Indian J. Pure Appl. Math., 41(4): 607-623, August 2010
c Indian National Science Academy
°
(Z2 )k -ACTIONS WITH FIXED POINT SET OF CONSTANT
CODIMENSION 2k + 81
Jingyan Li∗ and Yanying Wang∗∗
∗ Department
of Mathematics and Physics, Shijiazhuang Railway Institute,
Shijiazhuang, 050043, Peoples’ Republic of China
e-mail : [email protected]
∗∗ College of Mathematics and Information Science,
Hebei Normal University, Shijiazhuang, 050016, Peoples’ Republic of China
e-mail : [email protected]
(Received 19 September 2008; after final revision 26 March 2010;
accepted 19 May 2010)
k
2 +8
The ideal J∗,k
of cobordism classes in the unoriented cobordism ring
M O∗ containing a representative admitting a (Z2 )k -action with fixed
point set of constant codimension 2k + 8 is determined for k ≥ 4.
Key words : (Z2 )k -action, indecomposable cobordism class, fixed
point set, constant codimension, Dold manifold.
1. Introduction
Let φ : (Z2 )k × M n → M n be a smooth action of the group (Z2 )k =
{T1 , T2 , · · · , Tk | Ti2 = 1, Ti Tj = Tj Ti } on a closed n-dimensional manifold.
The fixed point set F of the action is a disjoint union of closed submanifolds
1
Project supported by the Academic Research Fund from Shijiazhuang Railway Institute and NSFC (10971050).
608
JINGYAN LI AND YANYING WANG
of M n . If each component of F is (n − r)-dimensional, then F has constant
r denote the set of n-dimensional cobordism classes
codimension r. Let Jn,k
containing a representative M n admitting a (Z2 )k -action with fixed point
P
r =
r
set of constant codimension r and J∗,k
n≥r Jn,k . From [1] we know that
r is a subgroup in the unoriented cobordism group M O and J r forms
Jn,k
n
∗,k
P
an ideal of the cobordism ring M O∗ = n≥0 M On . It is not difficult to see
r1 r2
r1 +r2
r ⊂ Jr
that J∗,k
.
∗,k+1 and J∗,k J∗,k ⊂ J∗,k
1 = (0) in [2]. Inspired in
In 1965, Conner and Floyd showed that J∗,1
r and
this fact, Stong introduced the question of determining the ideal J∗,1
2
3
4
computed J∗,1 in [3]. Soon after, Capobianco computed J∗,1 in [4] and J∗,1
5 , J 6 , J 7 and
in [5]. After this, Iwata, Wada, Wu and Kikuchi obtained J∗,1
∗,1 ∗,1
8 respectively in [6-9]. In 1992, Pergher introduced the same question of
J∗,1
r with k ≥ 2 in [10]. Also, he computed J 1 and J 2 for
determining J∗,k
∗,k
∗,k
k ≥ 2, and made a lot of characteristic numbers calculations to determine
3 defined in terms of certain restrictions on the fixed
certain sub-ideals of J∗,2
data of the actions. In 1994, Shaker got the most important advance in
r in [11]. The crucial point was the discovery
the direction of computing J∗,k
of a method for constructing models of (Z2 )k −actions on certain manifolds
so that one can control two involved properties: the indecomposability of
the manifold and the codimension of the fixed point set of the action, thus
providing a method to construct generators for the unoriented cobordism
r for
ring with desired properties. Using this method, Shaker computed J∗,k
r < 2k , and concluded that


r = 1,
 (0),
r
∞
J∗,k =
⊕n=r M On ,
r even,

 ⊕∞ M O ∩ Kerχ, r odd, r ≥ 3,
n
n=r
where χ : M O∗ → Z2 denotes the mod 2 Euler characteristic. Later, in
[12], Shaker solved the case r = 2k . The key point was a way to equip Dold
manifolds with certain (Z2 )k −actions and the fact that there is a method
to recognize the indecomposability of Dold manifolds. These two articles
of Shaker gave a clear scheme to attack the next cases. Using this scheme
and the complicated formula of Kosniowski and Stong given in [13], Wang,
Wu and Ma solved the case r = 2k + 1 in [14], leaving open some particular
cases. In [15], Wang, Wu and Ding gave a crucial contribution concerning
the determination of the next cases and made an important advance in the
r for r > 2k . The authors showed that, for a given
setting of computing J∗,k
(Z2 )k -ACTIONS WITH FIXED POINT SET
609
r > 2k , there exists a number g(r) much bigger than r so that, if n ≥ g(r),
r
then Jn,k
is explicitly computed. For r > 2k , this completely solves the
r
question of computing Jn,k
for all sufficiently large n. In other words, for
k
r to a finite and explicit list of
r > 2 , this reduces the computation of Jn,k
values of n. The authors also solved the case r = 2k + 3 for k ≥ 4 and
n ≥ 2k + 4. In [16], Liu and Wu studied the case r = 2k + 2. They used the
generalized Dold manifold given in [17] to construct certain indecomposable
manifolds equipped with (Z2 )k −actions. This allowed to obtain, besides the
case r = 2k + 2, the solution for certain cases left previously open. For
r have been determined in [18-21]
r = 2k + 4, 2k + 5, 2k + 6 and 2k + 7, J∗,k
respectively.
