ON FIBREWISE SIMPLICIAL MONOIDS AND
MILNOR-CARLSSON'S CONSTRUCTIONS
JIE WU
Abstract. We give a categorical view of Milnor-Carlsson's constructions. *
* The
word length filtration is studied. Certain natural maps are combinatoria*
*lly con-
structed. An application to the homology of (FP1 ^ X) is given for F = R,*
* C
or H.
1. Introduction
In [4], G. Carlsson introduced a simplicial group construction which gives a *
*gen-
eralization of Milnor's F (K) construction [10]. Roughly speaking, if we const*
*ruct
a simplicial group which is a free product of a simplicial group G over a point*
*ed
simplicial set X, then we get a simplicial group construction for (BG ^ X), whe*
*re
BG is the classifying space of G. In this article, we give a categorial view *
*of 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 [8] or sheaf theory. If the category C has coproducts, then the abstra*
*ct F-
construction is defined to be certain coadjoint functor from the category of fi*
*brewise
simplicial objects over C to the category of simplicial objects over C. Suppose*
* 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 there is an induced functor T from the
category of fibrewise simplicial objects over C to the category of pointed bisi*
*mplicial
sets. Theorem 4.7 shows that T is homotopy equivalent to T OF . Let C be a cate*
*gory
of monoids. Notice that the bar-construction B preserves coproduct up to homoto*
*py
[6]. 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 suspension spaces to certa*
*in
combinatorial groups as centers [15]. Applications of Carlsson's construction t*
*o min-
imal simplicial groups are given in [16]. In this paper, we pay more attention *
*to the
geometry of the Carlsson construction. The word length filtration is considered*
*. The
resulting cofibres are certain smash product pinched out certain reduced diagon*
*al
elements (Proposition 5.2). Our construction in the monoid case is a generaliza*
*tion
of the James construction [8]. We construct certain natural map Hn : (Y ^ X) !
___________
Research at MSRCI is supported in part by NSF grant DMS-9022140.
1
2 JIE WU
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 = {(x1 ^ : :^:xn) 2 X(n)| xi = xi+1for some i}
(Theorem 6.11). A direct application of these natural maps is to give a decompo*
*sition
of H*(F P 1 ^ X) for F = R, C or H. Let F P21 = F P 1=F P 1.
Theorem 1.1. Let F = R, C or H and let X be a pointed space. Suppose that H*
is a multiplicative homology theory such that (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 induce*
*s a
monomorphism in the homology. Then there is a product filtration {FrH*(F P 1 ^
X)}r0 of H*(F P 1 ^ X) such that F0 = H*(pt) and
Fr=Fr-1 ~=(d-1)rH*(X(r)=4r );
where d = dimRF and is the suspension. Furthermore, this filatraion is natural*
* for
X.
In particular, if F = R, this theorem holds for the mod 2 homology and if F =*
* C
or H, this theorem holds even for integral homology.
The article is organized as follows. In Section 2, we recall some basic prop*
*erties
of the free products of monoids. In Section 3, we recall some basic properties*
* of
the bar constructions. The fibrewise simplicial monoids are introduced in Secti*
*on 4.
The word length filtration is studied in Section 5. Theorem 1.1 is given in Sec*
*tion 5
without proof, where Theorem 5.8 is Theorem 1.1. Some natural maps are combina-
torially constructed in Section 6. The proof of Theorem 1.1 (Theorem 5.8) is gi*
*ven
in Section 7. In Section 8, we give a description of the group ring Fp(F Z=p(X)*
*).
The author is indebted to helpful discussions with J. C. Moore. The abstract *
*F-
construction was suggested by him and he conjectured the main results in Sectio*
*n 4.
2. Free Products of Monoids
In this section, we recall some basic properties of the free products of mono*
*ids.
Most theorems in this section are well known. We only give the theorems which w*
*ill
be used in next sections.
Notation 2.1. Let {Mx | x 2 X} be a given set of the monoids Mx, where X is an
(indices) set (pointed set).
We write Xa as the element a in Mx with index x. A word w is an ordered system
of elements
w = (xa11xa22: :x:ann);
where xaii2 Mxi. A reduced word w is an ordered system of elements
w = (xa11xa22: :x:ann)
ON FIBREWISE SIMPLICIAL MONOIDS AND MILNOR-CARLSSON'S CONSTRUCTIONS 3
such that aiis not the unit element 1 in the monoid Mxifor i i n and xi6= xi+*
*1for
1 i n-1 (xi6= * for 1 i n if X is a pointed set with a point *). In additio*
*n, we
regard the case n = 0 as corresponding to the empty (reduced) word. The set of *
*all of
the words is denoted by M". Let M = M"= denote by the quotient set of M" modulo
the equivalent relation generated by (1) (xa11xa22: :x:ann) (xa11: :x:ai-1i-1x*
*ai+1i+1:x:a:nn)
if ai= 1 for some 1 i n (or xi= * for some 1 i n if x is a pointed set with
a point *); and (2) (xa11xa22: :x:ann) (xa11: :x:ai-1i-1xaiai+1ixai+2i+2:x:a:n*
*n) if xi = xi+1 for
some i. Furthermore, let M* denote by the set of all of the reduced words.
Recall that the free product of monoids is defined to be the coproduct of the
monoids in the category of monoids and homomorphisms. We will show that there
exist natural multiplications in the sets M and M* such that both M and M* are
free products of the monoids Mx. In the case of groups, the proof can be found *
*out
in the standard text book of group theory.
Proposition 2.2. The binary operation in M induced by juxtaposition make M to
be a monoid. Furthermore, M is a free product of the monoids Mx with the natural
inclusion Mx ! M
Proof: The juxtaposition given by
(xa11: :x:ann)(yb11: :y:bmm) = (xa11: :x:annyb11: :y:bmm)
preserves the equivalent relation. Thus the binary operation is well defined i*
*n the
quotient set M = M"= . It is easy to check that this operation is associative w*
*ith
unit element 6 0. The natural inclusion Mx ! M is given by
a ! {(xa)}
which is a monoid homomorphism. Furthermore, it is straightforward to check that
M is a coproduct of the monoids Mx.
Now we define a product in M* as follows: Let w = (xa11: :x:annand w0= yb11: *
*:y:bmm)
be two reduced words. There exists a unique integer 0 i min{n; m} such that
(1) xn-j+1 = yj and an-j+1 . bj = 1 for z j i and (2) xn-i 6= yi+1, or xn-i =*
* yi+1
but an-ibi+16= 1. If xn-i 6= yi+1, we put
w . w0= (xa11: :x:an-in-iybi+1i+1:y:b:mm)
. But if xn-i = yi+1and an-ibi+16= 1, then
w . w0= (xa11: :x:an-i-1n-i-1xan-ibi+1n-iybi+2i+2:y:b:mm):
This product gives a binary operation in M*. The empty word is the unit eleme*
*nt.
The proof of associativity is technically complicated. We start with the follo*
*wing
lemma which follows directly from the definition.
Lemma 2.3. Let w = (xa11: :x:ann) and w0= yb11: :y:bmmbe reduced words. Then
w . w0= (xa11. ((xa22: :x:ann) .:w0)
4 JIE WU
Lemma 2.4. Let w = (xa11: :x:ann) and w0= yb11: :y:bmmbe reduced words. Then
(w . w0) . (xa) = w . (w0. (xa))
for any reduced word (xa) of length 1.
