Filtering the fiber of the pinch map
Brayton Gray
Dept. of Mathematics, Statistics
and Computer Science (M/C 249)
University of Illinois at Chicago
851 South Morgan Street
Chicago, IL 60607-7045
brayton@uic.edu
1 Introduction
In [CMN1 ], the authors analyzed the homotopy type of the mod pr Moore space
P 2n+1= S2n [pre2n+1
by constructing a fibration sequence
2S2n+1-@n! Fn -! P 2n+1-! S2n+1.
A key result is that both Fn and P 2n+1contain a factor which is the loop *
*space on a
one point union of Moore spaces that are at least 4n - 2 connected. Removing th*
*is factor
(which can be inductively considered later) leaves a fibration sequence:
_@__
. .!. 2S2n+1- ! F n- ! T 2n+1-! S2n+1
and they show that __ Y
k-1 r+1
F n~=S2n-1x S2np {p }.
k 1
__
An important remaining question is to understand the components of @:
__ 2 2n+1 2n-1
@0 : S -! S
__ 2 2n+1 2npk-1 r+1
@k : S -! S {p }.
__
@0 is well understood_and plays a key role in homotopy theory. There is much i*
*nterest
in understanding @k for k > 0; in particular, it is not known whether they are *
*all null
1
homotopic [AG], [CMN1],_[G2], [G3], [GT], [N1], [N2]. In case r > 1, Neisendor*
*fer ([N1],
[N2]) has shown that @k = 0 for k 1, so our emphasis will be on the case r = *
*1.
A remarkable observation of [CMN2] is that there is an isomorphism of Hopf a*
*lgebras
(see appendix): __
H*( 2S2n+1; Z=p) ~=H*(F n; Z=p).
__
However, the induced homomorphism (@)* = 0. __
The intention of this work is to consider constructions involving Fn which a*
*re analogous_
to constructions involving 2S2n+1. We will be able to do this compatibly with *
*the map @.
In particular, we will look at the filtration of Selick [S] and the classifying*
* space of the double
suspension [G2]. Our results will follow from the construction of a new "Hopf i*
*nvariant" type
map:
Fn -h! Fnp
which induces an epimorphism in p-local homology. Recall that there is a p-loca*
*l fibration
sequence:
Hps 2nps
S2n(ps-1)! S2n1-! S1 .
By analogy, we compare F(ps-1)to the fiber s of the Hopf invariant hs:
s
s ! Fn -h! Fnps
Theorem 1.1 s ' F(ps-1)x Xs where Xs is a wedge of mod pr Moore spaces.
This will appear as 7.1. These same ideas lead to:
Theorem 1.2 If k > 1, there is a homotopy commutative diagram:
_@ k
2S2n+1 ---k! S2np -1{pr+1}
? x
flk?y~= ??_@1
Hpk-1 k-1
2S2n+1 - ---! 2S2np +1
where Hpk is a James-Hopf invariant and flk is a homotopy equivalence.
__ k
Theorem 1.3 The map @k : 2S2n+1- ! S2np -1{pr+1} is homotopic to a compositi*
*on:
k-1 r+1
2S2n+1- ! BWn -ffl!S2np {p }.
These appear as 8.2 and 9.3 in the sequel.
Throughout this paper all spaces will be assumed to be localized at a fixed *
*prime p > 2
and all homology will be p-local unless otherwise indicated.
2
2 Filtration of 2S2n+1
H*( 2S2n+1; Z=p) is a free commutative algebra on generators of dimensions 2npi*
* - 1,
2npi+1- 2 for i 0. Selick [S] has described a sequence of H-spaces whose hom*
*ology
filters H*( 2S2n+1; Z=p) by successively adding one generator at each stage. L*
*et us write
S2n(k)= Jk(S2n) for the kth filtration of the James construction J(S2n) = S2n1.*
* Then the space
capturing the first 2s generators of H*( 2S2n+1; Z=p) is precisely S2n(ps-1). *
*Selick [S] defines
spaces Gs which contain the first 2s + 1 generators. We define Gs by a diagram *
*of fibrations:
s-1 ff 2n
Gs - --! S2np ---! S(ps-1)
? ? fl
? ? fl
y y fl
s+1 2n 2n+1 Hnps 2nps+1
2S2n+1 - --! 2S2np ---! S(ps-1)---! S - --! S
? ?
