SYMPLECTIC REAL BOTT MANIFOLDSHIROAKI ISHIDA
Abstract. A real Bott manifold is the total space of an iterated RP1 -bundles
over a point, where each RP1 -bundle is the projectivization of a Whitney sumof two real line bundles. In this paper, we characterize real Bott manifolds
which admit a symplectic form. In particular, it turns out that a real Bottmanifold admits a symplectic form if and only if it is cohomologically symplectic. In this case, it admits even a Kähler structure. We also prove that any
symplectic cohomology class of a real Bott manifolds can be represented by asymplectic form. Finally, we study the ﬂux of a symplectic real Bott manifold.

1. Introduction
A real Bott tower (of height n) is a sequence of RP1 -bundles:
Mn → Mn−1 → · · · → M1 → M0 = {a point},
where each RP1 -bundle Mi → Mi−1 is the projectivization of a Whitney sum of
two real line bundles on Mi−1 . Each Mi is called a real Bott manifold. Clearly
M1 = RP1 and M2 = (RP1 )2 or a Klein bottle. If every bundle in the tower
is trivial, then Mn = (RP1 )n . However, there are many choices of non-trivial
bundles at each stage in the tower and it is known that there are many diﬀerent
diﬀeomorphism classes in real Bott manifolds ([5], [6]). A real Bott manifold is also
an example of a real toric manifold which admits a ﬂat Riemannian metric ([5]).
Although orientable ones occupy a small portion in all real Bott manifolds ([3]),the number of orientable ones of dimension n approaches inﬁnity as n approaches
inﬁnity. Among those orientable ones, some are symplectic, i.e., admit a symplectic
form. In this paper we give a complete characterization of symplectic real Bottmanifolds (Theorem 3.1). In particular, we prove that among real Bott manifolds
M the following are equivalent:(1) M is cohomologically symplectic,(2) M is symplectic,(3) M admits a Kähler structure.
We remark that the implication (3) ⇒ (2) ⇒ (1) always holds but the reverse
implications (1) ⇒ (2) and (2) ⇒ (3) do not hold in general as is well-known. Forexample, C P 2 # C P 2 is cohomologically symplectic but not symplectic because
it does not admit an almost complex structure and a certain T 2 -bundle over T 2
constructed in [8] is symplectic but does not admit a Kähler structure.
Date: January 19, 2010.
2000 Mathematics Subject Classiﬁcation. 57R17,57S25.
Key words and phrases. toric topology, symplectic topology, real Bott manifold.
12H. ISHIDAThis paper is organized as follows. In Section 2 we recall the quotient descriptionof real Bott manifolds. In Section 3 we state and prove our main theorem. In Section
4 we study the ﬂux group of a symplectic real Bott manifold.
Throughout this paper, all cohomology will be de Rham cohomology over R.
2. Quotient description of real Bott manifolds
In this section, we recall the quotient description of real Bott manifolds (see [5]and [6] for details) and observe the cohomology ring of a real Bott manifold.
Let B(n) be the the set of n × n upper triangular (0, 1) matrices with zerodiagonal entries. For a matrix A ∈ B(n), Aij denotes the (i, j) entry of A and Ai(respectively, Aj ) denotes the i-th row (respectively, j-th column) of A. Let S 1 bethe unit circle in C. For z ∈ S 1 and a ∈ Z /2 = {0, 1}, we set z(a) := a if a = 0and z̄ if a = 1. We then deﬁne the involution ai on T n := (S 1 )n by(2.1)