Proof: If n = 1, the assertion follows from the definition by checking the sev*
*eral
cases. For n > 1, the assertion follows by induction on n as follows.
b b a
(xa11: :x:ann) . (y11: :y:mm) . (x )
a a b b a
= (xa11) . (x22: :x:nn) . (y11: :y:mm) . (x()by Lemma 2.3)
a a b b a
= (xa11) . (x22: :x:nn) . (y11: :y:mm). (x()by induction)
a a a b b a
= (x11) . (x22: :x:nn) . (y11: :y:mm). (x ) (by the case n = 1)
a a a b b a
= (x11x22: :x:nn) . (y11: :y:mm). (x ) (by Lemma 2.3).
Proposition 2.5. The product in M* is associative, that is,
(w . w0) . w00= w . (w0. w00)
for any reduced words w, w0 and w00.
Proof: let w = xa11: :x:ann), let w0 = (yb11: :y:bmm) and let w00= (zc11: :z:c*
*pp). The
formula holds for the cases p = 0; 1. Now we show that the formula holds for an*
*y w00
by induction on p. This follows from the following equations.
a a b b c c
(x11: :x:nn) . (y11: :y:mm). (z11: :z:pp)
a a b b h c cp-1 c i
= (x11: :x:nn) . (y11: :y:mm). (z11: :z:p-1) . (zpp)
h i
= (xa11: :x:ann) . (yb11: :y:bmm). (zc11:.:z:cp-1p-1)(zcpp) (by the case*
* p = 1)
h h i i
= (xa11: :x:ann) . (yb11: :y:bmm) . (zc11:.:z:cp-1p-1)(zcpp) (by induc*
*tion)
hh i i
= (xa11: :x:ann) . (yb11: :y:bmm) . (zc11: :z:cp-1p-1).((zcpp)by the case*
* p = 1)
h h i i
= (xa11: :x:ann) . (yb11: :y:bmm) . (zc11: :z:cp-1p-1)(.b(zcpp)y the case*
* p = 1)
h i
= (xa11: :x:ann) . (yb11: :y:bmm) . (zc11: :z:cp-1p-1):
Theorem 2.6. The monoid M* is a free product of the monoids Mx
Proof: We have shown that M* is a monoid. The inclusion Mx ! M* is given by
a ! (xa). Now it is straight forward to check that M* satisfies the universal p*
*roperty
of the coproducts.
ON FIBREWISE SIMPLICIAL MONOIDS AND MILNOR-CARLSSON'S CONSTRUCTIONS 5
3. Classifying Spaces of Simplicial Monoids
There are many constructions for the classifying spaces of a simplicial monoi*
*d. We
recall two natural constructions which are used in this paper.
Let M be a simplicial monoid. Define a simplicial set E(M) as follows:
E(M)n = Mn x Mn-1 x : :x:M0
with face and degeneracy functions
dj(xn; : :;:x0) = djxn; dj-1xn-1; : :;:d0xn-j . xn-j-1; xn-j-2; : :;:x0)
and sj(xn; : :;:x0) = (sjxn; sj-1xn-1; : :;:s0xn-j; xn-j; : :;:x0) for 0 j n *
*and
xj 2 Mj.
The simplicial set E(M) is a contractible simplicial set with a (right) M-act*
*ion
given by
(xn; : :;:x0) . yn = (xnyn; xn-1; : :;:x0)
for (xn; : :;:x0 2 E(M)n and yn 2 Mn. An easy proof to show that E(M) is a
contractible simplicial set is to contract a simplicial map
F : C (E(M)) ! E(M);
which is an extension of the identity map 1E (M), given by
f (0; (xn; : :;:x0))= (xn; : :;:x0)
and
f (1; (xn-1; : :;:x0))= (1; xn-1; : :;:x0)
for the nondegenerate elements in C (E(M)) n, where, for any simplicial set X, *
*the
cone C(X) is defined by
C(X)n = {(q; xn-q) | q 0; xn-q 2 Xn-q}
ae
(q; dj-qxn-q) forj q
dj(q; xn-q) = (q - 1; x
n-q) forj < q
ae
(q; sj-qxn-q) forj q
and sj(q; xn-q) = (q + 1; x
n-q) forj < q;
(see [5]; also [12]).
Now the standard bar-construction B(M) of M is defined by
B(M) = E(M)=M;
the quotient simplicial set of E(M) modulo the M-action. Another bar-constructi*
*on
B4 (M) is constructed as follows.
Let M be a simplicial monoid. Define a bisimplicial set E(M)** by
E(M)n;m = E(Mn)m
6 JIE WU
with face and degeneracy functions induced by M and E(Mn) in the canonical way.
The simplicial set E4 (M) is defined to the diagonal simplicial set of the bisi*
*mplicial
set E(M)**. Notice that E(M)n;*is contractible for each n 0. By a result of A.
Bousfield and D. Kan ([3, Lemma 4.2, pp. 335]; also [2]), the diagonal simplici*
*al set
E4 (M) is also contractible. The (right) M-action on E4 (M) is given by
(xn; : :;:x0) . yn = (xnyn; xn-1; : :;:x0)
for (xn; : :;:x0) 2 E4 (M)n = Mnx: :x:Mn and yn 2 Mn. Now the bar-construction
B4 (M)is defined by
B4 (M) = E4 (M)=M;
the quotient simplicial set of E4 (M) modeulo the M-action.
The natural transformation O : B4 ! B is defined by
O(M) : B4 (M) ! B(M)
O(xn-1; : :;:x0) = (d0xn-1; d20xn-2; : :;:dn0x0)
for (xn-1; : :;:x0) 2 B4 (M)n = Mn x : :x:Mn.
Proposition 3.1. The simplicial map O : B4 (M) ! B(M) is a homotopy equiva-
lence for each simplicial monoid M.
This theorem can follow from a general theorem given in [14], [13] or [6]. We*
* give
an elementary proof as follows.
Proof: Notice that both Z(E4 (M)) and Z(E4 (M)) produce free resolutions over
Z(M). This shows that the map
O : B4 (M) ! B(M)
is a homology equivalence. By the standard excise, the canonical homomorphisms
i1:ss0(M) ! ss1(B4 (M)) and i2:ss0(M) ! ss1(B4 (M)) satisfy the universal condi*
*tion
of the group completion. Furthermore, there is a commutative diagram
ss1(B4 (M))
i1
%
ss0(M) # O*
i2
&
ss1(B(M)):
Thus O* : ss1(B4 (M)) ! ss1(B(M)) is an isomorphism and so O is a weak homoto*
*py
equivalence. This shows that O is a homotopy equivalence which is the assertion.
Recall that, for any pointed simplicial set X, the (reduced) suspension X is
defined by X = C(X)= (X [ C(*)), (rf: [5]; also [12]), where * is the base-poin*
*t of
X. The nondegenerate elements in (X)n are (1; xn-1) for nondegenerate elements
ON FIBREWISE SIMPLICIAL MONOIDS AND MILNOR-CARLSSON'S CONSTRUCTIONS 7
xn-1 2 xn-1. Let Mbe a simplicial monoid. The suspension map oe : M ! BM is
defined by
oe(1; xn-1) = (xn-1; 1; : :;:1) 2 B(M)n = Mn-1 x Mn-2x : :x:M0
for the nondegenerate elements (1; xn-1) in Xn.
Proposition 3.2. (1). The map oe* : H*(M) ! H*+1(BM) is the homology suspen-
sion in the bar spectral sequence for H*(BM).
(2). If M is a simplicial group, the map |oe| : |M| ! |BM| is the adjoint map*
* of
|M| -'! |BM| up to homotopy.
