  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. although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. ==main branches of algebraic topology== below are some of the main areas studied in algebraic topology ===homotopy groups===  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. ===homology===  in algebraic topology and abstract algebra, homology (in part from greek á½Î¼ÏÏ homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. ===cohomology===  in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. that is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries. cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. cohomology arises from the algebraic dualization of the construction of homology. in less abstract language, cochains in the fundamental sense should assign 'quantities' to the chains of homology theory. ===manifolds===  a manifold is a topological space that near each point resembles euclidean space. more precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the euclidean space of dimension n. lines and circles, but not figure eights, are one-dimensional manifolds. two-di
Algebraic_topology