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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
localized at a prime p as a product (up to homotopy) of other
spaces. Since as a Hopf algebra,
is isomorphic to the tensor algebra T(V), where
, 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
). 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
. 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)
where
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
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
and of the product
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 
representation theory of the symmetric group 
. It provides
some information about the important -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
. There
is an action of on (V) (and thus on T(V)) given by
for
. Let 
be
the subspace of
spanned by
. 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
. Let
and let
. We show that
is a
projective -submodule of
Lie(n) and that any projective
-submodule of
Lie(n) is a retract (up to isomorphism) of
. If n is invertible modulo p then it is well
known that
Lie(n) is itself projective and easy to see that
. In particularly, in characteristic 0,
for all n.
Next: Stephen D. Theriault - -2r Up: 2) Homotopy Theory / Théorie Previous: Laura Scull - Rational S-equivariant