? ?
y y
BWnps _______ BWnps
s-1 2n s 2n
where ff : S2np ! S(ps-1)is the attaching map for the 2np cell of S1 , Hnpsi*
*s the appro-
priate James-Hopf invariant, and BWnpsis the cassifying space for the double su*
*spension.
Since Gs is the fiber of O Hnps, it is an H space if p > 3 ([G2; Proposition*
* 6]). Gs could
0 s+1
also be described as the fiber of a Toda-Hopf invariant S2n(ps+1-1)H-! S2np *
*-1using the
techniques of ([G2], [MN]). __
Constructing an analogous filtration for F n is not difficult. The key resu*
*lt is in the
compatibility of the two filtrations.
Proof of Theorem CMN
In this section we give a brief summary of the proof of the main result of [CMN*
*1].
Theorem CMN There is a diagram of fibration sequences:
Fn ---! P 2n+1- --! S2n+1
? ? fl
? ? fl
y y fl =
S2n-1x Vn ---! T 2n+1- --! S2n+1
? ?
? ?
y i y i0
Pn --=-! Pn
? ?
? _ ?
y OE y
p 2n+1
Fn ---! P 2n+1- --! S
3
where the maps i and i0are null homotopic, p is the pinch map, Pn is a one poin*
*t union of
mod pr Moore spaces of dimension 4n and
Y k
Vn = S2np -1{pr+1}.
k 1
Here Sm {d} is the fiber of the degree d map on Sm . The inessentiality of i*
* and i0implies
that
Fn ' S2n-1x Vn x Pn
P 2n+1' T 2n+1x Pn.
Sketch of the proof. Both the mod pr homotopy and the mod p homology of P 2*
*n+1have
differential Lie algebra structures, and the Hurewicz map is a Lie algebra homo*
*morphism.
Furthermore Fn has an extended ideal structure. These structures are obtained *
*from the
Samelson product and the rth Bockstein fi(r)(fi(i)= 0 fori < r).
H*( P 2n+1; Z=p) = U(L) where L is a free Lie algebra on v 2 H2n( P 2n+1; Z=*
*p) and
u = fi(r)(v) and U(L) is the universal enveloping algebra. L(0) L is the Lie i*
*deal generated
by xi= adi-1(v)(u) for i > 1 and H*( Fn; Z=p) = U(L(0)).
Furthermore the suspensions of the xi,
oe(xi) 2 H2ni(Fn; Z=p) ~=Z=p
are non zero. u and v lie in the image of the mod pr Hurewicz homomorphism, so *
*all of L
and hence L(0)does as well
L(0) im{ss*( Fn; Z=pr) -! H*( Fn; Z=p)}
using the extended ideal structure in homotopy.
Both in homotopy and homology, fi(r)(xpk) = 0, so one may construct an exten*
*sion:
k-1 r+1 ffik
P 2np (p ) -! Fn
of the mod pr homotopy class k
P 2np -1-! Fn
representing xpkin homology. The only property of the maps ffik that is needed *
*is that their
Hurewicz image is xpk. From these maps the authors construct
k-1 r+1 e 2npk r+1 _ffik
S2np {p } -! P (p ) -! Fn
_
where ffikis the adjoint of ffik and e is a particular map described in the nex*
*t section. These
maps are assembled via loop multiplication to obtain:
: S2n-1x Vn -! Fn
4
The Bockstein fi(r)is trivial in H*(S2n-1 x Vn; Z=p) but not in H*( Fn; Z=p). *
*However
induces an isomorphism in the Bockstein homology of these homology groups (se*
*e the
appendix for a calculation of the Bockstein homology).
Next the authors construct a squence of sub-Lie algebras L(k+1) L(k) L(0)v*
*ia short
exact sequences of Lie algebras:
0 - ! L(1) - ! L(0) - ! -! 0
0 - ! L(k+1) - ! L(k) - ! -! 0.
Here and are free commutative Lie algebras generated by ok = xpk*
*of dimension
2npk - 1 and
pk-1` k'
1 X p (0)
oek = ___ [xi, xpk-i]fflL .
2p i=1 i
This is possible since ok is a generator and oek is decomposible in L(k-1)but n*
*ot in L(k)since
_1_ pk (k)
2p pk-1 is a p-local unit and xpk-1= ok-1 =2L .
