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2016 Localizations of abelian Eilenberg–Mac Lane spaces of finite type
Carles Casacuberta, José Rodríguez, Jin-yen Tai
Algebr. Geom. Topol. 16(4): 2379-2420 (2016). DOI: 10.2140/agt.2016.16.2379

Abstract

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.

Citation

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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

Information

Received: 30 September 2015; Accepted: 29 October 2015; Published: 2016
First available in Project Euclid: 28 November 2017

zbMATH: 1353.55002
MathSciNet: MR3546469
Digital Object Identifier: 10.2140/agt.2016.16.2379

Subjects:
Primary: 55P20, 55P60
Secondary: 16S10, 18A40

Rights: Copyright © 2016 Mathematical Sciences Publishers

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Vol.16 • No. 4 • 2016
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