Memoirs of the American Mathematical Society 2000; 109 pp; softcover Volume: 148 ISBN10: 0821821105 ISBN13: 9780821821107 List Price: US$49 Individual Members: US$29.40 Institutional Members: US$39.20 Order Code: MEMO/148/701
 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 PoincaréBirkhoffWhitt 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