ai (z1 , . . . , zn ) := (z1 , . . . , zi−1 , −zi , zi+1 (Aii+1 ), . . . , zn (Ain ))for i = 1, . . . , n. Let G(A) denote the transformation group on T n generated by ai ’s.
Then the quotient space M (A) := T n /G(A) is known to be a real Bott manifold andevery real Bott manifold can be obtained as M (A) for some A ∈ B(n). Although
A is not necessarily uniquely determined by a real Bott manifold, A contains all
geometrical information on M (A). For example,n∑(2.2)
M (A) is orientable ⇐⇒Aij = 0 in Z /2 for any ij=1(see [5]).
It is also helpful to describe M (A) as the quotient of Rn by aﬃne transformations.
In fact, let Γ(A) denote the aﬃne transformation group on Rn generated by si ’s
deﬁned byii1(2.3)
si (u1 , . . . , un ) := (u1 , . . . , ui−1 , ui + , (−1)Ai+1 ui+1 , . . . , (−1)An un )2√
for i = 1, . . . , n. Then, an exponential map from R to S 1 sending u to exp(2π −1u)induces a diﬀeomorphism from Rn /Γ(A) onto T n /G(A) = M (A).
Let du1 , . . . , dun denote the standard 1-forms on Rn . Since each duj is invariant
under parallel translations on Rn , it descends to a closed 1-form on T n ∼= Rn / Zn ,∗nwhich we also denote by duj . The (de Rham) cohomology ring H (T ) of T n is
the exterior algebra in n variables [du1 ], . . . , [dun ] over R, where [duj ] denotes thecohomology class represented by the 1-form duj . It follows from (2.1) or (2.3) thatthe endomorphism a∗i of H∗ (T n ) induced by ai ∈ G(A) is given by{[duj ]if Aij = 0,∗(2.4)ai ([duj ]) =−[duj ] if Aij = 1.
We note that since M (A) = T n /G(A) and G(A) is a ﬁnite group, we have(2.5)

H∗ (M (A)) = H∗ (T n )G(A)(see [2, Theorem 2.4 in p.120] for example), where the right hand side denotes theG(A)-invariants in H∗ (T n ).
∏Lemma 2.1. Let J be a subset of {1, . . . , n}. Then j∈J [duj ] ∈ H∗ (T n ) is G(A)∑invariant if and only if j∈J Aj = 0 in Z /2.

SYMPLECTIC REAL BOTT MANIFOLDSProof. By (2.4), we havea∗i (∏[duj ]) = (−1)

j∈J

Pij∈J Aj∏

3[duj ],j∈J∏∑Thus, j∈J [duj ] is ﬁxed by a∗i if and only if j∈J Aij = 0 in Z /2. This implies
the lemma since G(A) is generated by ai ’s.
¤
3. Main theorem
The following is our main theorem in this paper.
Theorem 3.1. Let A ∈ B(2n). The following conditions are equivalent:(1) M (A) is cohomologically symplectic, that is, there exists an α ∈ H2 (M (A))
such that αn is nonzero.
(2) There`exist n subsets {j1 , jn+1 }, . . . , {jn , j2n } of {1, 2, . . . , 2n} such thatn• k {jk , jk+n } = {1, 2, . . . , 2n} and• Aj1 = Ajn+1 , . . . , Ajn = Aj2n .
(3) There exists a symplectic form on M (A).
(4) There exists a Kähler structure on M (A).
Moreover, any α ∈ H2 (M (A)) in (1) can be represented by a symplectic form on
M (A).
Proof. Because any closed symplectic manifold is cohomologically symplectic andany Kähler manifold is a symplectic manifold, it suﬃces to prove implications (1)⇒ (2) and (2) ⇒ (4).
Proof of (1) ⇒ (2). Assume that there exists a de Rham cohomology classα ∈ H2 (M (A)) such that αn 6= 0. We identify H∗ (M (A)) with H∗ (T n )G(A) by(2.5). Then it follows from Corollary 2.1 that we can write α uniquely as∑(3.1)
α=cj,k [duj ∧ duk ] with some cj,k ∈ R.
j<k,Aj =AkThus α 6= 0 implies the condition (2).
Proof of (2) ⇒ (4). Assume that A ∈ B(2n) satisﬁes the condition (2), namely
Ajk = Ajk+n for k = 1, . . . , n. Then we identify R2n with Cn by√zk := ujk + −1ujk+nnfor k = 1, . . . , n. Consider the standard Hermitian metric on Cn . Then, Γ(A) acts
on Cn as biholomorphisms and isometries. In fact, through the above identiﬁcation,
it follows from (2.3) that the action of si ∈ Γ(A) on Cn is given by1
if i = jk ,zk + √2z + −1 if i = jkk+n ,2si (z1 , . . . , zn )k =izk
if Ajk = Aijk+n = 0 and i 6= jk , jk+n ,−zkif Aijk = Aijk+n = 1,
where the left hand side denotes the k-th component of si (z1 , . . . , zn ). Thus thequotient M (A) = Cn /Γ(A) inherits the standard Kähler structure on Cn .
Finally, we shall prove the last statement in the theorem. As observed above,α ∈ H2 (M (A)) is of the form (3.1). We then deﬁne the diﬀerential closed 2-form ω4H. ISHIDAon R2n by(3.2)∑ω :=cj,k duj ∧ duk .

