Editing Maximal compact subgroup (section)

===Topology===
The [[algebraic topology]] of the Lie groups is also largely carried by a maximal compact subgroup ''K''. To be precise, a connected Lie group is a topological product (though not a group theoretic product) of a maximal compact ''K'' and a Euclidean space – ''G'' = ''K'' × '''R'''<sup>''d''</sup> – thus in particular ''K'' is a [[deformation retract]] of ''G,'' and is [[homotopy equivalent]], and thus they have the same [[homotopy groups]]. Indeed, the inclusion <math>K \hookrightarrow G</math> and the deformation retraction <math>G \twoheadrightarrow K</math> are [[homotopy equivalence]]s.

For the general linear group, this decomposition is the [[QR decomposition]], and the deformation retraction is the [[Gram-Schmidt process]]. For a general semisimple Lie group, the decomposition is the [[Iwasawa decomposition]] of ''G'' as ''G'' = ''KAN'' in which ''K'' occurs in a product with a [[contractible]] subgroup ''AN''.