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 ==