In this paper, we will construct some special generators of M O∗ to de2k +8
termine the ideal J∗,k
. The main result is
k
2 +8
Theorem — For k ≥ 4, the ideal J∗,k
of M O∗ consists of all classes
k
in dimensions greater than 2 + 8 and the decomposable classes in dimension
2k + 8 which contain only factors with dimension less than 2k .
Throughout this paper manifolds will be smooth, compact and without
boundary but not necessarily connected. S m denotes the m-dimensional
sphere. The coefficient group is Z2 (the integers mod 2) and ≡ denotes
¡ ¢
congruence mod 2. Binomial coefficients are m
n = m!/n!(m − n)!.
2. Preliminaries
It is well known that the unoriented cobordism ring M O∗ is a Z2 -polynomial
algebra with a single generator in each dimension n which is not of the form
2u − 1 (see [1]). If the cobordism class [M n ] of the smooth closed manifold
M n can be expressed as a sum of products of lower dimensional cobordism
classes, then [M n ] is called decomposable. Otherwise it is indecomposable.
The indecomposable classes can be chosen as generators of the Z2 -polynomial
algebra M O∗ . For our purpose, the indecomposable classes will come from
the following sources.
Lemma 2.1 — [3; Lemma 3.4] Let RP (n1 , n2 , · · · , nl ) be the projective
space bundle of λ1 ⊕ λ2 ⊕ · · · ⊕ λl over RP (n1 ) × RP (n2 ) × · · · × RP (nl ),
where λi is the pullback of the canonical line bundle over the i-th factor.
Then for l > 1, [RP (n1 , n2 , · · · , nl )] is indecomposable in M O∗ if and only
610
JINGYAN LI AND YANYING WANG
if
Ã
n+l−2
n1
!
Ã
+
n+l−2
n2
!
Ã
+ ··· +
n+l−2
nl
!
≡ 1 mod 2,
where n = n1 + n2 + · · · + nl .
The manifold RP (n1 , n2 , · · · , nl ) has dimension n + l − 1. If ni+1 =
ni+2 = · · · = nl = 0, then RP (n1 , n2 , · · · , nl ) will sometimes be written as
RP (n1 , n2 , · · · , ni ; l).
In [17], Brown constructed the generalized Dold manifold. For a space
X and a positive integer m, let P (m, X) be formed from S m × X × X by
identifying (u, x, y) with (−u, y, x). If X is an n-dimensional manifold, then
P (m, X) is an m + 2n-dimensional manifold. He also gave
Lemma 2.2 — [17; Proposition 4.1] [P (m, M n )] is indecomposable in
M O∗ if and only if [M n ] is indecomposable in M O∗ and the binomial coefficient
!
Ã
m+n−1
≡ 1 mod 2.
m−1
To calculate binomial coefficients mod 2, we recall Kummer’s result: If
m=
l
X
mi 2i and n =
i=0
Ã
with 0 ≤ mi , ni ≤ 1, then
m
n
l
X
ni 2i
i=0
!
≡ 1 mod 2 if and only if ni ≤ mi for every
i.
Lemma 2.3 — [11; Lemma 3.1] Let λi −→ Xi be line bundles, and let
(Z2 )ki (ki ≥ 0) act on λi as bundle maps with fixed point set Fi on Xi for all
l
P
1 ≤ i ≤ l with
2ki ≤ 2k . Then RP (n1 , n2 , · · · , nl ) over X1 × X2 × · · · × Xl
i=1
admits a (Z2 )k -action with fixed point set F1 × F2 × · · · × Fl × E with E
being a set of l points.
Lemma 2.4 — [13] If (M n , φ) is a (Z2 )k -action on an indecomposable nmanifold, then some component of the fixed point set of M is of dimension
at least [n/2k ].
(Z2 )k -ACTIONS WITH FIXED POINT SET
611
Lemma 2.5 — [13] If (M n , φ) is a (Z2 )k -action on a manifold with
s(λ1 ,··· ,λi ) [M ] 6= 0,
then some component of the fixed point set of M is of dimension at least
[λ1 /2k ] + · · · + [λi /2k ].
3. Existence of Indecomposables
k
2 +8
The main task of this section is to exhibit indecomposable classes in J∗,k
.
k
2 +8
Lemma 3.1 — There exist indecomposable classes xn ∈ J∗,k
for k ≥ 4,
k
u
n ≥ 2 + 9 odd and not of the form 2 − 1.