It follows that there is a split short exact sequence of differential Hopf a*
*lgebras:
k-1 r+1
0 -! U(L(k+1)) -! U(L(k)) -! H*(S2np {p }; Z=p) ! 0
and hence
U(L(1)) Hp(S2n-1x Vn; Z=p) ~=U(L(0)) = H*( Fn; Z=p)
T
where L(1)= L(k).
k 0
Consequently Ht(U(L(1)); fi(r)) = 0 for t > 0 and hence L(1) has a basis con*
*sisting of
classes {xff, fi(r)xff}. Since xff2 L(1) L(0)one can choose maps
OEff: P nff-! Fn
whose homology image is xffand fi(r)xffusing the extended ideal structure. Asse*
*mbling these
one produces a one point union of mod pr Moore spaces, Pn and a map:
__
OE: Pn -! Fn
such that the homology image of
__
( OE)* : H*( Pn; Z=p) -! H*( Fn; Z=p)
is exactly U(L(1)). From this the authors produce a homotopy equivalence
S2n-1x Vn x Pn -! Fn
__ __
by_multiplying the maps and OEvia the H space structure in Fn. Let Fn be the*
* fiber of
OE. Then
__
S2n-1x Vn -! Fn -@! Fn
is a homotopy equivalence and i is inessential. Thus completes the outline of t*
*he proof.
5
3 Combinatorial description of Fn
In [G1 ] a combinatorial description of the fiber of the pinch map:
F -! X [ CA -ss!SA
was described in the spirit of the James construction X1 for SX. The model, de*
*signated
(X, A)1 consists of all words in X1 with the property that all letters after *
*the first letter
are required to lie in A, where A X. Alternatively, this can be described by *
*a push out
diagram:
X x A1 ---! (X, A)1
x x
(3.0) ?? ??
A x A1 ---! A1
Proposition 3.1 ([G1]). There is a map (X, A)1 ! F which is a homotopy equiv*
*alence
when the inclusion A X is a cofibration.
There is an action of the monoid A1 on (X, A)1 and (X, A)1 can be thought of*
* as a
universal space in the following sense. If Y is any space on which A1 acts and *
*g : X ! Y is
any map such that g(a) = a . * for some point * 2 Y , there is a unique A1 equi*
*variant map
g1 : (X, A)1 ! Y.
(See [G; 3.2]). The map (X, A)1 ! F is constructed from the action A1 xF ! SAx*
*F !
F .
The orbit space of (X, A)1 under the action of A1 is X=A and we may use the *
*universal
property to establish the following diagram
(X, A)1 --e-! X1
? ?
(3.2) ae?y ?y
X=A ---! (X=A)1
Note that the inclusion e : (X=A)1 ! X1 is the unique A1 equivariant extensio*
*ns of the
inclusion of X in X1 .
Example 3.3 Let A = S2n-2 P(2n-1pt)= X. Then (X, A)1 is the homotopy fiber *
*of the
map of degree pt
pt 2n-1
(X, A)1 = S2n-1{pt} -! S2n-1- ! S
and the map e : S2n-1{pt} -! (P 2n-1(pt))1 ~= P 2n(pt) is uniquely determined *
*as a S2n-1
equivariant map extending the inclusion of P 2n-1(pt).
6
One of the main features of the construction (X, A)1 is that we can define *
*functorial
Hopf invariants using the same formulas as in James [J]. In particular we have *
*a pointwise
commutative diagram:
Hk (k)
A1 ---! (A )1
? ?
? ?
y y
Hk (k-1)
(X, A)1 ---! (X ^ A )1
? ?
? ?
y y
Hk (k)
X1 ---! (X )1
It would be desireable to construct functorial compressions of these maps:
h : (X, A)1 - ! (X ^ A(k-1), A(k))1 (X ^ A(k-1))1
but I have been unable to do this. In the next section we will construct a map*
* h of this
form, but we have no knowledge of how it relates to Hk.
4 Construction of h : Fn - ! Fnp
In this section we will define a kind of Hopf invariant which is key for the re*
*sults of this
paper.
