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 ( 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 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
- Y is A-local; this means that all its homology groups 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 isomorphisms 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
- Adams, Frank (1978), Infinite loop spaces, Princeton, N.J.: Princeton University Press, pp. 74–95, ISBN 0-691-08206-5
- Sullivan, Dennis P. (2005), Ranicki, Andrew (ed.), Geometric Topology: Localization, Periodicity and Galois Symmetry: The 1970 MIT Notes (PDF), K-Monographs in Mathematics, Dordrecht: Springer, ISBN 1-4020-3511-X
 | This commutative algebra-related article is a stub. You can help Wikipedia by expanding it. |
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 ( 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 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
- Y is A-local; this means that all its homology groups 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 isomorphisms 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
- Adams, Frank (1978), Infinite loop spaces, Princeton, N.J.: Princeton University Press, pp. 74–95, ISBN 0-691-08206-5
- Sullivan, Dennis P. (2005), Ranicki, Andrew (ed.), Geometric Topology: Localization, Periodicity and Galois Symmetry: The 1970 MIT Notes (PDF), K-Monographs in Mathematics, Dordrecht: Springer, ISBN 1-4020-3511-X
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