In algebraic topology, a branch of mathematics, a homotopy sphere is an n-manifold that is homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology groups as the n-sphere, and so every homotopy sphere is necessarily a homology sphere.[1]
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 conjecture in dimensions 5 and larger implies that homotopy spheres in those dimensions are precisely exotic spheres. It is open whether non-trivial smooth homotopy spheres exist in dimension 4.
^A., Kosinski, Antoni (1993). Differential manifolds. Academic Press. ISBN 0-12-421850-4. OCLC 875287946.{{cite book}}: CS1 maint: multiple names: authors list (link)
branch of mathematics, a homotopysphere is an n-manifold that is homotopy equivalent to the n-sphere. It thus has the same homotopy groups and the same homology...
the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They...
an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappings...
be infinite; see the section on Gluck twists. All homotopy n-spheres are homeomorphic to the n-sphere by the generalized Poincaré conjecture, proved by...
identified (regular homotopy) classes of immersions of spheres with a homotopy group of the Stiefel manifold. Since the homotopy group that corresponds...
being called a homotopy (/həˈmɒtəpiː/, hə-MO-tə-pee; /ˈhoʊmoʊˌtoʊpiː/, HOH-moh-toh-pee) between the two functions. A notable use of homotopy is the definition...
optical properties of a mirror sphere Hoberman sphere Homology sphereHomotopy groups of spheresHomotopysphere Lenart Sphere Napkin ring problem Orb (optics)...
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic...
is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger...
of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter...
of spheres as the homotopy groups of Stiefel manifolds, and Hirsch's generalization of this to immersions of manifolds being classified as homotopy classes...
In stable homotopy theory, a branch of mathematics, the sphere spectrum S is the monoidal unit in the category of spectra. It is the suspension spectrum...
the homotopy quotient (or Borel construction) of X by G, where EG is the universal bundle of G. homotopy spectral sequence homotopysphere A homotopy sphere...
number of linearly independent sections of the tangent bundle of any homotopysphere. The case of n {\displaystyle n} odd is taken care of by the Poincaré–Hopf...
In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic"...
for n {\displaystyle n} sufficiently large. In particular, the homotopy groups of spheres π n + k ( S n ) {\displaystyle \pi _{n+k}(S^{n})} stabilize for...
more examples see 4-manifold. Brieskorn manifold Exotic sphere Homology sphereHomotopysphere Lens space Spherical 3-manifold Einstein manifold Ricci-flat...
The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological...
concept of a sphere bundle, is a fibration whose fibers are homotopy equivalent to spheres. For example, the fibration BTop ( R n ) → BTop ( S n )...
algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category Ho ( Top ) {\displaystyle...
a homotopysphere, remove two balls, apply the h-cobordism theorem to conclude that this is a cylinder, and then attach cones to recover a sphere. This...
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups...
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