Theorem 4.1 Suppose (X, A) is a suspension pair and r 1. Then there is a ma*
*p:
h : (X, A)1 - ! (X ^ A(r-1), A(r))1
such that the composition:
p (r-1) (r)
X x A1 - ! (X, A)1 - h!(X ^ A(r-1), A(r))1 - ! X ^ A =A
is homotopic to the composition:
1xHr-1 (r-1) (r-1) (r-1) (r-1) (r)
X x A1 - ! X x A1 -! X ^ A1 -! X ^ A -! X ^ A =A
1^ffl
where Hr-1: A1 ! A(r-1)1is any map and ffl uses the suspension structure of X t*
*o collapse
of X ^ A(r-1)1to X ^ A(r-1).
Corollary 4.2 Let Fn be the fiber of the pinch map
p : P 2n-1(pr) -! S2n+1
for each n and Hr-1 be any choice of James-Hopf invariants. Then there is a map:
h : Fn -! Frn
7
such that the composition:
S2n x S2n+1- ! Fn -h! Frn- ! P 2rn+1
is homotopic to:
1xHr-1 2n 2(r-1)n+1 2rn 2rn+1
S2n x S2n+1 -! S x S -! S - ! P ;
in particular, h* : H2rn(Fn) -! H2rn(Frn) is an isomorphism.
Note: It is an easy calculation to see that
ae
Z(p) n | i
Hi(Fn) =
0 n - i
and the map Fn -ss!P 2n+1(pr) is reduction mod pr in homology. In particular w*
*e have
defined a map:
Fn -! P 2nr+1(pr)
for each r 1 which is onto in p-local homology.
Proof of 4.1 We begin by constructing a map
~ : (X, A)1 - ! (X, A)1 =A1 - ! X x A1 =A o A1 X o A1 =A o A1
which follows from the push out diagram (3.0). This map is functorial and we h*
*ave a
commutative diagram:
~
(X, A)1 ---! X o A1 =A o A1
x x
? ?
? ?
X x A1 ---! X o A1
using the functorial property, we have the commutative diagram:
(X, A)1 - --! X o A1 =A o A1
? ?
? ?
y y
(CX, A)1 - --! CX o A1 =A o A1 ' S(A o A1 )
however (CX, A)1 is the fiber of CX=A -! SA and hence is contractible. Conseque*
*ntly ~
lifts to (X o A1 , A o A1 )1 which is the homotopy fiber of
X o A1 =A o A1 - ! SA o A1 ;
we now choose a map Hr-1 : A1 -! A(r-1)1and use the suspension structure of th*
*e pair
(X, A) to obtain a map
(X o A1 , A o A1 )1 -! (X ^ A(r-1)1, A ^ A(r-1)1)1
-! (X ^ A(r-1), A(r))1 ;
composing with ~0defines h.
8
Proposition 4.3 The map h : Fn -! Fnpsinduces an epimorphism in p-local homolo*
*gy.
Proof: We use the map S2n+1- @!Fn which has degree p in H2nkfor each k > 0 to *
*calculate
the cup product structure in H*(Fn). We choose generators ei2 H2ni(Fn) such tha*
*t @*(ei)
is p times the generator of H2ni( S2n+1) dual to the ith power of a homology ge*
*nerator in
H2n( S2n+1). Then it is easy to see that
` '
i + j
eiej = p ei+j.
i
s
Let us designate di2 H2nip(Fnps) for the corresponding generator; then
h*(d1) = u1eps
for some p local unit u1 by 4.2. We show that
h*(di) = uieips
where uiis a p-local unit for each i 1 by induction. Using the product struct*
*ure we have
pidi= d1di-1,
so pih*(di) = h*(d1di-1) = h*(d1)h*(di-1)
= (u1eps)(ui-1e(i-1)ps)
` '
ips
= pu1ui-1 eips.
ps
ips
It suffices to show that 1_ipsis a p-local unit.
Now let vp(m) be the exponent of p in m and [x] be the greatest integer less*
* than or
equal to x. Then ~ ~
X n
vp(n!) = __i .
i 1 p
Consequently vp((psi)!) = ps-1i + ps-2i + . .+.i + vp(i!) so
` ` ''
psi s s s
vp = vp((p i)!) - vp((p (i - 1))!) - vp(p !)
ps
= vp(i!) - vp((i - 1)!)
= vp(i).
and we are done.
By 4.2, the composition
2+1
S2n+1- @!Fn -h! Fnps-! P 2np
9
is null homotopic, so there is a lifting eHps
Heps s
S2n+1 ---! S2np +1
? ?
@?y ?y@
Fn --h-! Fnps.
s+1
Proposition 4.4 (Heps)* : H*( S2n+1) ! H*( S2np ) is an epimorphism.
