AN IMPROVEMENT OF THE FIVE HALVES THEOREM OF
J. BOARDMAN
PEDRO L. Q. PERGHER
Abstract. Let (M m , T ) be a smooth involution on a closed smooth mn
dimensional manifold and F = j=0 F j (n ≤ m) its fixed point set, where
F j denotes the union of those components of F having dimension j. The famous Five Halves Theorem of J. Boardman, announced in 1967, establishes
that, if F is nonbounding, then m ≤ 52 n. In this paper we obtain an improvement of the Five Halves Theorem when the top dimensional component
of F , F n , is nonbounding. Specifically, let ω = (i1 , i2 , ..., ir ) be a non-dyadic
partition of n and sω (x1 , x2 , ..., xn ) the smallest symmetric polynomial over
Z2 on degree one variables x1 , x2 , ..., xn containing the monomial xi11 xi22 ...xirr .
Write sω (F n ) ∈ H n (F n , Z2 ) for the usual cohomology class corresponding to
sω (x1 , x2 , ..., xn ), and denote by ℓ(F n ) the minimum length of a non-dyadic
partition ω with sω (F n ) = 0 (here, the length of ω = (i1 , i2 , ..., ir ) is r). We
will prove that, if (M m , T ) is an involution for which the top dimensional
component of the fixed point set, F n , is nonbounding, then m ≤ 2n + ℓ(F n );
roughly speaking, the bound for m depends on the degree of decomposability
of the top dimensional component of the fixed point set. Further, we will
give examples to show that this bound is best possible.
1. Introduction
Let (M m , T ) be a smooth involution on a closed smooth m-dimensional mann
ifold and F =
F j (n ≤ m) its fixed point set, where F j denotes the union
j=0
of those components of F having dimension j. If F is nonbounding (which
means that at least one F j is nonbounding) then n cannot be too small with
respect to m: this intriguing fact was firstly evidenced from Theorem 27.1 of
the old book (1964) [5] of P. E. Conner and E. E. Floyd, which stated: for
each natural number n, there exists a number ϕ(n) with the property that, if
1991 Mathematics Subject Classification. (2.000 Revision) Primary 57R85; Secondary
57R75.
Key words and phrases. involution, Five Halves Theorem, projective space bundle, indecomposable manifold, splitting principle, Stiefel-Whitney class, characteristic number .
The author was partially supported by CNPq and FAPESP.
1
(M m , T ) is an involution fixing F =
n
F j and if m > ϕ(n), then (M m , T )
j=0
bounds equivariantly.
Later (1967), in [4], J. Boardman announced the best result in this direction
in terms of generality, explicitly confirming the previous result of Conner and
Floyd:
Five Halves Theorem of J. Boardman : If (M m , T ) is an involution for which
5
F is nonbounding, then m ≤ n (in fact, the first version of the Five Halves
2
Theorem was proved under the hypothesis that M m is nonbounding; the result
in question is a consequence of the following strengthened version of this first
version, obtained by C. Kosniowski and R. E. Stong in [8]: if (M m , T ) is a
5
nonbounding involution, then m ≤ n). Further, this bound is best possible
2
(more detailed proofs for the Five Halves Theorem can be found in [3] and [9]).
The aim of this paper is to obtain improvements of the Five Halves Theorem
when the top dimensional component of the fixed point set, F n , is nonbounding. To precisely state it, first recall from [19] that one has an algebraic scheme
to determine the cobordism class of F n , given by the set of Stiefel-Whitney
numbers of F n : write W (F n ) = 1 + w1 + ... + wn for the total Stiefel-Whitney
class of F n . A general Stiefel-Whitney number of F n is a modulo 2 number obtained by evaluating an n-dimensional Z2 -cohomology class of the form
wi1 wi2 ...wir ∈ H n (F n , Z2 ) (that is, with i1 + i2 + ... + ir = n) on the fundamental homology class [F n ] ∈ Hn (F n , Z2 ). There is an useful numerical
alternative to analyze the cobordism class of F n : from a given homogeneous
symmetric polynomial over Z2 of degree n on degree one variables x1 , x2 , ..., xn ,
P (x1 , x2 , ..., xn ), we get a cohomology class in H n(F n , Z2 ) by identifying each
wi to the ith elementary symmetric function in the variables x1 , x2 , ..., xn , and
next by expressing P (x1 , x2 , ..., xn ) as an n-dimensional polynomial in the wi′ s .
