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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

We prove that every homotopical localization of the circle S1 is an aspherical space whose fundamental group A is abelian and admits a ring structure with unit such that the evaluation map End(A) 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 LfK(G,n) of Eilenberg–Mac Lane spaces with respect to any map f, where n 1 and G is any abelian group, and we show that many properties of G are transferred to the homotopy groups of LfK(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 πm(LfX) are modules over the ring π1(LfS1) in a canonical way. This explains and generalizes earlier observations made by other authors in the case of homological localizations.

homotopy, localization, Eilenberg–Mac Lane space, solid ring, rigid ring
Mathematical Subject Classification 2010
Primary: 55P20, 55P60
Secondary: 18A40, 16S10
Received: 30 September 2015
Accepted: 29 October 2015
Published: 12 September 2016
Carles Casacuberta
Institut de Matemàtica
Universitat de Barcelona
Gran Via de les Corts Catalanes, 585
08007 Barcelona
José L Rodríguez
Departamento de Matemáticas
Universidad de Almería
04120 Almería
Jin-yen Tai
Department of Mathematics
Dartmouth College
Hanover, NH 03755-3551
United States