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 (Sullivan 2005).

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.

References

• Adams (1978), Infinite loop spaces, pp. 74–95
• Sullivan, Dennis P. (2005), Ranicki, Andrew (ed.), Geometric Topology: Localization, Periodicity and Galois Symmetry: The 1970 MIT Notes, K-Monographs in Mathematics, ISBN 140203511X