Abstract. We consider functorial decompositions of \(\Omega\Sigma X\) in the case where \(X\) is a \(p\)-torsion suspension. By means of a geometric realization theorem, we show that the problem can be reduced to the one obtained by applying homology: that of finding natural coalgebra decompositions of tensor algebras. We solve the algebraic problem and give properties of the piece \(A^{\mathrm{min}}(V)\) of the decomposition of \(T(V)\) which contains \(V\) itself, including verification of the Cohen conjecture that in characteristic \(p\) the primitives of \(A^{\mathrm{min}}(V)\) are concentrated in degrees of the form \(p^t\). The results tie in with the representation theory of the symmetric group and in particular produce the maximum projective submodule of the important \(S_n\)-module \(\mathrm{Lie}(n)\).

Readership

Graduate students and research mathematicians interested in topology and representation theory.

Table of Contents

• Introduction
• Natural coalgebra transformations of tensor algebras
• Geometric realizations and the proof of Theorem 1.3
• Existence of minimal natural coalgebra retracts of tensor algebras
• Some lemmas on coalgebras
• Functorial version of the Poincaré-Birkhoff-Whitt theorem
• Projective \(\mathbf{k}(S_n)\)-submodules of Lie\((n)\)
• The functor \(A^{\mathrm{min}}\) over a field of characteristic \(p>0\)
• Proof of Theorems 1.1 and 1.6
• The functor \(L^\prime_n\) and the associated \(\mathbf{k}(\Sigma_n)\)-module \(\mathrm{Lie}^\prime(n)\)
• Examples
• References