Proof: (1). is obvious. Now suppose that M is a simplicial group. The twisting *
*func-
tion t : BM ! M; (xn-1; xn-2; : :;:x0) ! xn-1 induces a simplicial homomorphism
"t: G(BM) ! M, where G(X) is Kan's G-construction [9] for a reduced simplicial
set X, which is a homotopy equivalence. Notice that G(M) ~=F (M); where F (X)
is Milnor's F(K)-construction [10], and the composite
"t
M ! G(M) -Goe!G(BM) -! M
is the identity map 1M . The assertion folows.
4. Fibrewise Simplicial Objects
In this section, we introduce fibrewise simplicial monoid to give a categoric*
*al under-
standing of Milnor's F(K)-construction and Carlsson's construction. We start wi*
*th
an abstract construction.
Definition 4.1. Let C be a category. A fibrewise simplicial object A over C is*
* a
diagram of C over a non-empty simplicial set S.
More precisely, A = {Ax|x 2 Sn}n0 together with morphisms in C
dj : Ax ! Adjx
and
sj : Ax ! Asjx
for 0 j n and x 2 Sn such that the following simplicial identities hold:
didj = dj-1di8fori < j
< sj-1di fori < j
disj = identity fori = j; J + 1
: s
jdi-1for i > j + 1
sisj = sj+1si fori j
Example 4.2. Let C be a pointed category and let S be a simplicial set. Let Ax*
* = *,
the point in C, for each x 2 S. Then 0(S) = {Ax|x 2 Sn}n0 together with canoni*
*cal
face and degeneracy morphisms is a fibrewise simplicial object over C.
8 JIE WU
Example 4.3. Let C be a category and let X be a simplicial object over C. Let
(Xn)pt= Xn. Then X = {(Xn)pt}n0 is a fibrewise simplicial object C, where S =
{pt}. Thus the category of simplicial objects over C embeds into the category *
*of
fibrewise simplicial objects over C in this canonical way.
The category of fibrewise simplicial objects over C has enough structure prov*
*ided C
has. It is straight forward to show that the category of fibrewise simplicial o*
*bjects over
C is complete (cocomplete, bicomplete) if C is complete (cocomplete, bicomplete*
*).
Definition 4.4. Let T be a functor from the category C to the category of point*
*ed
simplicial sets.
Let A be a fibrewise simplicial object over C. The bisimplicial set T(A)**is *
*defined
by
T (A)n;m = VX2Sn T (Ax)m
where V is the wedge of pointed sets. The face and degeneracy functions are ind*
*uced
by the ones in A in the canonical way.
The F-construction on the category of fibrewise simplicial objects over C is *
*defined
as follows.
Definition 4.5. Let C be a category with coproducts and let A be a fibrewise si*
*mpli-
cial object over C.
The fibrewise simplicial object over C, F (A), is defined by
a
F (A)n = Ax
x2Sn
`
where is the coproduct in the category C. The face and degeneracy functions *
*are
inducedby the ones in A in the canonical way.
An important property of the construction F (A) is that the Seifert-van Kampen
theorem holds.
Proposition 4.6. Suppose that C is a cocomplete category. Then the construction
F (A) preserves the colimits.
Proof: Let f:s:C denote by the category of the fibrewise simplicial objects ov*
*er
C. The objects in f:s:C are fibrewise simplicial objects over C. The morphism*
*s in
f:s:C are the morphisms of the diagrams over C. Let E : s:C ! f:s:C denote by
the embedding functor given in Example 4.3, where s:C is the category of simpli*
*cial
objects over C.` Let A = {Ax|x 2 Sn}n0 be an object in f:s:C. The morphisms
Ax ! F (A)n = x2SnAx gives a natural transformation i : A ! EF (A). Now
let {Aff} be a diagram over f:s:C and let C be an object over s:C with a morphi*
*sm
{F (Aalpha)} ! C of diagrams over s:C. The composite
{Aff} ! {EF (Aff)} ! EC
ON FIBREWISE SIMPLICIAL MONOIDS AND MILNOR-CARLSSON'S CONSTRUCTIONS 9
is a morphism of the diagrams over f:s:C. Thus there is a unique extension morp*
*hism
colim Aff! EC and so there is an extension morphism F ( colim Aff) ! F EC = C
which is unique because F ( colim Aff)n is a coproduct of the objects of dimens*
*ion n
in the diagram colim Aff. The assertion follows.
The geometry of F (A)is given as follows:
Theorem 4.7. Let C be a pointed category with coproducts and let T : C ! s:se*
*ts
be a functor, where s:sets is the category of pointed simplicial sets. Suppose*
* that
T preserves the corproducts up to homotopy. Then
T (A)**' T(F (A))**
as bisimplicial sets for any fibrewise simplicial object A over C.
Proof: The morphism A ! EF (A), where Eis the embedding functor, gives a
simplicial map g : T (A)** ! T (EF`(A))** = T (F (A))**. Notice that T (A)n;*=
Vx2SnT (Ax)* and T(F (A))n;*= T ( x2SnAx)*. The maps g : T(A)n;*! T(F (A))n;*
are homotopy equivalences for each n. By the Bousfied-Kan Theorem [3, Lemma 4.2,
pp.335], the map g : T (A)** ! T (F (A))** is a homotopy equivalence which is t*
*he
assertion.
Now we come back to study fibrewise simplicial monoids, i.e., the fibrewise s*
*impli-
cial objects over the category of monoids.
Definition 4.8. Let M be a simplicial monoid and let S be a pointed simplicial *
*set
with a base point x0.
The simplicial monoid F M(S)is defined by
a*
F M(S)n = (Mn)x= ;
x2Sn
` *
where (Mn)x is a copy of Mn, where is the free product of monoids and where
the product equivalent relation is generated by
gxo 1
for the elements g in Mn with base point index x0. The face and degeneracy func*
*tions
are induced by the homomorphism
dj : (Mn)x ~=Mn ! Mn-1 ~=(Mn-1)djx
and
sj : (Mn)x ~=Mn ! Mn+1 ~=(Mn+1)sjx
for 0 j n.
Theorem 4.9. There is a homotopy equivalence
BM ^ S ! BF M(S)
for any pointed simplicial set S and any simplicial monoid M.
10 JIE WU
Proof: The second coordinate projection p : M x S=M x x0 ! S, where x0 is
the base point, induces a fibrewise simplicial monoid A with A = {(Mn)x; (1)x0|*
*x 2
Sn-{x0}}n0 . By the definition of F M construction, F M(S) ~=F (A). Notice that*
* the
bar-construction B preserves the coproducts up to homotopy [6]. By Theorem 4.7,
there is a homotopy equivalence B (A)**- '!B F (A)**. Notice that
B (A)n;*= (Vx2SnB((Mn)x)*)=B ((Mn)x0)*~=B(Mn)* ^ Sn
for each N 0. Thus there is an isomorphism
B(A)**~= B(M)**^ "S**;
where "S**is a bisimplicial set defined by "Snm= Sn. By taking the diagonal sim*
*plicial
set, we have a homotopy equivalence
B4 (M) ^ S ! B4 (F M(S)):
The assertion follows by Proposition 3.1.
Remark 4.10. (1). If M is the monoid of non-negative integers, then F M(S) is
exact James' construction J(S) [7] or Milnor's construction F +(S) [10].
(2). If M is the abelian group Z, then F M(S) produces a (reduced) free simpl*
*icial
group generated by S which is exact Milnor's F(K)-construction [10].
(3). In the case for a group G, the F G(S) construction here is a special ca*
*se of
Carlsson's construction [4].