Proof : Since n is not of the form 2u − 1, n−1
is not of the form
2
n−1
2k +9−1
u
k−1
2 − 1. We have 2 ≥
= 2
+ 4. From [11; 5.1], there exists
2
2k−1 +4
an indecomposable class X ∈ J n−1 ,k (for 2k−1 + 4 < 2k ), so that X has
2
a representative M admitting a (Z2 )k -action with fixed point set F 0 , where
n−1
n − 2k − 9
dim(F 0 ) =
− (2k−1 + 4) =
. Let Ti0 (i = 1, 2, · · · , k) denote
2
2
the (Z2 )k -action on M . Then we can define a (Z2 )k -action on S 1 × M × M
as follows:
T1 (u, x, y) = (u, T10 (x), T10 (y)),
T2 (u, x, y) = (u, T20 (x), T20 (y)),
······ ,
Tk (u, x, y) = (u, Tk0 (x), Tk0 (y)).
These involutions commute with the map T (u, x, y) = (−u, y, x), so that
induce a (Z2 )k -action on P (1, M ) with fixed point set F = S 1 × F 0 × F 0 /T =
P (1, F 0 ), where dimF = n − 2k − 8. Taking xn = [P (1, M )], we have that
2k +8
xn ∈ J∗,k
. By Lemma 2.2, xn is indecomposable.
¤
k
2 +8
Lemma 3.2 — There exist indecomposable classes xn ∈ J∗,k
for k ≥ 4,
k
k+1
2 + 10 ≤ n ≤ 2
and n even.
Proof : We will consider the following cases.
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JINGYAN LI AND YANYING WANG
Case 1 — n = 2k + 2r1 + 2r2 + · · · + 2rm + 2, k > r1 > r2 > · · · > rm ≥ 2
and m ≥ 1.
k−1
From [11; 5.1], there exists an indecomposable class X ∈ J 2n−2 ,k+4 (for
2
2k−1 + 4 < 2k ), so that X has a representative M admitting a (Z2 )k -action
k−1 + 4) = n−2k −10 . Take
with fixed point set F 0 , where dim(F 0 ) = n−2
2 − (2
2
xn = [P (2, M )], with the (Z2 )k -action given in the proof of Lemma 3.1.
The fixed point set of this action is F = S 2 × F 0 × F 0 /T = P (2, F 0 ) with
2k +8
. By Lemma 2.2, xn is indecomposable.
dimF = n − 2k − 8, then xn ∈ J∗,k
Case 2 — n = 2k + 2r1 + 2r2 + · · · + 2rm , k > r1 > r2 > · · · > rm ≥ 2 and
m ≥ 2.
(1) If m ≥ 3 or m = 2 and r1 ≥ 5, then 2k−1 − 2r1 −1 − · · · − 2rm−1 −1 + 8 <
2k−1 .
From [11; 5.1], there exists an indecomposable class
2k−1 −2r1 −1 −···−2rm−1 −1 +8
X ∈ J n−2rm
, so that X has a representative M admitting
2
,k−1
rm
a (Z2 )k−1 -action with fixed point set F 0 , where dim(F 0 ) = n−22 − (2k−1 −
2r1 −1 −· · ·−2rm−1 −1 +8) = 2r1 +2r2 +· · ·+2rm−1 −8. Let Ti0 (i = 1, 2, · · · , k−1)
denote the (Z2 )k−1 -action on M . Then we can define a (Z2 )k -action on
rm
S 2 × M × M as follows:
T1 (u, x, y) = (u, y, x),
T2 (u, x, y) = (u, T10 (x), T10 (y)),
······ ,
0
0
Tk (u, x, y) = (u, Tk−1
(x), Tk−1
(y)).
These involutions commute with the map T (u, x, y) = (−u, y, x), so that
rm
induce a (Z2 )k -action on P (2rm , M ) with fixed point set F = S 2 × ∆(F 0 ×
F 0 )/T , where dimF =dim(F 0 ) + 2rm = n − 2k − 8. Taking xn = [P (2rm , M )],
2k +8
we have xn ∈ J∗,k
. By Lemma 2.2, xn is indecomposable.
(2) If m = 2 and r1 = 4, we take xn = [RP (19, 2r2 − 1; 2k − 1)]. From
2k +8
[15; Lemma 2.4], xn ∈ J∗,k
.
Ã
! Ã
!
k
4
r
2 +2 +2 2 −1
2k + 24 + 2r2 − 1
+
+ 2k − 3 ≡ 1 mod 2.
24 + 2 + 1
2r2 − 1
By Lemma 2.1, xn is indecomposable.