Proof: This follows since both maps labeld @ have degree p in homology and h*
** is an
epimorphism.
In particular, the fiber of Hepsis S2n(ps-1)as if Hepswere the James-Hopf in*
*variant Hps.
Both Hepsand Hps can be placed in the fibration sequence induced by the inclu*
*sion
S2n(ps-1) S2n+1, so there is an equivalence fls : 2S2n+1! 2S2n+1 such that
Heps= HpsO fls
hence we have
Corollary 4.5 There is a homotopy commutative diagram
2S2n+1 - @n--! Fn
? ?
Heps?y ?y hs
s+1 @np
2S2np - --! Fnps
where Heps~ HpsO fls for some homotopy equivalence fls : 2S2n+1- ! 2S2n+1.
5 A Filtered Decomposition
Let s be the homotopy fiber of hs : Fn ! Fnps. In this section we will compar*
*e the
decompositions of Fn and Fnpsand prove
Q k s
Theorem 5.1 s ~=S2n-1x S2np -1{pr+1} x S2np -1x Rs where Rs is a wed*
*ge
1 k~~From this we construct
_
k+1-1 r+1 e 2npk+1r+1 ffik+1
S2np {p } ---! P (p ) - --! Fn
? ? ?
? ? ?
y= y= y h
_
k+1-1 r+1 e 2npk+1r+1 ffik(np)
S2np {p } ---! P (p ) ----! Fnp
where k > 0, and _
S2np-1{pr+1}---! P 2np(pr+1)--ffi1-! Fn
? ? ?
? ? ?
y y y h
S2np-1 ---! S2np ---! Fnp
using diagram 3.2. Multiplying these together in order gives the result.
__
Lemma 5.3 The map OEn: Pn ! Fn can be chosen so that Wn = Pnp_ Qn and there i*
*s a
homotopy commutative diagram:
_OE
Pnp_ Qn ---n! Fn
? ?
p?y ?yh
_OE
np
Pnp ---! Fnp
where p is the projection.
Proof: Since h : H*(Fn; Z=p) ! H*(Fnp; Z=p) is onto the same holds for
U(L(0)(n)) ~=H*( Fn; Z=p) ! H*( Fnp; Z=p) ~=U(L(0)(np))
in fact the generators xip2 L(0)(n) satisfy
h*(xip(n)) = uixi(np)
11
where uiis a p-local unit (see 4.3).
Now given a basis {xff, fi(r)xff} for L(1)(np), each xffis a Lie bracket in *
*the xi(np) and
this element consequently lifts to the same bracket in xip(n). Thus these lifti*
*ngs are linearly
independent and can be choosen as part of a basis. They are all in the image of*
* the mod
pr Hurewicz homomorphism. Thus we have choosen generators for L(1)(n) which sor*
*t into
those which are lifting of the generators_for L(1)(np) and the others. Realizin*
*g these via the
Hurewicz homomorphism gives the maps OEn. By a change in basis for Qn, we can a*
*ssume
that the map p : Pn ! Pnp is trivial on Qn.
We now use 5.3 to construct the following ladder of fibrations:
_@ __ _OE
Fn --n-! F n ---! Pn - -n-! Fn
? ? ? ?
(5.4) ?y ?yfls ?y ?yhs
_@ _
nps __ OEnps
Fnps ---! Fnps ---! Pnps- --! Fnps
and we use 5.2 to construct compatible equivalences:
__
S2n-1x Vn - --! Fn -@n--! Fn
? ? ?
(5.5) ?yOEs ?y hs ?yfls
s-1 __
S2np x Vnps- --! Fnps ---! Fnps.
Taking fibers vertically in (5.4), we have a fibration sequence:
s ! Ks ! Rs ! s
where Rs = (Qn _ . ._.Qpns-1) o Pnpsis a wedge of mod pr Moore spaces and
Y k s
Ks = S2n-1x S2np -1{pr+1} x S2np -1.
1 k~~~~ -! 0
ok+1
0 -! L(k+1) -! L(k) -! -! 0
where ssk+1 is any map of Lie algebras such that
ssk+1(oek)= oek
ssk+1(ok)= ok
where oek, ok 2 L(k) L(0)are given by the formulas:
ok = xpk
pk-1` k'
1 X p
oek= ___ [xi, xpk-i].