Example 4.11. Let M be a simplicial monoid and let S be a pointed simplicial s*
*et
with a right pointed M-action. The simplicial monoid C+M(S) is defined by
C+M(S) = F +(S x M)=
where F +(X) is the reduced free simplicial monoid generated by a pointed simpl*
*icial
set, i.e. Milnor's F +-construction, and where the product equivalent relation*
* is
generated by requiring the relations
[x; g][x; gh] [x; gh]
for x 2 S and g; h 2 M. The associate fibrewise simplicial monoid C+M(S) for C+*
*M(S)
is given by
+ -1
C+M(S) = CM p (a) ; (1)x0 | a 2 Sn=Mn - {p(x0)};
where p : S ! S=M is the quotient map and x0 is the base point. If G is a simpl*
*icial
group, then the simplicial group CG (S) is defined to be the group completion o*
*f the
simplicial monoid C+G(S).
The Carlsson Theorem is as follows.
Theorem 4.12 (Carlsson). Let G be a simplicial group and let S be a pointed si*
*m-
plicial G-set.
ON FIBREWISE SIMPLICIAL MONOIDS AND MILNOR-CARLSSON'S CONSTRUCTIONS 11
Then the classifying space BCG (S) is homotopy equivalent to the homotopy cof*
*ibre
of the map S ! EG xG S=EG xG x0, where x0 is the base point [4, Theorem 9].
We should point out that the group completion UM of a simplicial monoid M, i.*
*e.,
U(M)n = U(Mn), is not a homotopy group completion BM of M in general even
if M is a connected simplicial monoid.
Example 4.13. Let M be discrete monoid such that BM ' S2. By a result of
Fiedorowicz [6], any connected CW-complex has a homotopy type of the classifying
space of a discrete monoid. Notice that ss1(BM) = 0. The group completion UM =
{1}. Now, for any n > 0, F M(Sn) ' Sn+2. But UF M(Sn) ~=F U(M)(Sn) = {1} for
each n.
For any fibrewise simplicial monoid A, there is a simplicial set A defined by*
* An =
Ux2SnAx, the disjoint union, together with a simplicial surjection pA : A ! S, *
*gx ! x
for gx 2 Ax. The simplicial set A has a property that the face and degeneracy
functions, in A , restricted to each fibre p-1A(x) are homomorphiss. Conversely*
*, given
a sipmlicial surjection f : X ! S such that each fibre f-1(x) is a moniod and t*
*he
face and degeneracy functions, in X, restricted to each fibre are homomorphisms*
* of
monoids. Then the diagram A = {F -1(x) | x 2 Sn}n0 is a fibrewise simplicial
monoid such that A = X. This gives a geometric description.
Proposition 4.14. Let A be a fibrewise simplicial monoid. Then the geometric r*
*e-
alization |pA | : |A| ! |S| is a fibre-wise topologial monoid under the weak to*
*pology
(rf: [8]).
Proof: It is easy to check that A is fibrewise simplicial monoid if and only i*
*f the
simplicial surjection pA : A ! S has a fibrewise multiplication, i.e., there e*
*xists a
commutative diagram of simplicial sets
A 4 A -m! A
# pA 4 pA # pA
~=
4(S) -! S;
where 4(S) is the diagronal subsimplicial set of S x S and A 4 A = (pA x
pA )-1 (4(S)) A x A , such that the fibrewise multiplication m : A 4 A ! A is
fibrewise associative with a fibrewise unit element. The assertion follows by a*
*pplying
the geometric realization [11].
Remark 4.15. (a). Any nontrivial principal G-bundle is not a fibrewise simpli*
*cial
monoid since any fibrewise simplicial monoid has a cross-section. (b). The cons*
*truct
F M(X) is the tensor product X M.
5. The Word Length Filtration of F M(X)
In this section, we study some natural filtrations in the constructions F (A)*
* and
F M(X).
12 JIE WU
Definition 5.1. Let A be a fibrewise simplicial monoid. The word length filtrat*
*ion
{Fr(A)}r0 on the simplicial monoid F (A) is defined by
Fr(A)n = {! 2 F (A)n = q*x2SnAx | `(!) r};
where `(!) is the length of the reduced word ! in the free product q*x2SnAx. T*
*he
family of the subsimplicial sets, {Fr(A)}r0 , gives a product filtration of the*
* simplicial
monoid F (A) starting with F0(A) = * and F1(A) = A=S, where pA : A ! S is the
associate simplicial surjection of A and S is identified with its zero section.
Proposition 5.2. Let A be a fibrewise simplicial monoid with pA : A ! S. Then
there is an isomorphism of simplicial sets
Fr(A)=Fr-1(A) ~=(A =S)(r)=Dr
where X(r)is the r-fold self smash product of X and where the subsimplicial set*
* Dr
of (A =S)(r)is given by
Dr = {a1 ^ : :^:ar 2 (A =S)(r)| the indicespA (ai) = pA (ai+1) for some i}:
Proof: Consider the projection map ' : Ar ! Fr(A) defined by '(a1; : :a:r) =
a1: :a:rfor (a1; : :;:ar) 2 Ar = A x : :x:A. Then the map ' satisfies the prope*
*rty:
'(a1; : :;:ar) 2 Fr-1(A)
if ai = 1 for some i or pA (ai) = pA (ai+1) for some i. Thus there exists a sim*
*plicial
map ' : (A =S)(r)=Dr ! Fr(A)=Fr-1(A) such that the diagram
Ar -'! Fr(A)
# #
'
(A =S)(r)=Dr -! Fr(A)=Fr-1(A)
commutes. By the Reduce Word Theorem (Theorem 2.2), the map
' : (A =S)(r)=Dr ! Fr(A)=Fr-1(A)
is an isomorphism, which is the assertion.
Corollary 5.3. Let X be a pointed simplicial set and let M be a simplicial mono*
*id.
Let {FrM(X)}r0 be the word length filtration of F M(X). Then there is an isomo*
*r-
phism of simplicial sets
FrM(X)=FrM-1(X) ~=M(r)^ (X(r)=4 r);
where 4 r= {x1 ^ : :^:xr 2 X(r)| xi= xi+1 for some i}.
Proof: Let A = M x X=M x * with the second coordinate projection p : A ! X.
Then F (A) = F M(X). It is easy to check that
(A =X)(r)=Dr ~=M(r)^ (X(r)=4 r):
ON FIBREWISE SIMPLICIAL MONOIDS AND MILNOR-CARLSSON'S CONSTRUCTIONS 13
Corollary 5.4. Let F = R, C or H and let X be a pointed CW-complex. Then
there exists a simplicial group F G(S*(X)) such that |F G(S*(X))| is homotopy e*
*quiv-
alent to (F P 1 ^ X) together with a product filtration {FrG(S*(X))}r0 such th*
*at
{F0G(S*(X)) = {1} and
|FrG(S*(X)) =FrG-1(S*(X))| ' drX(r)=4 r;
where d = dimR F - 1.
By applying the word length filtration, we have
Theorem 5.5. Let X be a pointed space and let Y be a path connected pointed s*
*pace.
Then, for any homology theory H*, there exists a spectral sequence {En**(Y; X)}*
*n1
which has the following properties.
(1). {En**(Y; X)}n1 is natural for both X and Y .
(2). {En**(Y; X)}n1 is convergent to H*(Y ^ X).
(3). Epq1(Y; X) = 0 forp < 0, E10;*(Y; X) = H*(pt); and Epq1(Y; X) ~= Hq
(Y )(p)^ (X(p)=4 p) for p > 0.
