The answer to an email of Mr. Douglas C. Ravenel
Zhou Xueguang
Department of Mathematics, Nankai University
Tianjin 300071, China
October 27, 2003
Due to a mistake I made myself, I announce to withdraw the article Ä Reply"*
* that
appeared on the September 2003 submissions.
The following is the answer to an email of Mr Douglas C. Ravenel on January *
*12,2002.
I am sorry for replying so late.
(1) In your email, you said that the Serre spectral sequence of the followin*
*g sequence
! (S2m+1 ) ! (B(mp)) ! (S2m+2p+1) !
collapse and so we have the isomorphism H*( (B(mp))) = H*( (S2m+1 ) H*( (S2m+2p*
*+1))
as an additive group, but we do not know whether this is a ring isomorphism.
(2) In the second part, you said that you do not believe that as maps from M*
*(l+m+
n+2) to M(l)^M(m)^M(n), P ((ff fi) fl) = fl (ff fi) and id((ff fi) fl) = (ff fi*
*) fl
are homotopic. I agree that it needs a proof. Now, we will give a proof in the *
*last part of
this letter. The other part of the appendix on page 48 of my paper remain uncha*
*nged.
So, I still believe that the two proofs of Toda's result ff1fip1= 0 for p > 3 a*
*re incorrect.
(3) Besides above, there is still a gap in the proof of the statement that V*
* (3) does not
exist for n > 5. In page 290 of your book öC mplex Cobordism and Stable Homotopy
Groups of Spheres", you said that x761=< ff1fi3, fi4, fl2 > is a permanent cycl*
*e for p = 5.
But we have |ff1| = 7, |fi3| = 134, |fi4| = 82, |fl2| = 437, and so the Massey *
*product x761
exists modular a coset ufl2 + (ff1fi3)v with |u| = 324 and |v| = 620. From the *
*table in
page 291 we know that u and v may not be trivial. Then, how can you get the con*
*clusion
that x761is a permanent cycle?
(4) In what follows, we will prove that (ff fi) fl ff (fi fl). the*
* Moore space
M(l), M(m) and M(n) are the same as defined in my paper. Since ff (fi fl)
1
(-1)(l+1)(m+n+2)(fi fl) ff, if m = n, then ff (fi fl) (fi fl) ff.*
* So, we have
P ((ff fi) fl) (fi fl) ff ff (fi fl) (ff fi) fl for m = n*
* = l.
Firstly, we introduce the notions that will be used. It is assumed that all*
* maps
between spaces with base point keep the base point. For two spaces X, Y , X = Y
means that there is a given topological map f: X ! Y . Let n > 1. As usual, we
use Dn to denote the unit disk in Rn and Sn to denote the unit sphere in Rn+1. *
* So,
@(Dn+1) = Sn. The base point ] of Dn and Sn-1 is taken to be (-1, 0, . .,.0). *
* It is
obvious that there exists a topological map H from Sn to ^ni=1S1i, where S1iis *
*just a
copy of S1. We say that H is a decomposition of Sn. In general, we omit the map*
* H
and use the notion Sn = ^ni=1S1ito denote it. It is easy to check that Dn ^ D1 *
*= Dn+1.
By induction, we have Dn ^ Dl = Dn+l for n, l > 1. It is also obvious that Dn+*
*1 =
@(Dn+1) ^ D1 = Sn ^ D1. By the decomposition @(Dn+1) = Sn = ^ni=1S1i, any point*
* x
of Dn+1 can be expressed as x1 ^ . .^.xn ^ t with xi2 S1i, i = 1, . .,.n and -1*
* 6 t 6 1.
We say that x1, . .,.xn, t is the polar coordinates of x with respect to the de*
*composition
Sn = ^ni=1S1iand Dn+1 = Sn ^ D1. We also have that Dn+1 = (^ni=1S1i) ^ D1 is a
decomposition of Dn+1. In this appendix, p is always assumed to be a fixed odd *
*prime.
Let `: S1 ! S1 be defined by `(e2iix) = e2iipx. It is obvious that ` has degre*
*e p and
we call it the standard p-map of S1. Let `n: Sn ! Sn be the map defined by that
`n(x1^ . .^.xn) = `(x1) ^ . .^.xn for all xi2 S1i, i = 1, . .,.n. We call `n th*
*e standard
p-map of Sn with respect to the decomposition Sn = ^ni=1S1i. The identification*
* space
M~(n) = Dn+1 [ Sn=x ^ 1 in Dn+1 are identified with `n(x) 2 Sn for all x 2 Sn a*
*nd
M~(n) = Dn+1 [ Sn x (-1, 1]=x ^ t in Dn+1 are identified with (`n(x), t) for al*
*l x 2 Sn
and -1 < t 6 1 are respectively call S Moore space and SS Moore space. It is ob*
*vious
that M~(n) M~(n).We call Dn+1 the standard (n + 1)-cell of M~(n) and M~(n) an*
*d call
Sn the unit sphere of them. We use ~n+1 to denote both inclusion maps from Dn+1*
* to
M~(n) and M~(n).