2p i=1 i
Nothing is said about the value of ssk+1 on the other generators. We need to b*
*e more
specific at this point. Let us define the weight of an element in a free Lie a*
*lgebra to be
the minimal number of brackets in any term; in particular, L(0)is free on gener*
*ators xi, we
define !(xi) = 1, ![x, y] = !(k) + !(y) and !( ai) = min!(ai). For an element z*
* 2 L(k)we
define the weight of z to be the weight considered as an element of L(0). Thus*
* !(ok) = 1
and !(oek) = 2. We further specify the Lie algebra homomorphism ssk+1 by demand*
*ing that
13
ssk+1(z) = 0 if !(z) > 2. Now define L(k)s= Ls\ L(k)for k s. Since oek, ok 2 *
*L(k)s, we have
short exact sequences:
0 ---! L(1)s---! L(0)s---! ---! 0
? ? ? fl
? ? ? fl
y y y fl
0 ---! L(1) ---! L(0) ---! ---! 0
0 ---! L(k+1)s---! L(k)s---! ---!0
? ? ? fl
? ? ? fl
y y y fl
0 ---! L(k+1)---! L(k) ---! ---!0
for k < s.
Define L(s+1)sto be the kernel of L(s)s! ! 0. In fact we have L(s+1)s*
* L(1).
To see this we need to show that ssr+1(L(s+1)s) = 0 for r s. The generators *
*of L(s+1)sof
filtration 1 are of the form xi for i < ps and those of filtration 2 are of the*
* form [xi, xj]
for i, j < ps. None of these have dimension 2nps - 1, so L(s+1)slies in the ke*
*rnel of the
composition L(s+1)! ! . Consequently L(s+1)s L(s+1). Similarly f*
*or r > s the
generators of L(s+1)sof weight 1 and 2 have dimensions 4n(ps- 1) and conseque*
*ntly their
images are 0 in for r > s.
It follows that we may first choose a basis for L(s+1)sand then choose a bas*
*is for L(1)
containing these elements. This completes the proof.
Corollary 6.2 There is a commutative diagram of fibration sequences:
Fn ---! Vn - -*-! Pn ---! Fn
x x x x
? ? ? ?
? ? ? ?
F(ps+1-1)---! Ks+1 - -*-! Qs+1 ---! F(ps+1-1)
x x x x
? ? ? ?
? ? ? ?
F(ps-1) ---! Ks - -*-! Qs ---! F(ps-1)
where all the vertide maps are mod p homology monomorphisms and each Qs is a we*
*dge of
mod pr Moore spaces.
7 On the sequence F(ps-1) ! Fn ! Fnps
Recall that there is a p-local fibration sequence
Hps 2nps
S2n(ps-1)! S2n1-! S1 .
14
By analogy, we compare F(ps-1)to the fiber s of the Hopf invariant hs:
s
s ! Fn -h! Fnps
Theorem 7.1 s ' F(ps-1)x Xs where Xs is a wedge of mod pr Moore spaces.
Proof: For dimensional reasons, the inclusion F(ps-1) Fn lifts to s. Consi*
*der the pull
back diagram:
Ys ---! F(ps-1)- --! Ks ---! Ys - --! F(ps-1)
? ? ? ? ?
? ? ? ? ?
y y y= y y
Rs ---! s - --! Ks --*-! Rs - --! s.
The map Ks ! s ! Fn constructed in section 6 is obtained from maps:
_ k
ffik: P 2np-! Fn k < s
s-1 2nps _ffis
S2np - ! P -! Fn
and all of these maps factor through F(ps-1)for dimensional reasons. Consequen*
*tly the
retraction Ks ! s factors through F(ps-1)as well and hence the map Ks ! Ys i*
*s null
homotopic. Consequently Qs ' Ys and the map Qs ! F(ps-1)lifts to an equivale*
*nce
Qs ' Ys. Now the inclusion Qs ! Pn factors through Rs, so Rs = Qs _ Es and we h*
*ave a
diagram of fibrations:
Ks ---! Qs ---! F(ps-1)
? ? ?
=?y ?y ?y
Ks ---! Qs_ Es ---! s.
Now the fiber of Qs ! Qs_ Es is (Eso Qs) which is a retract of (Qs_ Es) = R*
*s and
hence a retract of s. This completes the proof with Xs = Eso Qs.
