Localizations of abelian Eilenberg–Mac Lane spaces of finite type

Abstract

We prove that every homotopical localization of the circle $S^1$ is an aspherical space whose fundamental group $A$ is abelian and admits a ring structure with unit such that the evaluation map $End\left(A\right)\to A$ 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 $L_{f}K\left(G,n\right)$ of Eilenberg–Mac Lane spaces with respect to any map $f$, where $n\ge 1$ and $G$ is any abelian group, and we show that many properties of $G$ are transferred to the homotopy groups of $L_{f}K\left(G,n\right)$. Among other results, we show that, if $X$ is a product of abelian Eilenberg–Mac Lane spaces and $f$ is any map, then the homotopy groups ${\pi }_m\left(L_{f}X\right)$ are modules over the ring ${\pi }_1\left(L_f{S}^1\right)$ in a canonical way. This explains and generalizes earlier observations made by other authors in the case of homological localizations.