Notice that in M~(n) and M~(n), @Dn+1 and the unit sphere Sn are not the same
space. For x 2 @Dn+1, we have x 62 `n(x) 2 Sn. In M~(n), any point can be expre*
*ssed
as x ^ t with x 2 Sn and -1 6 t 6 1.
For l > 0, m > 0, let Dl+1= Sl^D11= (^li=1)^D11and Dm+1 = Sm ^D12= (^mi=1)^D*
*12
__
be two decompositions of Dl+1 and Dm+1 . Let D2 = D1 ^ D2 = (@D2) ^ D 1, then
*
* __
Dl+1^ Dm+1 = (Sl^ D11) ^ (Sm ^ D12) = Sl^ Sm ^ (D11^ D12) = (^l+m+1i=1) ^ (@D2)*
* ^ D1.
2
Thus, we get a decomposition of Dl+1^ Dm+1 . We call this the natural decomposi*
*tion
of Dl+1^ Dm+1 with respect to the decomposition of the factors Dl+1 and Dm+1 .
Let M~(l) and M~(m) be two SS Moore spaces with respectively standard cell D*
*l+1
and Dm+1 and standard spheres Sl and Sm . Then, any point of M~(l) ^ M~(m) can *
*be
expressed as (x, t1) ^ (y, t2) with x 2 Sl, y 2 Sm and -1 6 t1, t2 6 1. We wr*
*ite it as
(x ^ y ^ (t1, t2). By the definition of M~(l) and M~(m), we have x ^ t = (`l(x*
*), t) and
y^t = (`m (y), t) for x 2 @Dl+1and y 2 @Dm+1 and -1 < t 6 1. Now, @(Dl+1^Dm+1 )*
* =
{x ^ t1 ^ y ^ t2 | x 2 Sl, y 2 Sm , t1 = 1, -1 6 t2 6 1 or t2 = 1, -1 6 t1 6 1}*
*. So, we
have the following
Proposition 1. For x 2 @Dl+1, y 2 @Dm+1 , t1 = 1, -1 6 t2 6 1 or t2 = 1, -1 *
*6 t1 6 1,
~l^ ~m (x ^ t1 ^ y ^ t2) = (`l(x), t1) ^ (`m (y), t2).
Let M~(l + m + 1) and M~(l + m + 1) be the S and SS Moore spaces defined abo*
*ve. We
defined a map ~ 0(l, m): ~M(l+m+1) ! M~(l)^M~ (m) as follows. For x 2 Dl+1, y 2*
* Dm+1 ,
~ 0(l, m)(Dl+m+1) = ~l(x) ^ ~m (y). Since Sl+m+1 = {x ^ t1^ y ^ t2 | x 2 Sl, y *
*2 Sm , t1 =
1, -1 6 t2 6 1 ort2 = 1, -1 6 t1 6 1}, we define a map ~ 00(l, m): Sl+m+1 ! M~(*
*l)^M~ (m)
as follows. For x 2 Sl and y 2 Sm , ~ 00(l, m)(x ^ t1 ^ y ^ t2) = x ^ `m (y) ^ *
*(t1, t2) for
t2 = 1, -1 6 t1 6 1 and t2 = 1, -1 6 t1 6 1. Let `l+m+1 be the p-map defined by
the decomposition Dl+m+2 = Dl+1^ Dm+1 with respect to the factors Dl+1and Dm+1 .
By Proposition 1, we have ~ 0(l, m)(x ^ y) = ~ 00(l, m)((`(x ^ y)) for x ^ y 2 *
*@Dl+m+2.
So, ~ 0(l, m) and ~ 00(l, m) define a map ~(l, m): ~M(l + m) ! M~(l) ^ M(m). We*
* extend
~ (l, m) to a map ~ (l, m): ~M(l + m + 1) ! M~(l) ^ M(m) as follows. For -1 6 t*
* 6 1,
let jt: [-1, 1] ! [-1, t] be the linear map such that jt = -1 and jt(1) = t. Le*
*t Dl+1t=
Sl^ (-1, t], Dm+1t= Sm ^ (-1, t]. Define M~t(l) = Dl+1t^ Slx t= x ^ t are ident*
*ified with
`l(xxt for all x 2 @Dl+1. M~t(m) is similarly defined. We have M~(l) = [t2(-1,1*
*]~Mt(l) and
M~(m) = [t2(-1,1]~Mt(m). Let jlt: ~M(l) ! M~t(l) and jmt: ~M(m) ! M~t(m) be res*
*pectively
the maps defined by jlt(x ^ ø) = x ^ jt(ø) for x 2 @Dl+1and jlt(x0^ 1) = (x, t)*
* for x02 Sl
and jmt(y ^ ø) = y ^ jt(ø) for y 2 @Dm+1 and jmt(y0^ 1) = (y, t) for y02 Sm . W*
*e extend
~ (l, m) to a map e~(l, m): ~M(l + m + 1) ! M~(l + m + 1) [ Sl+m+1 x (-1, 1] as*
* follows.
