Israel Journal of Mathematics

, Volume 66, Issue 1, pp 330-350

First online:

Simplicial group models for Ωn S n X

  • Jeffrey Henderson SmithAffiliated withDepartment of Mathematics, Johns Hopkins University

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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 Ω n S n |X|. Moreover, there is natural filtration
$$FX = \Gamma ^{(2)} X \subset \Gamma ^{(2)} X \subset \cdots \subset \Gamma ^{(n)} X \subset \cdots \subset \Gamma X,$$
corresponding to the filtration
$$\Omega S|X| \subset \Omega ^2 S^2 |X| \subset \cdots \subset \Omega ^2 S^2 |X| \subset \cdots \subset \Omega ^\infty S^\infty |X|.$$