(4). If H* is the ordinary homology theory with coefficients in a ring, then
{En**(Y; X)}n1
is a Hopf algebra spectral sequence.
(5). The homomorphism E11;*(Y; X) ~=H* (Y ^ X) ! E11;*(Y; X) is induced, up
to isomorphism by the adjoint map of the map oe ^ X : Y ^ X ! Y ^ X, where
oe : Y ! Y is the suspension map.
Proof: We may assume that X is a pointed simplicial set and Y is a reduced
pointed simplicial set. Let the simplicial group G = G(Y ), the Kan G-construct*
*ion
of Y . Then F G(X) ' (Y ^ X). The assertions (1) to (4) follow directly fromthe
word length filtration of F G(X). Consider the commutative diagram
G ^ X ff-'!(G ^ X) -! F G(X)
# oe ^ X # oe
fi G
BG ^ X -! BF (X)
" X ^ X " X
fi4 4 G
B4 G ^ X -! B F (X);
where
ff ((1; gn-1); xn)= ((1; gn-1); d0xn)
for gn-1 2 Gn-1 and xn 2 Xn,
(g; x) = xg; the element g inG with indexx;
for g 2 G and x 2 X
oe(1; gn-1) = (gn-1; 1; : :;:1)
14 JIE WU
for gn-1 2 Gn-1 or F G(X)n-1,
g 2 g n g
fi ((gn-1; : :;:g0);=xn)(d0xn)n-1; (d0xn) n-2; : :;:(d0xn) o
for gi2 Gi and xn 2 Xn,
fi4 ((ffn-1; : :;:ff0);=xn)(xffn-1n; xffn-2n:x:f:f0n)
for ffj 2 Gn and xn 2 Xn, and X is given in Proposition 3.1. Notice that the ma*
*p X is
a homotopy equivalence by Proposition 3.1 and the map fi4 is a homotopy equival*
*ence
by Theorem 4.7. By Proposition 3.2, the natural map oe is the suspension. Thus *
*the
inclusion map F1G(X) ! F G(X) is the adjoint of the map oe ^ X : Y ^ X ! Y ,
up to homotopy. The assertion follows.
The following proposition gives the spaces X(r)=4 r inductively by using cofi*
*bre
sequences.
Proposition 5.6. The spaces X(r)=4 r are built inductively as follows:
X(1)=4 1 ~=X
and X(r+1=4 r+1 is the cofibre of the map
fr : X(r)=4 r ! (X(r)=4 r) ^ X;
where fr(x1; : :;:xr) = (x1; : :;:xr-1; xr; xr), the diagonal map on the last c*
*oordinate.
Proof: The quotient map (X(r)=4 r) ^ X ! X(r+1)=4 r+1 induces a map
(r) (r) (r+1)
(X =4 r) ^ X =fr(X =4 r) ! X =4 r+1:
Now the resulting map is an isomorphism of simplicial sets. The assertion follo*
*ws.
Proposition 5.7. Suppose that the space X = Y , the suspension of a space Y .
Then the maps
fr : X(r)=4 r ! X(r)=4 r^ X
are null for each r.
Proof: Notice that X = Y S1 ^ Y . Define a map
' : S1 ^ Y ^ I ! (S1 ^ Y ) ^ (S1 ^ Y )
by setting
'(s; y; t) = ((s; y); ((1 - t)s;)y)
for 0 s 1, and y 2 Y . Now define a map
' : (X(r)=4 r) ^ I ! (X(r)=4 r) ^ X
by setting
'r(x1; x2; : :;:xr-1; xr; t) = (x1; x2; : :;:xr-1; '(xr; t))
ON FIBREWISE SIMPLICIAL MONOIDS AND MILNOR-CARLSSON'S CONSTRUCTIONS 15
for 0 t 1 and xj 2 X. Notice that '(x; 0) = (x; x). The map 'r : (X(r)=4 r)^I*
* !
(X(r)=4 r) ^ X is an extension of the map fr : X(r)=4 r ! (X(r)=4 r) ^ X. The
assertion follows.
We will show that the spectral sequence induced by the word length filtration*
* of
F M(X) collapses at E1-terms if the simplicial monoid M is homotopy equivalent *
*to
K(Z=2; 0), S1 or S3. Further study on the differentials in the word length spec*
*tral
sequences will be given in [17].
Theorem 5.8. Let F = R, C or H and let X be a pointed space. Suppose that H* *
*is
a multiplicative homology theory such that (1) both H* (F P 1) and H* (F P21) a*
*re free
H*(pt)-modules, where F P21 = F P 1=F P 1; and (2) the inclusion of the bottom *
*cell
Sd ! F P 1 induces a monomorphism in homology. Then there is a product filtrati*
*on
{FrH*(F P 1 ^ X)}r0 of H*(F P 1 ^ X) such that F0H*(F P 1 ^ X) = H*(pt)
and
FrH*(F P 1 ^ X)=Fr-1H*(F P 1 ^ X) ~=(d-1)rH*(X(r)=4 r);
where d = dimR F . Furthermore, this filtration is natural for X.
The examples of the homology theory satisfied the conditions (1) and (2) in t*
*he
Theorem can be found out in any standard text book. The proof of the theorem wi*
*ll
be given in Section 7.
6. Natural Extension Maps in the Word Length Filtration of F M(X)
In this section, we construct certain natural maps by using F M-construction.*
* These
maps are similar to James-Hopf maps. But they are not the extension of the Jame*
*s-
Hopf maps. A direct application of these maps is to give a proof of Theorem 5.8*
*. The
notation xg in this section means an element g in a monoid (group) with an inde*
*x x.
This new notation helps us to take some calculations in this section.
Let M and N be simplicial monoids and let X and Y be pointed simplicial sets.
We define a composition product
M
: F M(X) x F N(Y ) ! F NO F M(X ^ Y ) = F N F (X ^ Y )
by setting
(xg11: :x:gss)(yh11: :y:htt)
=[(x1^y1)g1: :(:xs^y1)gs]h1.[(x1^y2)g1: :(:xs^y2)gs]h2.[(x1^yt)g1: :(:xs^yt*
*)gt]ht
for words (xg11: :x:gss) 2 F M(X)n and (yh11: :y:htt) 2 F N(Y )n. It is a routi*
*ne work to
check the following statements.
(1). the functions : F M(X)n x F N(Y )n ! F NO F M(X ^ Y )n are well defined
by Proposition 2.2.
(2). the functions : F M(X)n x F N(Y )n ! F NO F M(X ^ Y )n induce a simplic*
*ial
map : F M(X) x F N(Y ) ! F NO F M(X ^ Y ).
16 JIE WU
Notice the composition product : F M(X) x F N(Y ) ! F NO F M(X ^ Y ) induces
a (reduced) composition product : F M(X) ^ F N(Y ) ! F NO F M(X ^ Y ).
Now let M be a simplicial monoid and let X and Y be pointed simplicial set. T*
*he
composition product
: F M(X) x F N(Y ) ! F M(X ^ Y )
is defined by setting
(xg11: :x:gss) y = (x1 ^ y)g1: :(:xs ^ y)gs
for word (xg11: :x:gss) 2 F M(X) and y 2 Y . Notice that the composition produ*
*ct
: F M(X)xY ! F M(X ^Y ) induces a (reduced) composition product : F M(X)^
Y ! F M(X ^ Y ). Inductively, we define the composition product
: (F M)n(X) ^ Y ! (F M)n(X ^ Y )
to be the composite
M n-1 M M n-1
(F M)n(X) ^ Y = F M (F ) (X)A ^ Y - ! F (F ) (X) ^ Y
FM () M M n-1 M n
- ! F (F ) (X ^ Y ) = (F ) (X ^ Y );
where (F M)n = F M O : :O:F M.