For a partition ω = (i1 , i2 , ..., ir ) of n, 1 ≤ r ≤ n, let sω (x1 , x2 , ..., xn ) be
the smallest symmetric polynomial containing the monomial xi11 xi22 ...xirr , and
denote by sω (F n ) ∈ H n (F n , Z2 ) the cohomology class which corresponds to
2
sω (x1 , x2 , ..., xn ) through the previous procedure. Since every symmetric polynomial P (x1 , x2 , ..., xn ) is a sum of such polynomials, the cobordism class of
F n is determined by the set of numbers of the form sω (F n)[F n ]. In fact, we
do not need all of these numbers: we say that a partition ω = (i1 , i2 , ..., ir ) is
non-dyadic if none of the it is of the form 2p − 1. From [8, Section 5], one has
the following
Fact. The cobordism class of F n is determined by the set of numbers of the
form sω (F n )[F n ], with ω non-dyadic.
If F n is nonbounding, denote by ℓ(F n ) the minimum length of a non-dyadic
partition ω with sω (F n )[F n ] = 0 (here, the length of ω = (i1 , i2 , ..., ir ) is r). We
will prove the following
m
Theorem. Let (M , T ) be an involution with fixed point set F =
n
F j,
j=0
and suppose that the top dimensional component F
n
is nonbounding. Then
n
m ≤ 2n + ℓ(F ). Further, this bound is best possible.
Roughly speaking, this result says that the bound for m depends on the
degree of decomposability of the top dimensional component of the fixed point
set. For example, if F n is indecomposable , which means that its cobordism
class cannot be expressed as a sum of products of lower dimensional cobordism
classes, then ℓ(F n ) = 1, and thus the above result complements and generalizes
the results of the recent paper [12]. We remark that the precise statement of
n
the Five Halves Theorem is: if (M m , T ) fixes the nonbounding F =
F j,
j=0
then: i) if n = 2k with k ≥ 1, m ≤ 5k, and ii) if n = 2k + 1 with k ≥ 0,
m ≤ 5k + 2. Among all non-dyadic partitions of n, the maximum length occurs
for the partition ω = (2, 2, ..., 2) if n = 2k, and ω = (2, 2, ..., 2, 5) if n = 2k + 1;
that is, 1 ≤ ℓ(F 2k ) ≤ k and 1 ≤ ℓ(F 2k+1 ) ≤ k − 1. Therefore, generally
speaking, our theorem gives the same bound of the Five Halves Theorem for n
even; however, for n odd one has the following
n
m
Corollary. If (M , T ) fixes F =
F j , where F n is nonbounding and
j=0
n = 2k + 1, then m ≤ 5k + 1.
3
In special cases, for n = 2k and ℓ(F n ) < k, and for n = 2k + 1 and ℓ(F n ) <
k−1, our theorem gives best possible improvements of the Five Halves Theorem.
Remark. The generality of the Five Halves Theorem, which is valid for every
n ≥ 1 and allows the possibility that fixed components of all dimensions 0 ≤
j ≤ n occur, suggests the question of finding better bounds for m (and ideally
the best possible bound) when we omit some components of F and restrict
the set of involved dimensions n. In this direction, C. Kosniowski and R. E.
Stong showed in [8] that if (M m , T ) is an involution for which F is nonbounding
and has constant dimension = n, then m ≤ 2n. This bound is best possible,
as can be seen by taking (F n × F n , twist), where F n is any nonbounding ndimensional manifold (with the exception of n = 1 and n = 3). In [18], D.
C. Royster showed that if (M m , T ) is an involution for which F has the form
F = F n ∪{point}, where n is odd, then m ≤ n+1. This bound is realized by the
involution (RP n+1 , T ), where RP n+1 is the (n + 1)-dimensional real projective
space, T [x0 , x1 , ..., xn+1 ] = [−x0 , x1 , ..., xn+1 ] and n is odd. Extending the two
components direction started with this result of Royster, recently some advances
have been obtained in the case in which F has the form F = F n ∪ F j , with
n > j. Specifically, we find best possible bounds for j = 0 in [10] and [17], j = 1
in [6] and [7], j = 2 in [13], [14] and [15], and j = n − 1 in [16]. In [6] and [7],
the case F = F n ∪ RP j , n = j, was also considered; more generally, in [11] we
find results on the case F = F n ∪ F j with F j indecomposable.
