We prove that every homotopical localization of the circle
is an aspherical space
whose fundamental group
is abelian and admits a ring structure with unit such that the evaluation map
at the
unit is an isomorphism of rings. Since it is known that there is a proper class of
nonisomorphic rings with this property, and we show that all occur in this way, it
follows that there is a proper class of distinct homotopical localizations of spaces (in
spite of the fact that homological localizations form a set). This answers a question
asked by Farjoun in the nineties.
More generally, we study localizations
of Eilenberg–Mac Lane spaces with respect to any
map ,
where
and
is any abelian group, and we show that many properties of
are transferred to the
homotopy groups of
. Among
other results, we show that, if
is a product of abelian Eilenberg–Mac Lane spaces and
is any map, then the
homotopy groups
are
modules over the ring
in a canonical way. This explains and generalizes earlier observations made by other
authors in the case of homological localizations.
Keywords
homotopy, localization, Eilenberg–Mac Lane space, solid
ring, rigid ring