Definition 6.1. Let M be a simplicial monoid and let X be a pointed simplicial *
*set.
The composition product
: (F M)n(X)(n)=4 n) x F M(X) ! (F M)n+1(X(n+1)=4 n+1);
for n 1, is defined to be the composite
M n (n)
(F M)n(X(n)=4 n) x F M(X) -! F M (F ) (X =4 n) ^ X
FM () M M n (n) M n+1 (n)
-! F (F ) (X =4 n ^ X) = (F ) (X =4 n) ^ X
(FM )n+1(p) M n+1 n+1
-! (F ) (X =4 n+1);
where X(n)is the n-fold self smash product of X, where 4 1= * and 4 n= {x1^ : :*
*^:
xn) 2 X(n)| xi= xi+1 for some i}, and where p : (X(n)=4 n) ^ X ! X(n+1)=4 n+1 is
the pinch map.
Definition 6.2. Let M be a simplicial monoid and let X be a pointed simplicial
set. The simplicial maps hn : F M(X) ! (F M)n(X(n)=4 n) are defined inductively*
* as
follows. The map h1 : F M(X) ! F M(X=4 1) = F M(X) is the identity map and the
map
hn : F M(X) ! (F M)n(X(n)=4 n)
ON FIBREWISE SIMPLICIAL MONOIDS AND MILNOR-CARLSSON'S CONSTRUCTIONS 17
is defined by the equations
8
< 1 x < n
hn(xg11: :x:gss) = (: :(:x1 ^ : :^:xn)g1)g2):):g:n s = n
: h g1 gs-1 g1 gs-1 gs
n(x1 : :x:s-1) . [hn-1(x1 : :x:s-1) xs]s > n
for each word xg11: :x:gss2 F M(X).
Example 6.3. The map h2 : F M(X) ! F M . F M(X(2)=4 2) is given by
h2(xg11: :x:gss) = [(x1^x2)g1]g2.[(x1^x3)g1.(x2^x3)g2]g3: :[:(x1^xs)g1: :x:s-1^*
*xs)gs-1]gs
for each word (xg11: :x:gss) 2 F M(X).
Lemma 6.4. The simplicial maps hn : F M(X) ! (F M)n(X(n)=4 n) are well define*
*d.
Proof: The proof is given by induction on n. The identity map h1 is already a
simplicial map. By checking the relations (1) and (2) in Notations 2.1, it is e*
*asy to
show that the functions h2 : F M(X)q ! F M O F M(X(2)=4 2)q preserves the relat*
*ions
(1) and (2) for each q 0. Thus the functions h2 : F M(X)q ! F M O F M(X(2)=4 2*
*)q
are well-defined. By the naturality, these functions preserve the simplicial st*
*ructure
and therefore the simplicial map h2 : F M(X) ! F M O F M(X(2)=4 2) is well-defi*
*ned.
Now suppose that hk : F M(X) ! (F M)k(X(k)=4 k) is well-defined for k < n with
n > 2. Consider hn : F M(X) ! (F M)n(X(n)=4 n). By the naturality, it suffices *
*to
show that
hn : F M(X)q ! (F M)n(X(n)=4 n)q
preserves the relations (1) and (2) in Notations 2.1 for each z 0.
It is easy to check that the function hn preserves the relation (1). To show*
* that
the function hn preserves the relation (2). We start with a second induction on*
* the
length s of the word (xg11: :x:gss) 2 F M(X)q. If s n, it is obvious that the *
*function
hn preserves the relation (2). Suppose that the function hn preserves the relat*
*ion (2)
for the words in F M(X)q with length < s and s > n. Let (xg11: :x:gss) be a wor*
*d in
F M(X)q such that xi= xi+1for some i. If i s - 2, then
hn(xg11: :x:gss)=hn(xg11: :x:gs-1s-1) . [hn-1(xg11: :x:gs-1s-1) xs]gs
gigi1gi+2 gs
= hn(xg11: :x:gi-1i-1xi xi+2 : :x:s)
by induction.
18 JIE WU
If i = s - 1, then
hn(xg11: :x:gss)=hn(xg11: :x:gs-1s-1) . [hn-1(xg11: :x:gs-1s-1) xs]gs
= hn(xg11: :x:gs-2s-2) . [hn-1(xg11: :x:gs-2s-2) xs-1]gs-1.
g gs-2 g gs-2 g gs
[hn-1(x11: :x:s-2) [hn-2(x11: :x:s-2) xs-1] s-1 xs
= hn(xg11: :x:gs-2s-2) . [hn-1(xg11: :x:gs-2s-2) xs-1]gs-1
g gs-2 g gs-2 g
[hn-1(x11: :x:s-2) xs] . [hn-2(x11: :x:s-2) xs-1] s-1 xs
= hn(xg11: :x:gs-2s-2) . [hn-1(xg11: :x:gs-2s-2) xs-1]gs-1
h gs *
*g i
[hn-1(xg11: :x:gs-2s-2) xs] . [hn-2(xg11: :x:gs-2s-2) xs-1*
*]s-xs1
= hn(xg11: :x:gs-2s-2) . [hn-1(xg11: :x:gs-2s-2) xs-1]gs-1
g gs-2 g gs-2 ggs
[hn-1(x11: :x:s-2) xs] . [hn-2(x11: :x:s-2) (xs-1^ xs)] s-1
Notice that xs-1 = xs. We need the following lemma.
Lemma 6.5. Let M be a simplicial monoid and let X be a pointed simplicial se*
*t.
Let : (F M)n(X(n)=4 n) ^ X(2)! (F M)n(X(n+2)=4 n+2) defined by the composite
i j(FM )n(p)
(F M)n(X(n)=4 n) ^ X(2)-! (F M)n (X(n)=4 n) ^ X(2) -! (F M)n(X(n+2)=4 n+2*
*);
where p : (X(n)=4 n) ^ X(2)! X(n+2)=4 n+2 is the pinch map. Then the element
ff (x ^ x) = 1
for any element ff in (F M)n(X(n)=4 n) and x 2 X.
Continuation of the proof of Lemma 6.4:
By Lemma 6.5, the element hn-2(xg11: :x:gs-2s-2) (xs-1^ xs) = 1 in
(F M)n-2(X(n)=4 n)
and so [hn-2(xg11: :x:gs-2s-2) (xs-1^ xs)]gs-1= 1 in (F M)n-1(X(n)=4 n). Thus
hn(xg11: :x:gss)= hn(xg11: :x:gs-2s-2) . [hn-1(xg11: :x:gs-2s-2) xs-1]*
*gs-1
= hn(xg11: :x:gs-2s-2) . [hn-1(xg11: :x:gs-2s-2) xs-1]*
*gs-1gs
= hn(xg11: :x:gs-2s-2xgs-1gss-1):
The second induction is finished and therefore the first induction is finished.
