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Home > Homotopy sphere

In 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 sphere.

The generalized Poincaré conjecture is that any homotopy sphere is a sphere; see the article Poincaré conjecture for more details.

Topics: Homotopy Sphere, Homotopy Group, Homotopy Lifting Property...

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 ma	Homotopy 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 fundament	Homotopy 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