j<k,Aj =Ak

Comparing (3.1) with (3.2), one sees that the condition αn 6= 0 implies that ω nis nowhere zero. Thus ω is a symplectic form on R2n . Since ω is invariant underthe Γ(A)-action on R2n , ω descends to a symplectic form on the quotient M (A) =R2n /Γ(A) and this represents the given class α.
¤Example 3.2. Let A ∈ B(4). If A is the zero matrix, then M (A) is the 4-dimensional
torus and symplectic. Suppose that A is non-zero and M (A) is symplectic. Thenit follows from Theorem 3.1 (2) that A is one of the following:)) () () () ((0000100010000000,000010000000

10000000,

0000100010000000,

00001
100

1100,

000000
00

0
1000
100.

Real Bott manifolds M (A) for A above are diﬀeomorphic to each other but notdiﬀeomorphic to the 4-dimensional torus ([5], [6]). One sees that M (A) is the total
space of a non-trivial T 2 -bundle over T 2 . On the other hand, T 2 -bundles over T 2
which are symplectic are classiﬁed in [4]. One can easily check that our M (A) is oftype {−I, I, (0, 0)} in [4, Table 1] .
Finally we note that if()
A=000010001
1000
100,

then M (A) is orientable by (2.2), but not symplectic. Therefore the class of symplectic real Bott manifolds is strictly smaller than that of orientable real Bottmanifolds.
4. The flux group
In this section, we will study the ﬂux group of a symplectic real Bott manifold.
For that, we recall the deﬁnition of a ﬂux group for a general symplectic manifold.
Let (M, ω) be a closed symplectic manifold. A diﬀeomorphism φ : M → M iscalled a symplectomorphism if φ∗ ω = ω and the group of symplectomorphisms of(M, ω) is denoted by Symp(M, ω). Associated to a smooth function f : M → R,the Hamiltonian vector ﬁeld Xf is deﬁned by iXf ω = df . For a one-parameterfamily {ft }0≤t≤1 of functions, we obtain a one parameter family {Xft }0≤t≤1 ofHamiltonian vector ﬁelds, and integrating {Xft }, we obtain a one-parameter family{φt }0≤t≤1 of diﬀeomorphisms deﬁned bydφt = Xft ◦ φt and φ0 = id .
dt
The time-one map φ1 is a symplectomorphism and called a Hamiltonian diﬀeomorphism. It is known that all Hamiltonian diﬀeomorphisms of (M, ω) form a subgroup,denoted Ham(M, ω), of the identity component Symp0 (M, ω) of Symp(M, ω). Fora symplectic isotopy {φt }, that is, an isotopy through symplectomorphisms, weobtain a one-parameter family {Xt } of vector ﬁelds deﬁne bydφ t = Xt ◦ φ t .
dt