Proof of Lemma 6.5:
We consider a slight general case. Let Y be a pointed simplicial set and l*
*et the
composition product
(2)
Y : (F M)n(Y ) ^ X(2)! (F M)n Y ^ (X =4 2)
given by the composite
(FM )n(p) M n (2)
(F M)n(Y ) ^ X(2)-! (F M)n(Y ^ X(2)) - ! (F ) (Y ^ X =4 2)
ON FIBREWISE SIMPLICIAL MONOIDS AND MILNOR-CARLSSON'S CONSTRUCTIONS 19
where p : Y ^ X(2)! Y ^ X(2)=(Y ^ 4 2) is the pinch map. If Y = X(n)=4 n, then
the composition product
: (F M)n(X(n)=4 n) ^ X(2)! (F M)n(X(n+2)=4 n+2)
is the composite
i (jFM )n(q)
(F M)n(X(n)=4 n)^X(2)Y! (F M)n (X(n)=4 n) ^ (X(2)=4 2) ! (F M)n(X(n+2)=4 n+*
*2);
where q : (X(n)=4 n) ^ (X(2)=4 2)to(X(n+2)=4 n+2 is the pinch map. Thus it su*
*ffices
to show that
ff Y (x ^ x) = 1
for any ff 2 (F M)n(Y ) and x 2 X. The proof of this statement is given by in*
*duction
on n. Suppose that n = 1. Let ff = (yg11: :y:gss) be a word in F M(Y ). Then
ff Y (x ^ x) = (y1 ^ x ^ x)g1: :(:ys ^ x ^ x)gs= 1
since yj^ x ^ x 2 Y ^ 4 2for each j. Suppose that this statement holds for th*
*e case
n - 1 with n > 1. Let
ff = (ffg11: :f:fgss)
be a word in (F M)n(Y ), where ffj is an element in (F M)n-1(Y ). Then
ff Y (x ^ x) = ((ff1 Y (x ^ x))g1:(:f:fs Y (x ^ x))gs)= 1
since ffj Y (x ^ x) = 1 for each j. The induction is finished and so the ass*
*ertion
follows.
Now let M be a simplicial monoid and let X be a pointed simplicial set. The
simplicial map : M ^ X ! F M(X) is defined by setting (g ^ x) = xg.
Lemma 6.6. The simplicial map : M ^ X ! F M(X) is the adjoint map of the
simplicial map oeM ^ X : M ^ X ! BM ^ X ' BF M(X) up to homotopy, where
the map oeM : M ! BM is the suspension defined in Section 3.
Proof: The assertion follows by the commutative diagram
M ^ X ff-'!(M ^ X) -! F M(X)
# oe ^ X # oe
fi M
BM ^ X -! BF (X);
where ff ((1; gn-1) ^ xn)= (1; (gn-1 ^ d0xn))for gn-1 2 Mn-1 and xn 2 Xn and
g 2 g n g
fi (gn-1; : :;:g0) ^ xn)= (d0xn) n-1; (d0xn) n-2; : :;:(d0xn) 0
for gi 2 Mi and xn 2 Xn. The simplicial map fi : BM ^ X ! BF M(X) is a
homotophy equivalence by the proof of Theorem 5.5.
20 JIE WU
Notations 6.7. Let M be a simplicial monoid and let X be a pointed simplicial s*
*et.
The simplicial maps n : M(n)^ X ! (F M)n(X) is defined by setting
n(gs ^ gs-1^ : :^:g1 ^ x) = (: :(:((x)g1)g2):):g:n
for gj 2 M and x 2 X. The simplicial map j(M) : M ! G(BM) is defined by
j(gn-1 = (gn-1; 1; : :;:1) for gn-1 2 M, where G(X) is the Kan G-construction of
a reduced siplicial set X. It is easy to show that the simplicial map j(M) : M*
* !
G(BM) is the adjoint of the suspension oe : M ! BM ' BH(BM) up to homotopy.
Let X be a reduced simplicial set and let Y be a pointed simplicial set. The si*
*mplicial
map t : Y ^ G(X) ! G(Y ^ X) is defined to be the adjoint of (Y ^ G(X)) '
Y ^G(X) Y-^oe!Y ^BG(X) ' Y ^X. The simplicial map tn : Y ^Gn(X) ! Gn(Y ^X)
is defined inductively to be the composite
G(tn-1)n
Y ^ Gn(X) = Y ^ G(Gn-1(X)) -t! G(Y ^ Gn-1(X)) -! G (Y ^ X):
Definition 6.8. Let M be a simplicial monoid and let X be a pointed simplicial *
*set.
The simplicial maps
(n)
n : (F M)n(X) ! Gn (BM) ^ X
are defined inductively as follows: The map 1 : F M(X) ! G(BM ^ X) is defined
to be the composite
j(FM (X)) M G(fi)-1
F M(X) -! G(BF (X)) -! G(BM ^ X);
where the map G(fi)-1 is a monomoty inverse of the simplicial map G(fi) : G(BM ^
X) -'! G(BF M(X)) and where the simplicial map fi : BM ^ X ! BF M(X) is de-
fined in the proof of Lemma 6.6. Now the map n : (F M)n(X) ! Gn (BM)(n)^ X
is defined to be the composite
M n-1 FM ( n-1)M n-1 (n-1)
(F M)n(X) = F M (F ) (X) - ! F G (BM) ^ X
1 n-1 (n-1) G(tn-1)n (n)
-! G(BM ^ G (BM) ^ X -! G (BM) ^ X :
Lemma 6.9. The composite
n n (n)
M(n)^ X -n! (F M)n(X) -! G (BM) ^ X
is the adjoint of the simplicial map
(n)^X (n)
nM(n)^ X ' (M)(n)^ X oe-! (BM) ^ X;
up to homotopy for n 1.
ON FIBREWISE SIMPLICIAL MONOIDS AND MILNOR-CARLSSON'S CONSTRUCTIONS 21
Proof: The proof is given by induction on n. If n = 1, the assertion follows by*
* the
commutative diagram
j M '
F M(X) -! G(BF (X)) -! G(BM ^ X)
1 " # oe
M ^ X oe^X-!BM ^ X -'! BF M(X) -'! BG(BF M(X)):
For the general case n, the assertion follows by induction and the following *
*homo-
topy commutative diagram
nFM ( n-1) n |
n(FM )n(X) -! nFM Gn-1(Yn-1) -1! nG BM ^ Gn-1(Yn-1) -! nGn(Yn) |
" " # oe # oe ||
nM ^ (FM )n-1(X) 1^-n-1! nM ^ Gn-1(Yn-1))oe^1-!n-1BM ^ Gn-1(Yn-1) |
" # # ||
(n-1) oe^1 |
M ^ n-1M(n-1)^ X 1^oe-! M ^ Yn-1 -! Yn = Yn; |
where we have set Yn-1 = (BM)(n-1)^ X and Yn = (BM)n ^ X.
Definition 6.10. Let M be a simplicial monoid and let X be a pointed simplicial
set. The maps
Hn : F M(X) ! Gn (BM)(n)^ (X(n)=4 n)
is defined to be the composite
n n (n) (n)
F M(X) -hn!(F M)n(X(n)=4 n) -! G (BM) ^ (X =4 n) :
Theorem 6.11. The natural maps Hn : F M(X) ! Gn (BM)(n)^ (X(n)=4 n) is a
homotopy extension of the pinch map FnM(X) ! FnM(X)=FnM-1(X), i.e., there is a
homotopy commutative diagram
F M(X) -Hn! Gn (BM)(n)^ (X(n)=4 n)
" " j1n
FnM(X) ! FnM(X)=FnM-1(X) ~=M(n)^ (X(n)=4 n)
for n 1, where {FnM(X)}n0 is the word length filtration of F M(X) and where t*
*he
simplicial map j1n: M(n)^ (X(n)=4 n) ! Gn (BM)(n)^ (X(n)=4 n) is the adjoint *
*of
the multiple suspension.
(n)^1 (n) (n)
nM(n)^ (X(n)=4 n) ' (M)(n)^ (X(n)=4 n) oe-! (BM) ^ (X =4 n)
up to homotopy.
