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, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
To define the n-th homotopy group, the base point preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the n-th homotopy group, πn(X), of the given space X with base point. Topological spaces with differing homotopy groups are never equivalent (homeomorphic), but the converse is not true.
The notion of homotopy of paths was introduced by Camille Jordan.
Read more about Homotopy Group: Introduction, Definition, Long Exact Sequence of A Fibration, Methods of Calculation, A List of Methods For Calculating Homotopy Groups, Relative Homotopy Groups, Related Notions
Famous quotes containing the word group:
“No other group in America has so had their identity socialized out of existence as have black women.... When black people are talked about the focus tends to be on black men; and when women are talked about the focus tends to be on white women.”
—bell hooks (b. c. 1955)