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 $(A) \rightarrow A$ 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 $L_{f} K(G, n)$ of Eilenberg-Mac Lane spaces with respect to any map $f$, where $n \geq 1$ and $G$ is any abelian group, and we show that many properties of $G$ are transferred to the homotopy groups of $L_{f} K(G, n)$. 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.