Proof: The assertion follows by the construction of Hn and Lemma 6.9.
Remark 6.12. The Kan G-construction is actually the loop functor in the homo-
topy category. For any pointed simplicial set X, the simplicial group G(X) is d*
*efined
to be the Kan G-construction on the base point connect component of X. Thus all
of the simplicial groups Gn(X) = G O : :O:G(X) are well defined for any pointed
simplicial set X.
22 JIE WU
7. Proof of Theorem 1.1 (Theorem 5.5)
Now we are ready to give a proof of Theorem 1.1. We need two lemmas.
Lemma 7.1. Let F = R, C or H and let H* be a multiplicative homology theory s*
*uch
that (1) both H*(F P 1) and H*(F P21) are free H*(pt)-modules; and (2) the incl*
*usion
of the bottom cell j : Sd ! F P 1 induces a monomorphism in the homology. Then
the homomorphism
1 (n)
(j(n)^ X)* : H*(Sdn ^ X) ! H* (F P ) ^ X
is a monomorphism for each n and any space X.
Proof: The proof is given by induction on n. Suppose that n = 1. Consider the
short exact sequence
0 ! H*(Sd) ! H*F P 1 ! H*F P21 ! 0
Notice that the H*(pt)-modules H* (Sd), H* (F P 1) and H* (F P21) are free modu*
*les.
There is a short exact sequence
0 ! H*(Sd) H*(pt)H*(X) ! H*F P 1 H*(pt)H*(X) ! H*(F P21) H*(pt)H*(X) ! 0
and the isomorphisms H*(Sd)H*(pt)H*(X) H*(Sd^X), H*(F P 1)H*(pt)H*(X)
H*(F P 1 ^ X) and H* (F P21) H*(pt)H*(X) H* (F P21 ^ X). Thus j ^ X)* :
H*(Sd ^ X) ! H* (F P 1 ^ X) is a monomorphism. The general case n follows by
induction and the following commutative diagram.
(n)^1)
H* (Sdn ^ X) (j-! *H* (F P 1)(n)^ X
d(j(n-1)^ 1)* # ||
-!
H* Sd ^ (F P 1)(n-1)^ X (j ^ 1)*H* (F P 1)(n)^ X
Let X be a pointed simplicial set. We denote 4 1= * and 4 n= {(x1^ : :^:xn) 2
X(n)| xi = xi+1 for some i}, a subsimplicial set of the n-fold self smash produ*
*ct of
X.
Lemma 7.2. Let G be a simplicial group, let X be a pointed simplicial set and*
* let
H* be a homology theory. Suppose that
(n) (n) (n) (n)
(oe(n)^ 1)* : H* (G) ^ (X =4 n) ! H* (BG) ^ (X =4 n)
is a monomorphism for each n 1, where oe : G ! BG is the suspension. Then
the map
(jn)* : H* FnG(X) ! H* F G(X)
is a monomorphism for each n 0, where jn : FnG(X) ! F G(X) is the injection in
the word length filtration {FnG(X)}n0 of F G(X).
ON FIBREWISE SIMPLICIAL MONOIDS AND MILNOR-CARLSSON'S CONSTRUCTIONS 23
Proof: The proof is given by induction on n. The assertion holds for the case
n = 0, where FoG(X) = *. Suppose that (jq)* : H* FqG(X) ! H* F G(X) is a
monomorphism for q < n and consider (jq)* : H* FnG(X) ! H* F G(X) . Let
ff 2 Ker(jn)*. By Theorem 6.11, there is a commutative diagram
n(H )
H* nF G(X) -n!* H* nGn (BG)(n)^ (X(n)=4 n) -! H* (BG)(n)^ (X(n)=4 n)
" (jn)* " (oe(n)^ 1)*
H* nFnG(X) (pn)*-! H* nG(n)^ (X(n)=4 n) ;
where the map
i j
pn : nFnG(X) ! nFnG(X)=FnG-1(X) ~=n G(n)^ (X(n)=4 n)' (g)(n)^ (X(n)=4 n)
is the pinch map. We already assume that
(n) (n) (n) (n)
(oe(n)^ 1)* : H* (G) ^ (X =4 n) ! H* (BG) ^ (X =4 n)
jn-1;n
is a monomorphism. Thus (pn)*(nff) = 0. By the cofibre sequence FnG-1(X) - !
FnG(X) ! FnG(X)=FnG-1(X), there exists an element fi 2 H* FnG-1(X) such that
jn-1;n*(fi) = ff. Thus (jn-1)*(fi) = (jn)* O (jn-1;n)*(fi) = (jn)*(ff) = 0 By i*
*nduction,
fi = 0 and so ff = 0. The assertion follows.
Proof of Theorem 5.8:
Let G be a simplicial group such that G ' (F P 1): The assertion follows dire*
*ctly
from Lemmas 7.1 and 7.2.
8. Simplicial Group Rings Fp (Fp(X))
Let Fp be a field with p elements, where p is a prime, and let x be a simplic*
*ial set.
It is well known that H*(X; Fp) ~=ss*(Fp(X)), where Fp(X) is the free sipmlicia*
*l Fp-
module generated by X. In this section, we only give a description of the simpl*
*icial
group ring Fp(Fp(X)). Further study will be given in [18].
Definition 8.1. Let V be a (graded) Fp-vector space. The algebra Ap(V ) is defi*
*ned to
be the quotient algebra of the tensor algebra T (V ) modulo the two sided ideal*
* generated
by the elements xp for x 2 V . If V is a simplicial Fp-vector space. Then Ap(V *
*) is a
simplicial algebra. Let X be a pointed simplicial set. The simplicial algebra A*
*p(X) is
defined to be the quotient simplicial algebra of Ap(Fp(X)) modulo the two sided*
* ideal
generated by the element *, where * is the base point of X. If p = 2, the alge*
*bra
Ap(V ) is a non-commutative view of exterior algebras.
Proposition 8.2. Let X be a pointed siplicial set. Then there is an isomorphism*
* of
simplicial algebras
~= Z=p
Ap(X) -! Fp F (X) :
24 JIE WU
Proof: Notice that the simplicial group F Z=p(X) is the quotient simplicial gro*
*up of
F (X) modulo the normal subsimplicial group generated by xp for x 2 X. Consider
the simplicial map i : X ! Fp F Z=p(X) i(x) = x - 1 for x 2 X. Let "-: T (Fp(X)*
*) !
Fp F Z=p(X) be a simplicial homomorphism of simplicial algebras which is a (un*
*ique)
extension of i. Notice that we have the following equations: (1). "-(xp) = (x -*
* 1)p =
xp - 1 = 0 and (2). "-(*) = * - 1 = 0:
Thus there is a (unique) simplicial homomorphism
Z=p
OE : Ap(X) ! Fp F (X)
of simplicial algebras which is a (unique) extension of i. Conversely, conside*
*r the
simplicial map j : X ! AP (X) given by j(x) = x + 1 for x 2 X. Similarly, there*
* is
a (unique) simplicial homomorphism
Z=p p
: Fp F (X) ! A (X)
of simplicial algebras which is a (unique) extension of the map j. It is easy t*
*o check
that ' O = 1Fp(FZ=p(X))and ' O = 1Ap(X), which is the assertion.
Corollary 8.3. Let X be a pointed simplicial set. Then there is an isomorphism *
*of
algebras
ss*(Ap(X)) ~=H*( (K(Z=p; 1) ^ X); Fp) :
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MSRI, 1000 Centennial Drive, Berkeley, CA 94720, USA