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Simplicial group models for Ωn S n X

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Abstract

LetX be a pointed simplicial set. The free group functorsF [10] and Γ [1] provide simplicial models of ΩS |X| and Ω∞ S ∞ |X|. The simplicial groupFX is a simplicial subgroup of ΓX, and this corresponds to the inclusion ΩS |X| ⊂ ⊂Ω∞ S ∞ X. In this paper we define free group functors Γ(n) such that Γ(n) X is a model of Ωn S n |X|. Moreover, there is natural filtration

$$FX = \Gamma ^{(2)} X \subset \Gamma ^{(2)} X \subset \cdots \subset \Gamma ^{(n)} X \subset \cdots \subset \Gamma X,$$
(1)

corresponding to the filtration

$$\Omega S|X| \subset \Omega ^2 S^2 |X| \subset \cdots \subset \Omega ^2 S^2 |X| \subset \cdots \subset \Omega ^\infty S^\infty |X|.$$
(1)

.

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Authors and Affiliations

  1. Department of Mathematics, Johns Hopkins University, 21218, Baltimore, MD, USA

    Jeffrey Henderson Smith

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  1. Jeffrey Henderson Smith
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Submitted to the Department of Mathematics on May 1, 1981 in partial fulfillment of the requirements for the Degree of Doctor of Philosophy in Mathematics.

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Smith, J.H. Simplicial group models for Ωn S n X . Israel J. Math. 66, 330–350 (1989). https://doi.org/10.1007/BF02765902

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  • DOI: https://doi.org/10.1007/BF02765902

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