Editing Homotopy sphere

{{Short description|Concept in algebraic topology}}

In [[algebraic topology]], a branch of [[mathematics]], a '''homotopy sphere''' is an ''n''-[[manifold]] that is [[Homotopy#Homotopy equivalence and null-homotopy|homotopy equivalent]] to the ''n''-[[Sphere#Topology|sphere]]. It thus has the same [[homotopy group]]s and the same [[homology (mathematics)|homology]] groups as the ''n''-sphere, and so every homotopy sphere is necessarily a [[homology sphere]].<ref>{{Cite book|last=A.|first=Kosinski, Antoni|url=http://worldcat.org/oclc/875287946|title=Differential manifolds|date=1993|publisher=Academic Press|isbn=0-12-421850-4|oclc=875287946}}</ref>

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 sphere]]s.  It is open whether non-trivial smooth homotopy spheres exist in dimension 4.

Homotopy spheres form an abelian group known as [[Kervaire–Milnor group]]. Its composition is the [[connected sum]] and its [[neutral element]] is the sphere, while inversion is given by opposite [[Orientability|orientation]].

==See also==
*[[Homology sphere]]
*[[Homotopy groups of spheres]]
*[[Poincaré conjecture]]

==References==

{{reflist}}

==External links==
*{{mathworld|HomotopySphere|author = Hedegaard, Rasmus}}

[[Category:Homotopy theory]]
[[Category:Topological spaces]]


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