__
8 Factorization of @
In this section we will consider the components of
__ 2 2n+1 2n-1 2n-1 Y 2npk-1 r+1
@n : S - ! S x Vn = S x S {p }
k 1
k-1 r+1 *
*__
we will write @n(k) : 2S2n+1- ! S2np {p } for the kth component, k > 0 and *
*@n(0) for
the projection 2S2n+1- ! S2n-1.
We begin by combing the diagram from 4.5:
__
2S2n+1 --@n-! Fn - --! Fn
? ? ?
Heps?y ?y hs ?yh0
s+1 __
2S2np ---! Fnps - --! Fnp
15
with the equivalences from 5.2
__
S2n-1x Vn --n-! Fn - --! Fn
? ? ?
? ? ?
y_ y h y h0
np __
S2np-1x Vnp ---! Fnp - --! F np
where h0is defined in 5.3, to get a homotopy commutative diagram:
_@
2S2n+1 --n-! S2n-1x Vn
? ?
Heps?y ?y_
_
s+1 @nps 2nps-1
2S2np ---! S x Vnps
>From this we see that
__ __
@n(k + 1) = @npk(1) O HpkO flk, k 0
while
_@(1)
2S2n+1 --n-! S2np-1{pr+1}
? ?
(8.1) Hep?y ?yP
_@
np(0) 2np-1
2S2np+1 ---! S .
Putting these together gives:
__ __ 1X __
(8.2) @n = @n(0) + @npk(1) O HpkO flk
k=0
where flk : 2S2n+1! 2S2n+1 is an equivalence.
Remark: If we had better control over the relationship between the Hopf invaria*
*nts hs and
the James-Hopf invariant, we could remove the factor flk.
__
Proposition 8.3 Suppose @n(1) : 2S2n+1 ! S2np-1{pr+1} is null homotopic for e*
*ach n.
Then __
a) @n factors through S2n-1
b) The loops on the pth James-Hopf invariant:
Hp : 2S2n+1- ! 2S2np+1
is homotopic to a composition:
2p 2 2np+1
2S2n+1- ! 2P 2np+1-! S
__
c) The fiber B of @np(0) : 2S2np+1! S2np-1is a retract of 2P 2np+1
d) B ' BWn.
16
Proof: a) follows directly from 7.2. To verify b), note that by hypothesis
Hp : 2S2n+1- ! 2S2np+1
__
lifts wo the fiber B of @np(0). The following diagram provides a map from B to *
* 2P 2np+1:
B ---! 2P 2np+1
? ?
? ?
y y
2S2np+1 --1-! 2S2np+1
? ?
?_ ?
y@np(0) y
S2np-1 ---! Fnp
c) follows since Fn ' S2np-1x Vnpx Pnp, so
2P 2np+1' B x Vnpx 2Pnp.
To verify d), note that S2BWn is a retract of S2 2S2n+1by [G2], so there is a m*
*ap S2BWn !
P 2np+1which is non zero in homology, It's adjoint provides an equivalence:
BWn ! 2P 2np+1! B.
9 Relationship with BWn
__
In this section we describe a factorization of @using the classifying space BWn*
* for the double
suspension. Recall from [G2]
Proposition 9.1 Suppose ff : S2n ! Y and fi : S2n-1Y ! S4n-1 are maps such t*
*hat
fi OS2n-1ff : S4n-1! S2n-1Y ! S4n-1 is homotopic to the identity. Then there is*
* a fibration
sequence: _
S2n-1- ff! Y - ! B.
This is natural in the following sense: Suppose we are also given
ff0: S2n -! Z
fi0: S2n-1Z -! S4n-1
f : Y - ! Z
such that fi ~ fi0O S2n-1f and ff0~ fff.
Then we have a homotopy commutative diagram;
_ff
S2n-1 ---! Y ---! B
? ? ?
1?y ?y f ?y_
_ff0
S2n-1 ---! Z ---! B0
17
Corollary 9.2 There is a homotopy commutative diagram:
S2n-1 ---! 2S2n+1 ---! BWn
? ? ?
? ? ?
y y y
S2n-1 ---! Fn ---! B
Since S2n-1 is a retract of Fn, B ' Vn x Wn and we have
Corollary 9.3 There is a homotopy commutative diagram:
2S2n+1 ---! BWn
? ?
? ?
y@ y
__
Fn ---! Fn.