SYMPLECTIC REAL BOTT MANIFOLDS

5The ﬂux of {φt } is then deﬁned to be∫ 1[iXt ω]dt ∈ H1 (M ).
(4.1)
0

It is known that the ﬂux depends only on the homotopy class of symplectic isotopies
with ﬁxed end points φ0 = id and φ1 , so that it deﬁnes a homomorphism
Flux : Symp0 (M, ω) → H1 (M )/Γω ,where Γω is the image of the fundamental group π1 (Symp0 (M, ω)) by the ﬂux andcalled the ﬂux group of (M, ω). The solution of the ﬂux conjecture ([7]) says thatthe subgroup Γω of H1 (M ) is closed and discrete. According to [1], the kernel ofFlux is exactly equal to Ham(M, ω), in other words, we have an exact sequence
Flux{1} → Ham(M, ω) → Symp0 (M, ω) → H1 (M )/Γω .
Now, we consider the ﬂux of a symplectic real Bott manifold.
Theorem 4.1. Let M (A) be a real Bott manifold with a symplectic form ω givenby (3.2). Then, the ﬂux group Γω is a lattice group of H1 (M (A)) of full rank.
Proof. It follows from Lemma 2.1 that H1 (M (A)) is generated by [duj ] with Aj = 0,and since M (A) is symplectic, the number of zero columns in A is even by Theorem
3.1, so that H1 (M (A)) is even dimensional. Let 2r be the dimension of H1 (M (A)).
We may assume that H1 (M (A)) is generated by du1 , . . . , du2r by changing thesuﬃces of the coordinates. Moreover, through a linear coordinate change of theﬁrst 2r coordinates u1 , . . . , u2r , we may assume that the symplectic form ω onM (A) is of the form
(4.2)

ω=r
∑

dui ∧ dui+r +∑cj,k duj ∧ duk .

j<k,Aj =Ak 6=0

i=1

Since M (A) = T 2n /G(A) and Ap = 0 for p = 1, . . . , 2r, the multiplication of S 1
on the p-th coordinate on T 2n for 1 ≤ p ≤ 2r descends to an S 1 -action on M (A)and deﬁnes a symplectic isotopy {φpt }. The one-parameter family {Xtp } of vectorﬁelds associated with {φpt } is then ∂/∂up (possibly up to a non-zero constant), sothat it follows from (4.1) and (4.2) that
∫ 1
∫ 1
the ﬂux of {φpt } =[iXtp ω]dt =[duq ]dt = [duq ]00

where q = p + r if 1 ≤ p ≤ r and q = p − r if r + 1 ≤ p ≤ 2r. This shows that Γωspans H1 (M (A)) over R. Since Γω is closed and discrete in H1 (M (A)) as remarked
before, it must be a lattice group of H1 (M (A)) of full rank.
¤Acknowledgment. The author would like to thank Professor Mikiya Masuda forstimulating discussion about toric topology and symplectic topology. He also wouldlike to thank Professor Kaoru Ono for explaining the ﬂux group of a symplectic
manifold, Professor Takahiko Yoshida for informing on the paper [4], and Shintaro
Kuroki and Yunhyung Cho for useful comments on the paper.

6H. ISHIDA

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[4] H. Geiges, Symplectic structures on T 2 -bundles over T 2 , Duke Math. J. 67 (1992), 539–555.
[5] Y. Kamishima and M. Masuda, Cohomological rigidity of real Bott manifolds, Alg. & Geom.
Top. 9 (2009), 2479–2502, arXiv:math.AT/0807.4263.
[6] M. Masuda, Classiﬁcation of real Bott manifolds, preprint, arXiv:math.AT/0809.2178.
[7] K. Ono, Floer-Novikov cohomology and the ﬂux conjecture, GAFA, Geom. funct. anal. 16(2006), 981–1020.
[8] W. P. Thurston, Some simple examples of symplectic manifolds, Proc. of Amer. Math. Soc.
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Graduate School of Science, Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
E-mail address: hiroaki.ishida86@gmail.com