Remark. In all the above discussion, each j-dimensional part of the fixed
point set of an involution can be assumed to be connected, since any involution
is equivariantly cobordant to an involution with this property.
2. Proof of the result
m
Let (M , T ) be an involution with fixed point set F =
n
F j , where the top
j=0
dimensional component F n is nonbounding. Write ℓ(F n ) = r, and take a nondyadic partition ω = (i1 , i2 , ..., ir ) of n for which sω (F n )[F n ] = 0. As announced
in Section 1, our aim is to show that m ≤ 2n + r.
4
For 0 ≤ j ≤ n, denote by ηj → F j the normal bundle of F j in M m , with
dim(ηj ) = m − j, and by λj → RP (ηj ) the line bundle over the projective
space bundle RP (ηj ), associated to the double cover S(ηj ) → RP (ηj ), S(ηj )
the sphere bundle.
In general, for a given vector bundle η → F , write
W (η) = 1 + w1 (η) + w2 (η) + ...
for the total Stiefel-Whitney class of η; in particular, if F is a manifold,
W (F ) = 1 + w1 (F ) + w2 (F ) + ...
means the Stiefel-Whitney class of the tangent bundle of F .
Each Fj has an invariant tubular neighborhood Nj , the disk bundle of ηj .
The boundary sphere bundle ∂(Nj ) = S(ηj ) is a manifold with free involution,
and the union of the S(ηj ) bounds the closure of the complement in M m of
the union of all the Nj , which is a manifold with free involution. In other
n
words,
S(ηj ), T bounds as an element of the cobordism group Nm−1 (BZ2 )
j=0
n
of manifolds with free involution; in this way,
λj → RP (ηj ) bounds as
j=0
a line bundle. From [5], this can be expressed by the following algebraic fact:
let P (w1 , w2 , ..., wm−1 , c) be any homogeneous polynomial over Z2 with degree
m − 1, where each variable wi has degree i and the variable c has degree 1. For
each 0 ≤ j ≤ n, we can evaluate the cohomology class
P w1 (RP (ηj )), w2 (RP (ηj )), ..., wm−1 (RP (ηj )), w1 (λj ) ∈ H m−1 RP (ηj ), Z2
on the fundamental homology class RP (ηj ) ∈ Hm−1 RP (ηj ), Z2 , thus getting
a modulo 2 number (called a characteristic number ),
P w1 (RP (ηj )), w2 (RP (ηj )), ..., wm−1 (RP (ηj )), w1 (λj ) RP (ηj ) ∈ Z2 .
Then
n
P w1 (RP (ηj )), w2 (RP (ηj )), ..., wm−1 (RP (ηj )), w1 (λj ) RP (ηj ) = 0. (∗∗∗)
j=0
5
The key point will consist in applying the above fact choosing a suitable
polynomial. To do this, we introduce a special cohomology class associated to
line bundles λ → W , where W is a closed (m − 1)-dimensional manifold; we will
use the non-dyadic partition ω = (i1 , i2 , ..., ir ) of n, mentioned in the beginning
of the section.
Let Fω (z1 , z2 , ..., zm ) be the smallest polynomial on degree one variables z1 , z2 ,
..., zm , which is symmetric in the variables z1 , z2 , ..., zm−1 and contains the polynomial
z1i1 (zm + z1 )i1 +1 z2i2 (zm + z2 )i2 +1 ...zrir (zm + zr )ir +1 .
As in Section 1, we then identify w1 (λ) to zm and each wi (M ) to the ith
elementary symmetric function in the variables z1 , z2 , ..., zm−1 ; next, we express
Fω (z1 , z2 , ..., zm ) as a polynomial of dimension 2n + r in the wi′ s (M ) and w1 (λ).
This class will be denoted by Fω (λ).
The crucial point is that Fω (λ) has a nice behavior with respect to the standard line bundles over projective space bundles. To see this, we will use the
splitting principle, which allows to write the Stiefel-Whitney class of any kdimensional vector bundle formally as
1 + w1 + w2 + ... + wk = (1 + y1 )(1 + y2 )...(1 + yk ),
where each yi has degree one, and effectively to see each wi as the ith elementary
symmetric function in the variables y1 , y2 , ..., yk .
