# On Natural Coalgebra Decompositions of Tensor Algebras and Loop Suspensions

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*Paul Selick; Jie Wu*

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 \(\mathrm{Lie}(n)\).

#### Table of Contents

# Table of Contents

## On Natural Coalgebra Decompositions of Tensor Algebras and Loop Suspensions

- Contents vii8 free
- 1. Introduction 110 free
- 2. Natural coalgebra transformations of tensor algebras 716 free
- 3. Geometric Realizations and the Proof of Theorem 1.3 1423
- 4. Existence of Minimal Natural Coalgebra Retracts of Tensor Algebras 1827
- 5. Some Lemmas on Coalgebras 2837
- 6. Functorial Version of the Poincaré-Birkhoff-Witt Theorem 3241
- 7. Projective k(S[sub(n)])-Submodules of Lie(n) 4655
- 8. The Functor A[sup(min)] over a Field of Characteristic p > 0 5463
- 9. Proof of Theorems 1.1 and 1.6 7988
- 10. The Functor L'[sub(n)] and the Associated k(Σ[sub(n)])-Module Lie'(n) 8392
- 11. Examples 96105
- 11.1. The functor A[sup(min)][sub(n)] for n ≤ p 96105
- 11.2. The functor B[sup(max)] 97106
- 11.3. The symmetric group module Lie[sup(max)](p) 99108
- 11.4. Calculations for small n when p = 2 100109
- 11.5. Decompositions of ΩΣ[sup(2)]X for two-cell complexes X 101110
- 11.6. The PBW map in characteristic 0 107116

- References 109118