Editing Algebraic topology (section)

==Main branches==
Below are some of the main areas studied in algebraic topology:

===Homotopy groups===
{{Main| Homotopy group}}
In mathematics, homotopy groups are used in algebraic topology to classify [[topological space]]s. 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===
{{Main|Homology (mathematics)|l1=Homology}}
In algebraic topology and [[abstract algebra]], '''homology''' (in part from [[Greek language|Greek]] ὁμός ''homos'' "identical") is a certain general procedure to associate a [[sequence]] of [[abelian group]]s or [[module (mathematics)|modules]] with a given mathematical object such as a [[topological space]] or a [[group (mathematics)|group]].<ref>{{harvtxt|Fraleigh|1976|p=163}}</ref>

===Cohomology===
{{Main|Cohomology}}
In [[homology theory]] and algebraic topology, '''cohomology''' is a general term for a [[sequence]] of [[abelian group]]s defined from a [[chain complex|cochain complex]]. That is, cohomology is defined as the abstract study of '''cochains''', [[chain complex|cocycle]]s, and [[coboundary|coboundaries]]. Cohomology can be viewed as a method of assigning [[algebraic invariant]]s to a topological space that has a more refined [[algebraic structure]] than does [[homology (mathematics)|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 ''[[chain (algebraic topology)|chains]]'' of homology theory.

===Manifolds===
{{Main|Manifold}}
A '''manifold''' is a [[topological space]] that near each point resembles [[Euclidean space]]. Examples include the [[Plane (geometry)|plane]], the [[sphere]], and the [[torus]], which can all be realized in three dimensions, but also the [[Klein bottle]] and [[real projective plane]] which cannot be embedded in three dimensions, but can be embedded in four dimensions. Typically, results in algebraic topology focus on global, non-differentiable aspects of manifolds; for example [[Poincaré duality]].

===Knot theory===
{{Main|Knot theory}}
'''Knot theory''' is the study of [[knot (mathematics)|mathematical knot]]s. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined so that it cannot be undone. In precise mathematical language, a knot is an [[embedding]] of a [[circle]] in three-dimensional [[Euclidean space]], <math>\mathbb{R}^3</math>. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of <math>\mathbb{R}^3</math> upon itself (known as an [[ambient isotopy]]); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

===Complexes===
{{Main|Simplicial complex|CW complex}}
[[File:Simplicial complex example.svg|thumb|200px|A simplicial 3-complex.]]
A '''simplicial complex''' is a [[topological space]] of a certain kind, constructed by "gluing together" [[Point (geometry)|point]]s, [[line segment]]s, [[triangle]]s, and their [[Simplex|''n''-dimensional counterparts]] (see illustration). Simplicial complexes should not be confused with the more abstract notion of a [[simplicial set]] appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an [[abstract simplicial complex]].

A '''CW complex''' is a type of topological space introduced by [[J. H. C. Whitehead]] to meet the needs of [[homotopy theory]]. This class of spaces is broader and has some better [[category theory|categorical]] properties than [[simplicial complex]]es, but still retains a combinatorial nature that allows for computation (often with a much smaller complex).