Let η be a k-dimensional vector bundle over a closed j-dimensional manifold
F , where j + k = m, and let λ → RP (η) be the standard line bundle; set
w1 (λ) = c. From [2; page 517] one has that
W RP (η) = W (F ).W (η ⊗ λ) = 1 + w1 (F ) + w2 (F ) + ... + wj (F ) (1 + c)k +
(1 + c)k−1 w1 (η) + ... + (1 + c)wk−1 (η) + wk (η) ,
where we are suppressing bundle maps. Using the splitting principle, write
W (F ) = (1 + x1 )(1 + x2 )...(1 + xj )
and
W (η) = (1 + y1 )(1 + y2 )...(1 + yk ).
6
Then
W RP (η) = (1 + x1 )(1 + x2 )...(1 + xj )(1 + c + y1 )(1 + c + y2 )...(1 + c + yk ).
To calculate Fω (λ), select ze1 , ze2 ,...,zer from z1 = x1 ,...,zj = xj , zj+1 = c +
y1 ,...,zj+k = c + yk ; we must analyse the corresponding term
zei11 (c + ze1 )i1 +1 zei22 (c + ze2 )i2 +1 ...zeirr (c + zer )ir +1
of Fω (λ). The product
zeihh (c + zeh )ih +1
contains a factor xith if zeh = xt , or ysih +1 if zeh = ys . Then every term of Fω (λ)
contains at least n factors xt or ys from H 1 (F ), which implies that Fω (λ) = 0 if
j < n. If j = n, the only terms that survive are those where zeh is taken from
{x1 , ..., xn }, and so Fω (λ) = sω (F )cn+r .
Returning to the line bundles λj → RP (ηj ) coming from the fixed data of
(M m , T ), set w1 (λj ) = cj . By contradiction, suppose m > 2n+r. Then m−1 ≥
of
2n + r, and thus it makes sense to consider the polynomial Fω (λj ).cm−1−2n−r
j
degree m − 1, which is zero if j < n. Thus equation (***) gives that
n
Fω (λj ).cm−1−2n−r
RP (ηj ) = sω (F n ).cm−1−n
RP (ηn ) = 0.
n
j
j=0
From the Leray-Hirsch Theorem (see [1]; pag. 129) H ∗ RP (ηn ), Z2 is the free
H ∗ (F n , Z2 )-module on 1, cn , c2n , ..., cm−1−n
. Then
n
sω (F n ).cm−1−n
RP (ηn ) = sω (F n )[F n].
n
Since sω (F n ) = 0, this gives the desired contradiction.
3. Maximal examples
In this section we will construct involutions to show that the result of Section
2 is best possible. As mentioned in Section 1, our result complements and generalizes the results of the recent paper [12], where it was shown that, if (M m , T )
is an involution for which the top dimensional component of the fixed point set,
F n , is indecomposable, then m ≤ 2n + 1. In fact, in [19, page 79] R. Thom
showed that F n is indecomposable if and only if sn (F n ) = sω (F n ) is nonzero,
7
where ω is the trivial partition, ω = (n). In this setting, in [12] we constructed,
for each n ≥ 2 not of the form 2t − 1, a special involution (W 2n+1 , Tn ) so that
the dimension of the top dimensional component of the fixed point set is n and
with this top dimensional component being indecomposable, thus showing that
the bound of [12] is best possible (we recall that indecomposable n-dimensional
manifolds occur only for these values of n). Here, our examples will be simply products of the examples of [12]. In fact, first suppose that F1 , F2 , .., Fr
are indecomposable manifolds, with dim(Fi ) = ni for i = 1, 2, ..., r. Then, by
using the splitting principle and dimensional considerations, we can see that
sω (F1 × F2 × ... × Fr ) = sn1 (F1 ).sn2 (F2 )...snr (Fr ) = 0, where ω = (n1 , n2 , ..., nr ).
Further, if ω = (j1 , j2 , ..., jt ) is a non-dyadic partition of n1 + n2 + ... + nr with
t < r, then sω (F1 × F2 × ... × Fr ) = 0. This means that ℓ(F1 × F2 × ... × Fr ) = r.
