Abstract
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.
Citation
Carles Casacuberta. José Rodríguez. Jin-yen Tai. "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
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