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Localizations of
abelian Eilenberg–Mac Lane spaces of finite type
Carles Casacuberta, José L Rodríguez and Jin-yen Tai |
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Algebraic & Geometric Topology 16 (2016) 2379–2420
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Abstract |
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We prove that every homotopical localization of the circle is an aspherical space whose fundamental group is abelian and admits a ring structure with unit such that the evaluation map 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 of Eilenberg–Mac Lane spaces with respect to any map , where and is any abelian group, and we show that many properties of are transferred to the homotopy groups of . Among other results, we show that, if is a product of abelian Eilenberg–Mac Lane spaces and is any map, then the homotopy groups are modules over the ring 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
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Mathematical Subject Classification 2010
Primary: 55P20, 55P60
Secondary: 18A40, 16S10
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References |
Publication
Received: 30 September 2015
Accepted: 29 October 2015
Published: 12 September 2016
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Authors |
| Carles Casacuberta | |
| Institut de Matemàtica Universitat de Barcelona Gran Via de les Corts Catalanes, 585 08007 Barcelona Spain |
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| José L Rodríguez | |
| Departamento de Matemáticas Universidad de Almería 04120 Almería Spain |
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| Jin-yen Tai | |
| Department of Mathematics Dartmouth College Hanover, NH 03755-3551 United States |
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