Algebraic topology

Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.

In algebraic topology and abstract algebra, homology (in part from Greek ὁμός 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.

[1] In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex.

In less abstract language, cochains in the fundamental sense should assign "quantities" to the chains of homology theory.

Typically, results in algebraic topology focus on global, non-differentiable aspects of manifolds; for example Poincaré duality.

A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts (see illustration).

A CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory.

The fundamental groups give us basic information about the structure of a topological space, but they are often nonabelian and can be difficult to work with.

In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here.