#### Volume 16, issue 4 (2016)

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Localizations of abelian Eilenberg–Mac Lane spaces of finite type

### Carles Casacuberta, José L Rodríguez and Jin-yen Tai

Algebraic & Geometric Topology 16 (2016) 2379–2420
##### 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.

##### Keywords
homotopy, localization, Eilenberg–Mac Lane space, solid ring, rigid ring
##### Mathematical Subject Classification 2010
Primary: 55P20, 55P60
Secondary: 18A40, 16S10