In [[mathematics]], well-behaved [[topological space]]s can be '''localized''' at primes, in a similar way to the [[localization of a ring]] at a prime. This construction was described by [[Dennis Sullivan]] in 1970 lecture notes that were finally published in {{harv|Sullivan|2005}}.

The reason to do this was in line with an idea of making [[topology]], more precisely [[algebraic topology]], more geometric. Localization of a space ''X'' is a geometric form of the algebraic device of choosing 'coefficients' in order to simplify the algebra, in a given problem. Instead of that, the localization can be applied to the space ''X'', directly, giving a second space ''Y''.

==Definitions== We let ''A'' be a [[subring]] of the [[rational number]]s, and let ''X'' be a [[simply connected]] [[CW complex]]. Then there is a simply connected CW complex ''Y'' together with a map from ''X'' to ''Y'' such that *''Y'' is ''A''-local; this means that all its [[homology group]]s are modules over ''A'' *The map from ''X'' to ''Y'' is universal for (homotopy classes of) maps from ''X'' to ''A''-local CW complexes. This space ''Y'' is unique up to [[homotopy equivalence]], and is called the '''localization''' of ''X'' at ''A''.

If ''A'' is the localization of '''Z''' at a prime ''p'', then the space ''Y'' is called the '''localization''' of ''X'' at ''p''.

The map from ''X'' to ''Y'' induces [[isomorphism]]s from the ''A''-localizations of the homology and homotopy groups of ''X'' to the homology and homotopy groups of ''Y''.

== See also == [[:Category:Localization (mathematics)]] * [[Local analysis]] * [[Localization of a category]] * [[Localization of a module]] * [[Localization of a ring]] * [[Bousfield localization]]

==References== *{{citation|first=Frank|last=Adams|authorlink=Frank Adams|year=1978|title=Infinite loop spaces|pages=74–95|isbn=0-691-08206-5|publisher=Princeton University Press|location=Princeton, N.J.}} *{{citation|title=Geometric Topology: Localization, Periodicity and Galois Symmetry: The 1970 MIT Notes |series=K-Monographs in Mathematics |first= Dennis P.|last= Sullivan|authorlink=Dennis Sullivan|editor-first= Andrew |editor-last=Ranicki|editor-link=Andrew Ranicki|isbn= 1-4020-3511-X|year=2005|url=http://www.maths.ed.ac.uk/~aar/surgery/gtop.pdf|publisher=Springer|location=Dordrecht}}

[[Category:Homotopy theory]] [[Category:Localization (mathematics)]]

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