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A homotopy idempotent construction by means of simplicial groups

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Abstract

We obtain a simplicial group model for localization of (not necessarily nilpotent) spaces at sets of primes by applying a suitable functor dimensionwise, as in earlier work of Quillen and Bousfield-Kan. For a set of primesP and any groupG, letG→L PG be a universal homomorphism fromG into a group which is uniquely divisible by primes not inP, and denote also byL P the prolongation of this functor to simplicial groups. We prove that, ifX is any connected simplicial set andJ is any free simplicial group which is a model for the loop space ΩX, then the classifying space\(\overline W LpJ\) is homotopy equivalent to the localization ofX atP. Thus, there is a map\(X \to \overline W LpJ\) which is universal among maps fromX into spacesY for which the semidirect products πκ(Y)⋊ π1(Y) are uniquely divisible by primes not inP. This approach also yields a neat construction of fibrewise localization.

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Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193, Bellaterra, Spain

    Gemma Bastardas & Carles Casacuberta

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  1. Gemma Bastardas
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  2. Carles Casacuberta
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Corresponding author

Correspondence to Gemma Bastardas.

Additional information

Both authors are supported by DGES under grant PB97-0202, and the first-named author has a DGR fellowship with reference code 1997FI 00467.

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Bastardas, G., Casacuberta, C. A homotopy idempotent construction by means of simplicial groups. Isr. J. Math. 121, 333–349 (2001). https://doi.org/10.1007/BF02802510

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  • DOI: https://doi.org/10.1007/BF02802510

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