# Localization of a topological space

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Localization_of_a_topological_space
> Markdown URL: https://mediated.wiki/source/Localization_of_a_topological_space.md
> Source: https://en.wikipedia.org/wiki/Localization_of_a_topological_space
> Source revision: 1214297586
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

In [mathematics](/source/Mathematics), well-behaved [topological spaces](/source/Topological_space) can be **localized** at primes, in a similar way to the [localization of a ring](/source/Localization_of_a_ring) at a prime. This construction was described by [Dennis Sullivan](/source/Dennis_Sullivan) in 1970 lecture notes that were finally published in ([Sullivan 2005](#CITEREFSullivan2005)).

The reason to do this was in line with an idea of making [topology](/source/Topology), more precisely [algebraic topology](/source/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](/source/Subring) of the [rational numbers](/source/Rational_number), and let *X* be a [simply connected](/source/Simply_connected) [CW complex](/source/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 groups](/source/Homology_group) 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](/source/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 [isomorphisms](/source/Isomorphism) 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)](https://en.wikipedia.org/wiki/Category:Localization_(mathematics))

- [Local analysis](/source/Local_analysis)

- [Localization of a category](/source/Localization_of_a_category)

- [Localization of a module](/source/Localization_of_a_module)

- [Localization of a ring](/source/Localization_of_a_ring)

- [Bousfield localization](/source/Bousfield_localization)

## References

- [Adams, Frank](/source/Frank_Adams) (1978), *Infinite loop spaces*, Princeton, N.J.: Princeton University Press, pp. 74–95, [ISBN](/source/ISBN_(identifier)) [0-691-08206-5](https://en.wikipedia.org/wiki/Special:BookSources/0-691-08206-5)

- [Sullivan, Dennis P.](/source/Dennis_Sullivan) (2005), [Ranicki, Andrew](/source/Andrew_Ranicki) (ed.), [*Geometric Topology: Localization, Periodicity and Galois Symmetry: The 1970 MIT Notes*](http://www.maths.ed.ac.uk/~aar/surgery/gtop.pdf) (PDF), K-Monographs in Mathematics, Dordrecht: Springer, [ISBN](/source/ISBN_(identifier)) [1-4020-3511-X](https://en.wikipedia.org/wiki/Special:BookSources/1-4020-3511-X)

This commutative algebra-related article is a stub. You can help Wikipedia by adding missing information.

- [v](https://en.wikipedia.org/wiki/Template:Commutative-algebra-stub)
- [t](/source/Template_talk%3ACommutative-algebra-stub)
- [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Commutative-algebra-stub)

---
Adapted from the Wikipedia article [Localization of a topological space](https://en.wikipedia.org/wiki/Localization_of_a_topological_space) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Localization_of_a_topological_space?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.