Now take ω = (i1 , i2 , ..., ir ) a non-dyadic partition of n. Since each ij is not of
the form 2t − 1, one has the example (W 2ij +1 , Tij ), with the top dimensional
component of the fixed point set, F ij , being indecomposable. The product
involution
(W 2i1 +1 , Ti1 ) × (W 2i2 +1 , Ti2 ) × ... × (W 2ir +1 , Tir )
has the n-dimensional manifold F i1 × F i2 × ... × F ir as the top dimensional
component of its fixed point set. Since ℓ(F i1 ×F i2 ×...×F ir ) = r and (2i1 +1)+
(2i2 +1)+...+(2ir + 1) = 2n+r, (W 2i1 +1 , Ti1 )×(W 2i2 +1 , Ti2 ) ×...×(W 2ir +1 , Tir )
is the desired maximal example.
Acknowledgement. I am very grateful to the referee for suggestions that
helped to clarify considerably the original version.
References
[1]
[2]
[3]
[4]
Borel, A.: Sur la cohomologie des espaces fibrés principaux et des espaces homogènes des
groupes de Lie compacts, Ann. of Math. 57, n. 2, (1953), 115-207
Borel, A. and Hirzebruch, F.: Characteristic classes and homogeneous spaces, I, Amer.
J. Math. 80, (1958), 458-538.
Boardman, J. M.: Cobordism of involutions revisited, Proceedings of the Second Conference on Compact Transformation Groups, Univ. Massachusetts, Amherst, Mass., Part I,
(1971), 131-151. Lecture Notes in Math., Vol. 298, Springer, Berlin (1972).
Boardman, J. M.: On manifolds with involution, Bulletin Amer. Math. Soc. 73, (1967),
136-138.
8
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
Conner, P. E. and Floyd, E. E. Differentiable periodic maps, Springer-Verlag, Berlin,
(1964).
Kelton, S. M.: Involutions fixing IRPj ∪ F n , Topology Appl. 142, (2004), 197-203.
Kelton, S. M.: Involutions fixing IRPj ∪ F n , II., Topology Appl. 149, n. 1-3, (2005),
217-226.
Kosniowski, C. and Stong, R. E.: Involutions and characteristic numbers, Topology 17,
(1978), 309-330.
Morava, J.: Cobordism of involutions revisited, revisited. Homotopy invariant algebraic
structures, Baltimore, MD, (1998), 15-18. Contemp. Math., 239, Amer. Math. Soc.,
Providence, RI, (1999).
Pergher, P. L. Q.: Bounds on the dimension of manifolds with certain Z2 fixed sets, Mat.
Contemp. 13, (1996), 269-275.
Pergher, P. L. Q.: Involutions fixing F n ∪ {Indecomposable} , Canadian Math. Bulletin
, to appear.
Pergher, P. L. Q.: Involutions whose top dimensional component of the fixed point set is
indecomposable, Geom. Dedicata , (2009), DOI 10.1007/s10711-009-9419-5.
Pergher, P. L. Q. and Figueira, F. G.: Dimensions of fixed point sets of involutions ,
Arch. Math. (Basel) 87, (2006), n. 3, 280-288.
Pergher, P. L. Q. and Figueira, F. G.: Bounds on the dimension of manifolds with
involution fixing F n ∪ F 2 , Glasg. Math. J. 50, n. 3, (2008), 595-604.
Pergher, P. L. Q. and Figueira, F. G.: Involutions fixing F n ∪ F 2 , Topology Appl. 153,
(2006), n. 14, 2499-2507.
Pergher, P. L. Q. and Figueira, F. G.: Two commuting involutions fixing F n ∪ F n−1 ,
Geom. Dedicata 117, (2006), 181-193.
Pergher, P. L. Q. and Stong, R. E.: Involutions fixing {point} ∪ F n , Transformation
Groups 6, (2001), 78-85.
Royster, D. C.: Involutions fixing the disjoint union of two projective spaces, Indiana
Univ. Math. J. 29, n. 2, (1980), 267-276.
Thom, R.: Quelques propriétés globales des variétés différentiables, Comment. Math.
Helv. 28, (1954), 17-86.
Pedro L. Q. Pergher
Departamento de Matemática
Universidade Federal de São Carlos
Caixa Postal 676
São Carlos, SP 13565-905, Brazil
E-mail: [email protected]
9