Simplicial group models for \(\Omega ^ nS^ nX\). (English) Zbl 0723.55004
Starting with a pointed simplicial set X, J. Milnor [in the book of J. F. Adams: Algebraic topology - A student’s guide (1972; Zbl 0234.55002)] on the one hand and M. G. Barratt and P. J. Eccles [Topology 13, 25-45 (1974; Zbl 0292.55010)] on the other defined free simplicial groups having the homotopy type of \(\Omega\) \(S| X|\) and \(\Omega^{\infty}S^{\infty}| X|\) respectively. The present author fills the gap between 1 and \(\infty\) by defining free simplicial groups \(\Gamma^{(n)}\) of the homotopy type of \(\Omega^ nS^ n| X|\) for every natural number n. These yield a filtration of the Barratt-Eccles group \(\Gamma\) X weakly equivalent to the corresponding filtration of \(\Omega^{\infty}S^{\infty}| X|\).
Reviewer: R.Fritsch (München)
References:
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