ON FIBREWISE SIMPLICIAL MONOIDSAND
MILNOR-CARLSSON'S CONSTRUCTIONS
JIE WU
In [?], G. Carlsson introduced a simplicial group construction which gives a *
*gen-
eralization of Milnor's F(K) construction [?]. Roughly speaking, if we construc*
*t a
simplicial group which is a free product of a simplicial group Gover a pointed *
*sim-
plicial set X, then we get a simplicial group construction for (BG ^ X), where
BG is the classifying space of G. In this article, we give a categorial view o*
*f this
construction.
Let C be a category. A fibrewise simplicial object over C, roughly speaking, *
*is a
diagram over C with indices in a simplicial set. This is an abstract view of fi*
*brewise
topology [?] or sheaf theory. If the category Chas coproducts, then the abstrac*
*t F-
construction is defined to be certain coadjoint functor from thecategory of fib*
*rewise
simplicial objects over C to the category of simplicial objects over C. Suppos*
*e that
there is a functor T from C to the category of pointed simplicial sets such tha*
*t T
preserves coproducts up to homotopy. Then thereis an induced functor T from the
category of fibrewise simplicial objects over C to the category of pointed bisi*
*mplicial
sets. Theorem ??shows that T is homotopy equivalent to T ffiF. Let C be a categ*
*ory
of monoids. Notice that the bar-construction Bpreserves coproduct up to homotopy
[?]. A corollary of this abstrct theorem is the Carlsson theorem.
An application of Carlsson's construciton to homotopy theory is to give a rep*
*re-
sentation of the homotopy groups of simply connected susp ension spaces to cert*
*ain
combinatorial groups as centers [?]. Applications of Carlsson's construction to*
* min-
imal simplicial groups are given in [?]. In this paper, we pay more attention t*
*o the
geometry of the Carlsson construction. The word length filtration is considered*
*. The
resulting cofibres are certain smash product pinched out certainreduced diagonal
elements (Proposition ??). Our construction in the monoid case is a generalizat*
*ion
of the James construction [?]. We construct certain natural map Hn : (Y ^X ) !
n(Y n^ (X(n)=4n )), which is similar to the James-Hopf map,for any path con-
nected CW-complex Y and any pointed CW-complex X, where X(n)is the n-th fold
self!smash product of X and 4n = f(x1 ^: :^:xn) 2 X(n)j xi= xi+1for some ig
(Theorem!??). A directapplication of these natural maps is to give a decomposit*
*ion
of!H!(F P 1^ X) for F =R, C or H. Let F P21= F P 1=F P 1.
!
!
!
!
Research at MSRCI is supported in part by NSF grant DMS-9022140.
2 JIE WU
Theorem 0.1. Let F = R,C or H and let X be a pointed space. Suppose that H
is a multiplicative homology theory suchthat (1) both H (F P 1) and H (F P21)*
*are
free H (pt)-modules;and (2) the inclusion of the bottom cell Sd !F P 1 induces a
monomorphism in the homology. Then there is a product filtration fFrH (F P 1^
X)gr0 of H(F P 1 ^X) such that F0 =H (pt) and
Fr=Fr1 = (d1)rH (X (r)=4r);
where d = dimRF and is the suspension. Furthermore, this filatraion isnatural *
*for
X.