(Z2 )k -ACTIONS WITH FIXED POINT SET
613
(3) If m = 2 and r1 = 3, we take xn = [RP (11, 11; 2k − 9)]. Let
T1 [x1 , x2 , · · · , x12 ] = [x1 , x2 , · · · , x6 , −x7 , −x8 , · · · , −x12 ],
T2 [x1 , x2 , · · · , x12 ] = [x1 , x2 , x3 , −x4 , −x5 , −x6 , x7 , x8 , x9 , −x10 , −x11 , −x12 ].
Then (T1 , T2 ) defines a (Z2 )2 -action on RP (11) with fixed point set being
four copies of RP (2). Let (Z2 )0 act as the identity on the rest of the base.
2k +8
Since 22 + 22 + 2k − 9 − 2 = 2k − 3 < 2k , by Lemma 2.3, xn ∈ Jn,k
. By
Lemma 2.1, xn is indecomposable.
Case 3 — n = 2k + 2r1 and k > r1 ≥ 4.
If r1 > 4, we take xn = [RP (2n − 2k+1 − 17, 1, 1; 2k+1 − n + 16)]. From
2k +8
[22; Lemma 2.3], xn ∈ Jn,k
.
!
n−1
+ 2k+1 − n + 16 − 3
+2
1
Ã
!
2k + 2r1 −1 + · · · + 2 + 1
≡
+1
2r1 + · · · + 25 + 23 + 22 + 2
Ã
n−1
2n − 2k+1 − 17
!
Ã
≡ 1 mod 2.
By Lemma 2.1, xn is indecomposable.
If r1 = 4, we take xn = [RP (19, 4; 2k − 6)]. Let
T1 [x1 , x2 , · · · , x20 ] = [x1 , x2 , · · · , x10 , −x11 , −x12 , · · · , −x20 ],
T2 [x1 , x2 , · · · , x20 ] = [x1 , · · · , x5 , −x6 , · · · , −x10 , x11 , · · · , x15 , −x16 , · · · , −x20 ].
Then (T1 , T2 ) defines a (Z2 )2 -action on RP (19) with fixed point set being
four copies of RP (4). Let (Z2 )0 act as the identity on the rest of the base.
2k +8
By Lemma 2.3, xn ∈ Jn,k
.
Ã
2k + 24 − 1
19
!
Ã
+
2k + 24 − 1
4
!
+ 2k − 8 ≡ 1 mod 2.
By Lemma 2.1, xn is indecomposable.
¤
k
2 +8
Lemma 3.3 — There exist indecomposable classes xn ∈ J∗,k
for k ≥ 4,
k+1
k+2
2
≤n<2
and n even.
614
JINGYAN LI AND YANYING WANG
Proof : We will consider the following cases.
Case 1 — 2k+1 ≤ n ≤ 2k+1 + 2k and k ≥ 5.
Let xn = [RP (2k + 1, 1, n − 2k − 2k−1 − 8; 2k−1 + 7)]. From [22; Lemma
2k +4
2.3], xn ∈ Jn,k
.
Ã
! Ã
! Ã
!
n−1
n−1
n−1
+
+
+ 2k−1 + 7 − 3
2k + 1
1
n − 2k − 2k−1 − 8
Ã
! Ã
!
n−1
n−1
≡
+
+ 1.
2k + 1
2k + 2k−1 + 7
If n = 2k+1 ,
! Ã
!
Ã
2k+1 − 1
2k+1 − 1
+
+ 1 ≡ 1 + 1 + 1 ≡ 1 mod 2.
2k + 1
2k + 2k−1 + 7
If n = 2k+1 + 2k ,
!
! Ã
Ã
2k+1 + 2k − 1
2k+1 + 2k − 1
+ 1 ≡ 0 + 0 + 1 ≡ 1 mod 2.
+
2k + 2k−1 + 7
2k + 1
If n = 2k+1 + 2r1 + · · · + 2rm and r1 < 2k ,
Ã
! Ã
!
2k+1 + 2r1 + · · · + 2rm − 1
2k+1 + 2r1 + · · · + 2rm − 1
+
+1
2k + 1
2k + 2k−1 + 7
≡ 0 + 0 + 1 ≡ 1 mod 2.
By Lemma 2.1, xn is indecomposable.
Case 2 — 2k+1 ≤ n ≤ 2k+1 + 2k and k = 4.
k
2 +8
If n = 25 , we take xn = [RP (26; 7)]. From [15; Lemma 2.5], xn ∈ Jn,k
.
By Lemma 2.1, xn is indecomposable.
If 25 < n ≤ 25 + 24 , we take xn = [RP (11, 11, n − 34; 13)]. From [22;
2k +8
Lemma 2.3], xn ∈ Jn,k
.
Ã
! Ã
!
Ã
!
n−1
n−1
n−1
2
+
+ 13 − 3 ≡
≡ 1 mod 2,
11
n − 34
25 + 1
By Lemma 2.1, xn is indecomposable.
(Z2 )k -ACTIONS WITH FIXED POINT SET
615
Case 3 — 2k+1 + 2k < n < 2k+2 and k ≥ 4.
