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The Hopf fibration is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere. This picture mimics part of the Hopf fibration, an interesting mapping from the three-dimensional sphere to the two-dimensional sphere. This mapping is the generator of the third homotopy group of the 2-sphere.
The topological generalized Poincaré conjecture is that any n-dimensional homotopy sphere is homeomorphic to the n-sphere; it was solved by Stephen Smale in dimensions five and higher, by Michael Freedman in dimension 4, and for dimension 3 (the original Poincaré conjecture) by Grigori Perelman in 2005. The resolution of the smooth Poincaré ...
Direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spheres (2-spheres) In mathematics, a hypersphere, 3-sphere, or glome is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central ...
Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes.
The opposite is also true: If X has a hole with a d-dimensional boundary, then there is a d-dimensional sphere that is not homotopic to a constant map, so the d-th homotopy group of X is not trivial. In short, X has a hole with a d -dimensional boundary, if-and-only-if π d ( X ) ≇ 0 {\displaystyle \pi _{d}(X)\not \cong 0} .The homotopical ...
3.1 Homotopy groups. ... -dimensional disk, , with ... However, its cover is a regular CW structure on the sphere, with 2 cells in every dimension; indeed ...
A homotopy between two embeddings of the torus into R 3: as "the surface of a doughnut" and as "the surface of a coffee mug".This is also an example of an isotopy.. Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function: [,] from the product of the space X with the unit interval [0, 1] to Y such ...
In particular, the lens spaces L(7,1) and L(7,2) give examples of two 3-manifolds that are homotopy equivalent but not homeomorphic. The lens space L(1,0) is the 3-sphere, and the lens space L(2,1) is 3 dimensional real projective space.