__ k
In particular, @n(k) : 2S2n+1! S2np -1{pr+1} factors through BWn, proving Theo*
*rem 1.2.
18
A Appendix
The following observation appears in the work of Cohen, Moore and Neisenderfer.*
* We
repeat it here because it sheds light on their results and is easily extended t*
*o the case of the
filtrations in section 6.
We begin by looking at the homology Serre spectral sequence for the fibering
Fn -! P 2n+1-! S2n+1.
ss
This is a multiplicative spectral sequence with E2p,q~=Hp( S2n+1; Z=p) Hq( Fn*
*; Z=p).
Since all elements of E2p,0are permanent cycles, E2 = E1 . Restricting this fib*
*ration to S2n(k)
leads to fibration
Fn -! Ek -! S2n(k)
and H*(Ek; Z=p) H*(Ek+1; Z=p) . . .H*( P 2n+1; Z=p).
Thus H*( P 2n+1; Z=p) is a filtered differential group using fi(r). It has a*
* spectral sequence:
E2p,q= Hp( S2n+1; fi(r)) Hq(H*( Fn; Z=p), fi(r))
converging to Hp+q( P 2n+1; Z=p) = 0 if p + q > 0. This gives a multiplicative*
* spectral
sequence:
E2p,q= Zp[u2n] Hq(H*( Fn; Z=p); fi(r)) ) 0.
This spectral sequence has the same form as the Serre Spectral sequence for the*
* homology
of the fibration:
2S2n+1- ! P S2n+1- ! S2n+1.
The differentials are forced by the multiplicative structure and consequently t*
*hese spectral
sequences are isomorphic. From this we conclude
Proposition A.1 ([CMN1]) H*(H*( Fn; Z=p); fi(r)) ~= H*( 2S2n+1; Z=p) as alegbr*
*as and
co-algebras.
Corollary A.2 H*(H*( F(ps-1); Z=p); fi(r)) ~=H*( S2n(ps-1); Z=p).
The later follows by restricting the spectral sequence to the first ps- 1 colum*
*ns.
References
[G] Anick, David; Gray, Brayton: Small H spaces related to Moore spaces. To*
*pology
34 (1995), no. 4, 859-881.
[CMN1] Cohen, F. R.; Moore, J. C.; Neisendorfer, J. A.: Exponents in homotopy *
*theory.
Algebraic topology and algebraic K-theory (Princeton, N.J.,1983), 3-34,*
* Ann. of
Math. Stud., 113, Princeton Univ. Press, Princeton, NJ, 198.
19
[CMN2] Cohen, F. R.; Moore, J. C.; Neisendorfer, J. A.: Torsion in homotopy gr*
*oups. Ann.
of Math. (2) 109 (1979), no. 1, 121-168.
[G1] Gray, Brayton: On the homotopy groups of mapping cones. Proc. London Ma*
*th.
Soc. (3) 26 (1973), 497-520.
[G2] Gray, Brayton: On the iterated suspension. Topology 27 (1988), no. 3,30*
*1-310.
[G3] Gray, Brayton EHP spectra and periodicity. I. Geometric constructions. *
*Trans.
Amer. Math. Soc. 340 (1993), no. 2, 595-616.
[G4] Gray, Brayton,; Theriault, Stephen: On the double suspension and the M*
*od-p
Moore space. Contemporary Mathematics Voll 399, 2006, 101-121.
[J] James, I. M.: Reduced product spaces. Ann. of Math. (2) 62 (1955), 170-*
*197.
[MN] Moore, John C.; Neisendorfer, Joseph A.: Equivalence of Toda-Hopf inva*
*riants.
Israel J. Math. 66 (1989), no. 1-3, 300-318.
[N1] Neisendorfer, Joseph: Product decompositions of the double loops on odd*
* primary
Moore spaces. Topology 38 (1999), no. 6, 1293-1311.
[N2] Neisendorfer, Joseph: James-Hopf invariants, Anick's spaces, and the do*
*uble loops
on odd primary Moore spaces. Canad. Math. Bull. 43 (2000), no. 2, 226-2*
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[S] Selick, Paul: A spectral sequence concerning the double suspension. Inv*
*ent. Math.
64 (1981), no. 1, 15-24.
[T] Theriault, Stephen D.: Proofs of two conjectures of Gray involving the*
* double
suspension. Proc. Amer. Math. Soc. 131 (2003), no. 9, 2953-2962.
20
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