Suppose n = 2k+1 +2r1 +2r2 +· · ·+2rs , k = r1 > r2 > · · · > rs ≥ 1, s ≥ 2.
Write S = {i|there exists some i in the 2-adic expansion of n such that ri >
ri+1 + 1}.
(1) If S 6= ∅, let l = min{i | i ∈ S} and p = 2rl+1 + 2rl+2 + · · · + 2rs .
(i) If p > 9, then k ≥ 4. Take xn = [RP (2p + 1, 1, n − 2k − p − 8; 2k −
2k +8
p + 7)]. From [22; Lemma 2.3], xn ∈ Jn,k
.
Ã
!
n−1
+
+
+ 2k − p + 7 − 3
n − 2k − 2p − 8
Ã
!
2k+1 + 2r1 + · · · + 2rl + 2rl+1 · · · + 2rs − 1
≡
+
2rl+1 +1 + 2rl+2 +1 + · · · + 2rs +1 + 1
Ã
!
2k+1 + 2r1 + · · · + 2rl + 2rl+1 · · · + 2rs − 1
+1
2k + 2r1 + · · · + 2rl − 1 − 7
n−1
2p + 1
!
Ã
n−1
1
!
Ã
≡ 1 mod 2.
By Lemma 2.1, xn is indecomposable.
(ii) If 2 ≤ p ≤ 8, the argument is divided into the following cases:
(a) p = 8 or p = 4.
Take xn = [RP (19, n − 2k − 17; 2k − 1)]. From [15; Lemma 2.4], xn ∈
Jn,k . If p = 8, then rl ≥ 5.
2k +8
Ã
Ã
!
n−1
19
Ã
+
!
n−1
n − 2k − 17
+ 2k − 1 − 2
!
2k+1 + 2r1 + · · · + 2rl + 8 − 1
≡
24 + 2 + 1
!
Ã
2k+1 + 2r1 + · · · + 2rl + 8 − 1
+1
+
2k + 2r1 + · · · + 2rl − 8 − 1
!
Ã
2k+1 + 2r1 + · · · + 2rl + 8 − 1
+1
≡ 0+
2k + 2r1 + · · · + 2rl −1 + · · · + 24 + 22 + 1
≡ 0 + 0 + 1 ≡ 1 mod 2.
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JINGYAN LI AND YANYING WANG
If p = 4, then rl ≥ 4.
Ã
! Ã
!
n−1
n−1
+
+ 2k − 1 − 2
19
n − 2k − 17
Ã
!
2k+1 + 2r1 + · · · + 2rl + 22 − 1
≡
24 + 2 + 1
Ã
!
2k+1 + 2r1 + · · · + 2rl + 22 − 1
+
+ 1. (∗)
2k + 2r1 + · · · + 2rl − 1 − 8 − 4
For rl > 4
Ã
(∗) ≡ 0 +
2k+1 + 2r1 + · · · + 2rl + 2 + 1
2k + 2r1 + · · · + 2rl+1 + 2rl −1 + · · · + 2 + 1 − 8 − 4
!
+1
≡ 0 + 0 + 1 ≡ 1 mod 2.
For rl = 4
Ã
(∗) ≡ 1 +
2k+1 + 2r1 + · · · + 24 + 2 + 1
2k + 2r1 + · · · + 2rl+1 + 2 + 1
!
+1
≡ 1 + 1 + 1 ≡ 1 mod 2.
From above, we know that xn is indecomposable.
(b) p = 6.
If p = 6, then rl ≥ 4. For rl > 4, we take xn = [RP (19, n−2k −17; 2k −1)].
2k +8
From [15; Lemma 2.4], xn ∈ Jn,k
.
Ã
Ã
n−1
19
!
Ã
+
n−1
n − 2k − 17
!
+ 2k − 1 − 2
!
2k+1 + 2r1 + · · · + 2rl + 4 + 1
≡
24 + 2 + 1
!
Ã
2k+1 + 2r1 + · · · + 2rl + 4 + 1
+1
+
2k + 2r1 + · · · + 2rl − 1 − 8 − 2
≡ 0 + 0 + 1 ≡ 1 mod 2.
By Lemma 2.1, xn is indecomposable.
(Z2 )k -ACTIONS WITH FIXED POINT SET
617
For rl = 4, we take xn = [RP (2k+1 + 1, 2, 2, 2, 2, n − 2k+1 − 16, 0, 0)].
2k +8
From [15; Lemma 2.4], xn ∈ Jn,k
.
Ã
!
2k+1 + 2r1 + · · · + 24 + 22 + 2 − 1
2k+1 + 1
Ã
!
2k+1 + 2r1 + · · · + 24 + 22 + 2 − 1
+
2r1 + · · · + 24 + 22 + 2 − 24
≡ 1 + 0 ≡ 1 mod 2
By Lemma 2.1, xn is indecomposable.
