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Home > Homotopy groups of spheres

 
In mathematics, the homotopy groups of spheres are the groups
πk(Sn)

in algebraic topology, more specifically homotopy theory, where πk(.) for k ≥ 1 denotes the homotopy group and Sn the n-sphere. From a geometric point of view these are fundamental invariants; on the other hand there is ample evidence from the algebraic aspect that they involve substantial complexity of structure, and intense study from around 1950 has not completely elucidated that.

The case k < n is trite: for example if n ≥ 2 the n-sphere is simply connected. The case k = n is always infinite cyclic, with mappings classified by their degree. It is the case k > n that is of real importance. Here, according to a suspension theorem of Hans Freudenthal, the group depends only on

k − n = i

for k large enough; and is a finite group (abelian), denoted by

πiS.

These are the stable homotopy groups of spheres. They have been computed in numerous cases, but the general pattern is still elusive. There are ad hoc methods for the cases of i small; a systematic tool is the J-homomorphism .

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Topics: Homotopy Groups Of Spheres, Homotopy Group, Homotopy Lifting...

HomotopyAlgebraic topology Homotopy theory In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.Homotopy groupIn mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The many different ways to (continuously) map an n dimensional sphere into a given space are collected into equivalence classes, called homotopy classes. Two maHomotopy lifting propertyHomotopy theory Algebraic topology In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property is a technical condition on a continuous function from a topological space X to another one, Y''. It is designed t
Homotopy groups of spheresIn mathematics, the homotopy groups of spheres are the groups :π S n in algebraic topology, more specifically homotopy theory, where π(. for k ≥ 1 denotes the homotopy group and S n the n sphere. From a geometric point of view these are fundamentHomotopy sphereIn algebraic topology, a branch of mathematics, a homotopy sphere is a topological space homotopy equivalent to a sphere. It thus has the same homotopy groups, and a fortiori the same homology groups, as a sphere. So every homotopy sphere is an homology s 
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