Israel Journal of Mathematics

, Volume 66, Issue 1, pp 330–350

Simplicial group models for ΩnSnX

  • Jeffrey Henderson Smith

DOI: 10.1007/BF02765902

Cite this article as:
Smith, J.H. Israel J. Math. (1989) 66: 330. doi:10.1007/BF02765902


LetX be a pointed simplicial set. The free group functorsF [10] and Γ [1] provide simplicial models of ΩS |X| and ΩS |X|. The simplicial groupFX is a simplicial subgroup of ΓX, and this corresponds to the inclusion ΩS |X| ⊂ ⊂ΩSX. In this paper we define free group functors Γ(n) such that Γ(n)X is a model of ΩnSn |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
$$\Omega S|X| \subset \Omega ^2 S^2 |X| \subset \cdots \subset \Omega ^2 S^2 |X| \subset \cdots \subset \Omega ^\infty S^\infty |X|.$$

Copyright information

© The Weizmann Science Press of Israel 1989

Authors and Affiliations

  • Jeffrey Henderson Smith
    • 1
  1. 1.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA