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.
"Localizations of abelian Eilenberg–Mac Lane spaces of finite type." Algebr. Geom. Topol. 16 (4) 2379 - 2420, 2016. https://doi.org/10.2140/agt.2016.16.2379