(c) p = 2.
If p = 2, then rl ≥ 3.
For rl > 4, we take xn = [RP (19, n − 2k − 17; 2k − 1)]. From [15; Lemma
2k +8
2.4], xn ∈ Jn,k
.
Ã
!
2k+1 + 2r1 + · · · + 2rl + 2 − 1
24 + 2 + 1
Ã
!
2k+1 + 2r1 + · · · + 2rl + 2 − 1
+
+1
2k + 2r1 + · · · + 2rl − 1 − 8 − 4 − 2
≡ 0 + 0 + 1 ≡ 1 mod 2.
By Lemma 2.1, xn is indecomposable.
For rl = 4 or 3, we take xn = [RP (2k+1 + 1, 2, 2, 2, 2, n − 2k+1 − 16, 0, 0)].
k
2 +8
From [15; Lemma 2.4], xn ∈ Jn,k
.
Ã
2k+1 + 2r1 + · · · + 2rl + 2 − 1
2k+1 + 1
!
Ã
+
2k+1 + 2r1 + · · · + 2rl + 2 − 1
2r1 + · · · + 2rl + 2 − 24
≡ 1 + 0 ≡ 1 mod 2.
By Lemma 2.1, xn is indecomposable.
(2) If S = ∅, i.e. n = 2k+1 + 2r1 + 2r2 + · · · + 2rs , k = r1 > r2 > · · · >
rs ≥ 1, ri = ri+1 + 1 and s ≥ 2.
!
618
JINGYAN LI AND YANYING WANG
2rs +1 − 8 2rs +1 − 8
,
, n − 2k+1 −
2
2
2k +8
2rs +1 ); 8]. From [15; Lemma 2.4], xn ∈ Jn,k
.

 Ã
Ã
!
!
n
−
1
n−1
n
−
1
+ 2  2rs +1 − 8  +
+8−4
2k+1 + 1
n − 2k+1 − 2rs +1 )
2
Ã
! Ã
!
k+1
r
r
1
2
2
+ 2 + 2 + · · · + 2rs − 1
2k+1 + 2r1 + 2r2 + · · · + 2rs − 1
≡
+
2k+1 + 1
2r1 + 2r2 + · · · + 2rs − 2rs +1
Ã
!
2k+1 + 2r1 + 2r2 + · · · + 2rs−1 + 2rs −1 + · · · + 2 + 1
≡ 1+
2r1 + 2r2 + · · · + 2rs−2 + 2rs
For rs ≥ 2, we take xn = [RP (2k+1 + 1,
≡ 1 + 0 ≡ 1 mod 2.
By Lemma 2.1, xn is indecomposable.
For rs = 1, we take xn = [RP (2k+1 + 1, n − 2k+1 − 8; 8)]. From [15;
2k +8
Lemma 2.4], xn ∈ Jn,k
.
Ã
! Ã
!
2k+1 + 2r1 + 2r2 + · · · + 2 − 1
2k+1 + 2r1 + 2r2 + · · · + 2 − 1
+
2k+1 + 1
2r1 + 2r2 + · · · + 2 − 8
≡ 1 + 0 ≡ 1 mod 2.
By Lemma 2.1, xn is indecomposable.
¤
Proposition 3.4 — If n ≥ 2k + 9 and n 6= 2u − 1, then there exist inde2k +8
composable classes xn ∈ Jn,k
for k ≥ 4.
Proof : Take xn as in Lemma 3.1 for n odd, xn as in Lemma 3.2 and
Lemma 3.3 for 2k + 10 ≤ n < 2k+2 even, and xn as in [15; Lemma 3.1] for
2k +8
n ≥ 2k+2 even. Then, every xn is indecomposable and xn ∈ Jn,k
.
¤
To complete the proof of the main theorem, we need the following two
lemmas.
Lemma 3.5 — For k ≥ 4 and n = 2k + 7, there exist indecomposable
2k +6
classes xn ∈ Jn,k
.
Proof : From [11; 5.1], there exists an indecomposable class X ∈
k−1
J 2n−1 ,k+3 (for 2k−1 + 3 < 2k ), so that X has a representative M admitting a
2
(Z2 )k -action with fixed point set F 0 , where dim(F 0 ) =
n−1
2
− (2k−1 + 3) = 0.
(Z2 )k -ACTIONS WITH FIXED POINT SET
619
Take xn = [P (1, M )] with the (Z2 )k -action as in Lemma 3.1. The fixed point
set of this action is F = S 1 ×F 0 ×F 0 /T = P (1, F 0 ) with dimF = 1 = n−2k −6,
2k +6
then xn ∈ J∗,k
. By Lemma 2.2, xn is indecomposable.
¤
Lemma 3.6 — For k ≥ 4 and n = 2k + 8, there exist indecomposable
2k +6
classes xn ∈ Jn,k
.