For x 2 Sl+m+1 and -1 6 t 6 1, e~(l, m)(x, t) = (jlt^ jmt) O ~(l, m)(x, 1) 2 M~*
*lt^ M~mt
M~(l) ^ M~(m). It can be easily seen that the map e~(l, m) so defined is contin*
*uous and
e~(l, m) l+m+1 l m+1 l+1 l+1 m ~ l+1 m+1
*(S ) = S ^D +(-1) D ^S and (l, m)*(Dl+m+2) = D ^D .
So, both e~(l, m) and ~(l, m) are ff fi maps.
3
Now, we prove the main proposition of this appendix (ff fi) fl = ff (f*
*i fl). Let
ff = M~(l), fi = M~(m), fl = M~(n) be three SS Moore spaces and Dl+1, Dm+1 , Dn*
*+1 be
respectively the standard cells of ff, fi and fl. Dl+1= (^li=1S1i)^D11, Dm+1 = *
*(^li=mS1i)^
D12, Dn+1 = (^li=nS1i) ^ D13. Now, there are two different decompositions of Dl*
*+m+n+3.
The first is Dl+m+n+3 = Dl+m+2 ^ Dn+1 and the second is Dl+m+n+3 = Dl+1^ Dm+n+2*
* .
Let `(1)l+m+n+2and `(2)l+m+n+2be respectively the standard p-maps with the firs*
*t and second
decomposition. Then, we have the following
Proposition 2. For all x 2 Sl+m+n+2 , `(1)l+m+n+2(x) = `(2)l+m+n+2(x), and *
*there exist
two disks ~D1and ~D1such that
D11^ D12= @(D11^ D12) ^ ~D1, D12^ D13= @(D12^ D13) ^ ~D2
Proof. We have Dl+m+2 = Dl+1^ Dm+1 = (^l+mi=1S1i) ^ (D11^ D12) = (^l+mi=1S*
*1i) ^
@(D11^ D12) ^ ~D1and Dl+m+n+3 = Dl+m+2 ^ Dn+1 = (^l+mi=1S1i) ^ @(D11^ D12) ^ ~D*
*1^ D13.
It can be easily seen that @(D11^ D12) ^ @(D12^ D13) = @(D11^ D12^ D13). Since*
* any
point x in Sl+m+n+2 can be expressed by ^l+m+ni=1xi^ T1 ^ T2 with T1 2 @(D11^ D*
*12) and
T1 2 @(D~11^ D13), we have by definition `(1)l+m+n+2(x) = `(x1) ^ (^l+m+ni=2xi)*
* ^ T1 ^ T2.
Since any point T in @(D11^ D12^ D13) can be expressed by T1 ^ T2 and any point*
* x can
be expressed by (^l+m+ni=1S1i) ^ T , we have `(1)l+m+n+2(x) = `(x1) ^ (^l+m+ni=*
*2xi) ^ T . By
the same method as above, we also have `(2)l+m+n+2(x) = `(x1) ^ (^l+m+ni=2xi) ^*
* T . So, we
have `(1)l+m+n+2(x) = `(2)l+m+n+2(x).
Proposition 3. (l, m) ^ id) O (l + m + 1, n) = id^ (m, n)) O (l, m + n +*
* 1).
Proof. Notice that in the definition of ~(l + m + 1, n) and ~(l, m + n + 1),*
* the two
standard p-maps are the same, so the two M~(l + m + n + 1 are the same. Now, we*
* have
the following commutative diagram
@(Dl+m+n+3) -id! @(Dl+m+2^ Dn+1) -id! @(Dl+1^Dm+1 ^Dn+1)
# ~`l+m+n+3 # ~l+m ^ ~n # ~l^ ~m ^ ~n
Sl+m+n+2 - ! ~M(l+ m+ 1)^ ~M(n) -! M~(l)^ ~M(m)^ ~M(n)
where the two maps on the bottom line are respectively ~(l+ m+ 1, n) and ~(l+ m*
*+ 1, n)^
id. For x 2 Sl+m+n+2 , there exists an x0 2 @Dl+m+n such that x = `(1)l+m+n+2(*
*x0) =
`(2)l+m+n+2(x0). Since
(~ (l, m)^ id)(~ (l + m + 1, n))(x)
= (~ (l, m)^ id)(~ (l + m + 1, n))~`l+m+n+2(x0)
4
= (~l^~m ^~n)(id)(id)(x0)
= (~l^~m ^~n)(x0)
Similarly, we have (id^ ~(m, n))(~ (l, m + n + 1))(x) = (~l^~m ^~n)(x0). So, w*
*e have
(k = l + m + n + 2)
( (l, m)^ id)(~ (l + m + 1, n))|Sk = (id^ ~(m, n))(~ (l, m + n + 1))|Sk*
* .
Since both maps on Dl+m+n+3 are ~l^~m ^~n, we have
( (l, m)^ id)(~ (l + m + 1, n)) = id^(~ (m, n))(~ (l, m + n + 1)).
5