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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

to the ''n''-

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...

. It thus has the same

homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...

s and the same homology groups as the ''n''-sphere, and so every homotopy sphere is necessarily a

homology sphere Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor *Sequence homology, biological homology between DNA, RNA, or protein sequences * Homologous chrom ...

. The topological

generalized Poincaré conjecture In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere a sphere. More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or di ...

is that any ''n''-dimensional homotopy sphere is

homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

to the ''n''-sphere; it was solved by

Stephen Smale Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics faculty ...

in dimensions five and higher, by Michael Freedman in dimension 4, and for dimension 3 (the original

Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured ...

) by

Grigori Perelman Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...

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 still an open question () whether or not there are non-trivial smooth homotopy spheres in dimension 4.

References

See also