Proof : Take xn = [RP (11; 2k −6)]. Just as in Lemma 3.2, there exsits a
(Z2 )2 -action on RP (11) with fixed point set being four copies of RP (2). Let
2k +6
(Z2 )0 act as the identity on the rest of the base. By Lemma 2.3, xn ∈ Jn,k
.
Ã
2k + 8 − 1
8+2+1
!
+ 2k − 6 − 1 ≡ 1 mod 2.
By Lemma 2.1, xn is indecomposable.
¤
4. Proof of the Main Theorem
Proof : We choose a system of generators xn as follows:
(a) Let x2 = [RP (2)]. Noticing that χ(x2 ) = 1, by [11; 5.1], we have
2 .
x2 ∈ J∗,k
(b) For 3 ≤ n ≤ 2k + 6 and n 6= 2u − 1, we can choose indecomposable
n/2
classes xn such that χ(xn ) = 0 (otherwise, replacing xn by xn + x2 ).
(c) Take xn as in Lemma 3.5 for n = 2k + 7 and xn as in Lemma 3.6 for
2k +8
n = 2k + 8. By Lemma 2.4, x2k +8 ∈
/ J∗,k
.
(d) For n ≥ 2k + 9, let xn be as in Proposition 3.4.
k
2 +8
Since J∗,k
is an ideal of M O∗ , to complete the proof it is necessary
only to show that it contains the decomposable classes xi1 xi2 · · · xim with
m ≥ 2, 2 ≤ i1 ≤ i2 ≤ · · · ≤ im ≤ 2k + 8 and i1 + i2 + · · · + im ≥ 2k + 8.
The argument is divided into two cases: Case 1. i1 +i2 +· · ·+im > 2k +8.
Case 2. i1 + i2 + · · · + im = 2k + 8.
Case 1 — i1 + i2 + · · · + im > 2k + 8.
(1) The case im > 2k .
620
JINGYAN LI AND YANYING WANG
8 . From
(i) If i1 + i2 + · · · + im−1 ≥ 8, from [11; 5.1] xi1 xi2 · · · xim−1 ∈ J∗,k
k
k
k
2 +8
2 and x x · · · x
8
2
[12], xim ∈ J∗,k
i1 i2
im ∈ J∗,k J∗,k ⊂ J∗,k .
4
(ii) If i1 + i2 + · · · + im−1 = 7, then xi1 xi2 · · · xim−1 ∈ J∗,k
T
7 and
J∗,k
k
2 +4
im > 2k +1. If im > 2k +4, by [18; Theorem] xim ∈ J∗,k
and xi1 xi2 · · · xim ∈
k
k
4 J 2 +4 ⊂ J 2 +8 .
J∗,k
∗,k
∗,k
k
2 +1
.
If im ≤ 2k + 4, from (b), χ(xim ) = 0. By [14; Theorem 1], xim ∈ J∗,k
k
k
7 J 2 +1 ⊂ J 2 +8 .
So xi1 xi2 · · · xim ∈ J∗,k
∗,k
∗,k
k
2 +2
(iii) If i1 + i2 + · · · + im−1 = 6, then im > 2k + 2. From [16], xim ∈ J∗,k
.
k
k
2 +8
2 +2
6 . Then x x · · · x
6
⊂ J∗,k
.
By [11; 5.1], xi1 xi2 · · · xim−1 ∈ J∗,k
i1 i2
im ∈ J∗,k J∗,k
T 5
4
(iv) If i1 + i2 + · · · + im−1 = 5, then xi1 xi2 · · · xim−1 ∈ J∗,k
J∗,k and
k
2 +4
im > 2k +3. If im > 2k +4, by [18; Theorem], xim ∈ J∗,k
and xi1 xi2 · · · xim ∈
k
k
4 J 2 +4 ⊂ J 2 +8 .
J∗,k
∗,k
∗,k
If im = 2k + 4, from (b), χ(xim ) = 0.
k
k
2 +3
5 J 2 +3 ⊂
By [15; Proposition 4.3], xim ∈ J∗,k
. Then xi1 xi2 · · · xim ∈ J∗,k
∗,k
k
2 +8
J∗,k
.
(v) If i1 +i2 +· · ·+im−1 = 4, then im > 2k +4. From [18; Theorem], xim ∈
2k +4
4 . Then x x · · · x
4
By [11; 5.1], xi1 xi2 · · · xim−1 ∈ J∗,k
⊂
i1 i2
im ∈ J∗,k J∗,k
2k +4
J∗,k
.
2k +8
J∗,k .
