bundle...
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around each other naturally due to the Hopf fibration. The collective of edges forms another discrete Hopf fibration of 12 rings with 10 vertices each. These...
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known as the Bloch sphere or, equivalently, the Riemann sphere. See Hopf fibration for details of the projectivization construction in this case. The Cartesian...
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be regarded as a family of circles parametrized by CPn: this is the Hopf fibration. Complex projective space carries a (Kähler) metric, called the Fubini–Study...
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the circular fibers of a Hopf fibration SU(2) → S2. As such, the Berger metrics can also be constructed via the Hopf fibration, by scaling the tangent...
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trivial. Recall there is a spectral sequence for any Serre fibration, such as the fibration K ( π n + 1 ( X ) , n + 1 ) ≃ F n + 1 → S n + 1 n → S n n ≃...
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skew polygons is a discrete Hopf fibration of the 120-cell (just as their 20 dual 30-cell rings are a discrete fibration of the 600-cell). The Petrie...
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of 4, the number of dimensions. Clifford torus Duoprism Flat torus Hopf fibration Manifold The Fourth Dimension Simply Explained, Henry P. Manning, Munn...
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action of the modular group. It also provides a canonical example of Hopf fibration, where the geodesic flow induced by the linear fractional transformation...
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conjecture. He also established the conjecture for perturbations of the Hopf fibration. The conjecture was disproven in 1974 by Paul Schweitzer, who exhibited...
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not all preimages of magnetisation have a nonzero linking. Skyrmion Hopf fibration Faddeev L, Niemi AJ (1997). "Stable knot-like structures in classical...
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to circle bundles. There are some celebrated ones, for example the Hopf fibrations of spheres to spheres. In algebraic geometry, an invertible sheaf (i...
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group of rotations. This quotient is realized explicitly by the famous Hopf fibration S1 → S2n+1 → CPn, the fibers of which are among the great circles of...
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Paul Dirac back in 1931 and Hopf fibration, a fibration of a 3-sphere proposed io the same year by mathematician Heinz Hopf. Mathematician Jim Simons discussing...
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"standard" braid corresponding to the Borromean rings gives rise to the Hopf fibration S3 → S2, and iteration of this (as in everyday braiding) is likewise...
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