Localization of a topological space
- Localization of a topological space
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".
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 localizationof "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".
References
*citation|last=Adams|year=1978|title=Infinite loop spaces|pages=74-95|isbn=0691082065
*citation|title=Geometric Topology: Localization, Periodicity and Galois Symmetry: The 1970 MIT Notes |series=K-Monographs in Mathematics
first= Dennis P.|last= Sullivan|editor-first= Andrew |editor-last=Ranicki|ISBN= 140203511X|year=2005
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