2 and
(vi) If i1 + i2 + · · · + im−1 = 2, then m = 2, i1 = 2, x2 ∈ J∗,k
k
k
k
2 +6
2 J 2 +6 ⊂ J 2 +8 .
2k +6 < i2 ≤ 2k +8. From (c), xi2 ∈ J∗,k
. Then x2 xi2 ∈ J∗,k
∗,k
∗,k
(2) The case im ≤ 2k and there exists some l(1 ≤ l ≤ m) such that
8 ≤ il < 2k .
il
From [11; 5.1], xil ∈ J∗,k
. Since i1 + i2 + · · · + im > 2k + 8, i1 +
i2 + · · · + il−1 + il+1 + · · · + im > 2k + 8 − il . Because 2k + 8 − il ≤ 2k ,
2k +8−il
from [11; 5.1] and [12], we have xi1 xi2 · · · xil−1 xil+1 · · · xim ∈ J∗,k
. Then
k
k
2 +8−il il
2 +8
xi1 xi2 · · · xim ∈ J∗,k
J∗,k ⊂ J∗,k
.
(3) The case im ≤ 2k , with no l(1 ≤ l ≤ m) satisfying 8 ≤ il < 2k .
In this case, ij = 2k , or ij ≤ 6(1 ≤ j ≤ m).
(Z2 )k -ACTIONS WITH FIXED POINT SET
621
2k −2 T 2k −1
(i) If im = 2k , then i1 + i2 + · · · + im−1 ≥ 9 and xim ∈ J∗,k
J∗,k .
If there are odd numbers in i1 , · · · , im−1 , then χ(xi1 xi2 · · · xim−1 ) = 0 and
2k −l
2k +8
9 . So x x · · · x
9
xi1 xi2 · · · xim−1 ∈ J∗,k
i1 i2
im ∈ J∗,k J∗,k ⊂ J∗,k . If i1 , · · · , im−1
10 . Then
are all even, i1 + i2 + · · · + im−1 ≥ 10 and xi1 xi2 · · · xim−1 ∈ J∗,k
k
k
10 J 2 −2 ⊂ J 2 +8 .
xi1 xi2 · · · xim ∈ J∗,k
∗,k
∗,k
im
(ii) If im ≤ 6 and there exists some t such that it 6= 2, then xim ∈ J∗,k
and i1 +i2 +· · ·+im−1 > 2k +8−im . From [15],16] and [18], xi1 xi2 · · · xim−1 ∈
2k +8−im
2k +8−im im
2k +8
J∗,k
. Then xi1 xi2 · · · xim ∈ J∗,k
J∗,k ⊂ J∗,k
.
(iii) If i1 = i2 = · · · = im = 2, then 2(m − 4) > 2k . From [11; 5.1] and
4 m−4 ∈ J 8 J 2k ⊂ J 2k +8 .
[12], xm
2 = x2 x2
∗,k ∗,k
∗,k
Case 2 — i1 + i2 + · · · + im = 2k + 8.
ir
If im < 2k , from [11; 5.1], xir ∈ J∗,k
(r = 1, 2, · · · , m) and
k
i1 i2
im
2 +8
xi1 xi2 · · · xim ∈ J∗,k
J∗,k · · · J∗,k
⊂ J∗,k
.
If im ≥ 2k , then s(i1 ,i2 ,··· ,im ) [xi1 xi2 · · · xim ] 6= 0. By Lemma 2.5, the decomposable classes x2 x2k +6 , x4 x2k +4 , x2 2 x2k +4 , x5 x2k +3 , x6 x2k +2 , x2 x4 x2k +2 ,
2k +8
x2 3 x2k +2 , x2 x5 x2k +1 , x8 x2k , x2 x6 x2k , x4 2 x2k , x2 2 x4 x2k , x2 4 x2k are not in J∗,k
.
For some linear combination of the above classes such as
xi1,1 xi2,1 · · · xim1 ,1 + xi1,2 xi2,2 · · · xim2 ,2 + · · · + xi1,j xi2,j · · · ximj ,j ,
we can find
(i1 , i2 , · · · , inj ) ∈ {(i1,1 , i2,1 · · · im1 ,1 ), (i1,2 , i2,2 · · · im2 ,2 ), · · · , (i1,j , i2,j , · · · imj ,j )}
such that (i1 , i2 , · · · , inj ) is not the refinement of other elements in
{(i1,1 , i2,1 · · · im1 ,1 ), (i1,2 , i2,2 · · · im2 ,2 ), · · · (i1,j , i2,j , · · · imj ,j )}.
Then, from [23],
s(i1 ,i2 ,··· ,inj ) [xi1,1 xi2,1 · · · xim1 ,1 +xi1,2 xi2,2 · · · xim2 ,2 +· · ·+xi1,j xi2,j · · · ximj ,j ] 6= 0.
By Lemma 2.5, any linear combination of the above classes are not in
2k +8
J∗,k
.
¤
622
JINGYAN LI AND YANYING WANG
Acknowledgement
We would like to thank the referee for a careful reading of the paper and
many helpful suggestions.
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