Editing Group theory (section)

===Algebraic topology===
{{Main|Algebraic topology}}
[[Algebraic topology]] is another domain which prominently [[functor|associates]] groups to the objects the theory is interested in. There, groups are used to describe certain invariants of [[topological space]]s. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some [[homeomorphism|deformation]]. For example, the [[fundamental group]] "counts" how many paths in the space are essentially different. The [[Poincaré conjecture]], proved in 2002/2003 by [[Grigori Perelman]], is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of [[Eilenberg–MacLane space]]s which are spaces with prescribed [[homotopy groups]]. Similarly [[algebraic K-theory]] relies in a way on [[classifying space]]s of groups. Finally, the name of the [[torsion subgroup]] of an infinite group shows the legacy of topology in group theory.

[[File:Torus.png|thumb|right|200px|A torus. Its abelian group structure is induced from the map {{nowrap|'''C''' → '''C'''/('''Z''' + ''τ'''''Z''')}}, where ''τ'' is a parameter living in the [[upper half plane]].]]