eBook ISBN:  9781470402921 
Product Code:  MEMO/148/701.E 
List Price:  $52.00 
MAA Member Price:  $46.80 
AMS Member Price:  $31.20 
eBook ISBN:  9781470402921 
Product Code:  MEMO/148/701.E 
List Price:  $52.00 
MAA Member Price:  $46.80 
AMS Member Price:  $31.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 148; 2000; 109 ppMSC: Primary 55; 20; 16; 17
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 \(\mathrm{Lie}(n)\).
ReadershipGraduate students and research mathematicians interested in topology and representation theory.

Table of Contents

Chapters

1. Introduction

2. Natural coalgebra transformations of tensor algebras

3. Geometric realizations and the proof of Theorem 1.3

4. Existence of minimal natural coalgebra retracts of tensor algebras

5. Some lemmas on coalgebras

6. Functorial version of the PoincaréBirkhoffWitt theorem

7. Projective $\mathbf {k}(S_n)$submodules of $\operatorname {Lie}(n)$

8. The functor $A^{\mathrm {min}}$ over a field of characteristic $p > 0$

9. Proof of Theorems 1.1 and 1.6

10. The functor $L’_n$ and the associated $\mathbf {k}(\Sigma _n)$module $\operatorname {Lie}’(n)$

11. Examples


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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 \(\mathrm{Lie}(n)\).
Graduate students and research mathematicians interested in topology and representation theory.

Chapters

1. Introduction

2. Natural coalgebra transformations of tensor algebras

3. Geometric realizations and the proof of Theorem 1.3

4. Existence of minimal natural coalgebra retracts of tensor algebras

5. Some lemmas on coalgebras

6. Functorial version of the PoincaréBirkhoffWitt theorem

7. Projective $\mathbf {k}(S_n)$submodules of $\operatorname {Lie}(n)$

8. The functor $A^{\mathrm {min}}$ over a field of characteristic $p > 0$

9. Proof of Theorems 1.1 and 1.6

10. The functor $L’_n$ and the associated $\mathbf {k}(\Sigma _n)$module $\operatorname {Lie}’(n)$

11. Examples