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.
Read more about this topic: Localization Of A Topological Space
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