In
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, a branch of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a homotopy sphere is an ''n''-
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
that is
homotopy equivalent
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...
to the ''n''-
sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. 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.
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.
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
orientation.
See also
*
Homology sphere
*
Homotopy groups of spheres
*
Poincaré conjecture
References
External links
*
Homotopy theory
Topological spaces
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