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In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group , denoted π 1 ( X ) , {\displaystyle \pi _{1}(X),} which records information about loops in a space .
A torus, one of the most frequently studied objects 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 invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function from a topological space E to another one, B.
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces The main article for this category is Algebraic topology . Contents
A CW complex is a space that has a filtration whose union is and such that . is a discrete space, called the set of 0-cells (vertices) in .; Each is obtained by attaching several n-disks, n-cells, to via maps ; i.e., the boundary of an n-disk is identified with the image of in .
In mathematics, especially in algebraic topology, the homotopy limit and colimit [1] pg 52 are variants of the notions of limit and colimit extended to the homotopy category (). The main idea is this: if we have a diagram:
This terminology is often used in the case of the algebraic topology on the set of discrete, faithful representations of a Kleinian group into PSL(2,C). Another topology, the geometric topology (also called the Chabauty topology ), can be put on the set of images of the representations, and its closure can include extra Kleinian groups that are ...
Allen Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. ISBN 0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the author's homepage. Kainen, P. C. (1971). "Weak Adjoint Functors". Mathematische Zeitschrift. 122: 1– 9.