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Paul Selick - Natural decompositions of loop suspensions and tensor algebras

PAUL SELICK, Department of Mathematics, University of Toronto, Toronto, Ontario  M5S 3G3, Canada
Natural decompositions of loop suspensions and tensor algebras

(Joint work with Jie Wu).

Consider the full subcategory of pointed topological spaces whose objects are simply connected suspensions of finite type. For X in the above category we examine natural decompositions of $\Omega\Sigma
X$ localized at a prime p as a product (up to homotopy) of other spaces. Since as a Hopf algebra, $H_\ast(\Omega\Sigma X;\bbd Z/p\bbd
Z)$ is isomorphic to the tensor algebra T(V), where $V={\tilde
H}_\ast(X;\bbd Z/p\bbd Z)$, any such decomposition yields a natural coalgebra decomposition of T(V) (which need not be a Hopf algebra decomposition since we have not required our decomposition to respect the H-space structure on $\Omega\Sigma
X$). We have shown that the converse is true: every natural decomposition of T(V) can be geometrically realized as a natural decomposition of the space $\Omega\Sigma
X$. Having thus translated the problem to algebra, we next consider the algebraic problem of finding natural coalgebra decompositions of tensor algebras. We show that there is a natural coalgebra decomposition of T(V) (natural with respect to the vector space V) $T(V) = A^{\min}(V)\otimes B_2(V)\otimes B_3(V)\cdots$where $A^{\min}(V)$ contains V itself and is minimal in the sense that it is (up to isomorphism) a retract of any coalgebra containing V which is a natural retract of T(V). The coalgebra Bn(V) is the smallest natural coalgebra retract containing a certain submodule $L^{\max}_n(V)$ described below. This decomposition generalizes that given by the Poincaré-Birkhoff-Witt Theorem, except it is natural with respect to maps of vector spaces, whereas PBK is natural only with respect to maps of ordered vector spaces.

Some properties of $A^{\min}(V)$ and of the product $B(V)=B_2(V)\otimes
B_3(V)\cdots$ of all the other factors are as follows. B(V) is a sub-Hopf-algebra of T(V) which is a retract as a coalgebra. We show that, as conjectured by Cohen, the only primitives in A(V) occur in weights of the form pt. Also, A(V) has a filtration where each of the filtration quotients is a polynomial algebra. A description of the generators for these polynomial algebras is given for the first p2-1filtration quotients, computation of the others remaining beyond our present capabilities.

One important aspect of this work is its relationship to the $\mod p$representation theory of the symmetric group $\Sigma_n$. It provides some information about the important $\Sigma_n$-module Lie(n)described below which has arisen in many contexts and appears in current work of Cohen, Dwyer, Arone, and others. To define Lie(n)consider the vector space V with basis $\{v_1,\ldots, v_n\}$. There is an action of $\Sigma_n$ on (V) (and thus on T(V)) given by $v_j\mapsto v_{\sigma(j)}$ for $\sigma\in\Sigma_n$. Let $\gamma(V)$be the subspace of $V^{\otimes n}$ spanned by $\{v_{\sigma(1)}\cdots
v_{\sigma(n)} \}$. Let Ln(V) be the primitives of ``weight'' nin T(V) which are indecomposable (i.e. not p-th powers). Explicitly Ln(V) consists of commutators of length n in the elements of V. Let $L^{\max}_n(V)=L_n(V)\cap B$. Let $\textrm{Lie}(n)=L_n(V)\cap\gamma$ and let $\textrm{Lie}^{\max}(n)=
L^{\max}_n(V)\cap\gamma$. We show that $\textrm{Lie}^{\max}(n)$ is a projective $\Sigma_n$-submodule of Lie(n) and that any projective $\Sigma_n$-submodule of Lie(n) is a retract (up to isomorphism) of $\textrm{Lie}^{\max}(n)$. If n is invertible modulo p then it is well known that Lie(n) is itself projective and easy to see that $L^{\max}_n(V)=L_n(V)$. In particularly, in characteristic 0, $L^{\max}_n(V)=L_n(V)$ for all n.

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Next: Stephen D. Theriault - -2r Up: 2)  Homotopy Theory / Théorie Previous: Laura Scull - Rational S-equivariant

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