| 词条 | Localization of a topological space |
| 释义 |
In mathematics, well behaved topological spaces 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. DefinitionsWe let A be a subring of the rational numbers, 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
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 isomorphisms from the A-localizations of the homology and homotopy groups of X to the homology and homotopy groups of Y. See alsoCategory:Localization (mathematics)
References
|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}} 2 : Homotopy theory|Localization (mathematics) |
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