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.
See also
editReferences
edit- ^ A., Kosinski, Antoni (1993). Differential manifolds. Academic Press. ISBN 0-12-421850-4. OCLC 875287946.
{{cite book}}: CS1 maint: multiple names: authors list (link)
External links
edit- Hedegaard, Rasmus. "Homotopy sphere". MathWorld.
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