Ishida, Hiroaki

Osaka City University Repository

Ishida, H.
Osaka J. Math.
49 (2012), 515–522
FILTERED COHOMOLOGICAL RIGIDITY OF BOTT TOWERS
HIROAKI ISHIDA
(Received July 5, 2010, revised November 19, 2010)
Abstract
A Bott tower is an iterated CP1-bundle over a point, where each CP1-bundle
is the projectivization of a rank 2 decomposable complex vector bundle. For a
Bott tower, the filtered cohomology is naturally defined. We show that isomorphism
classes of Bott towers are distinguished by their filtered cohomology rings. We even
show that any filtered cohomology ring isomorphism between two Bott towers is in-
duced by an isomorphism of the Bott towers.
1. Introduction
A Bott tower of height n is a sequence of CP1-bundles
(1) Bn n ! Bn 1 n 1  !    2 ! B1 1 ! B0 D {a point},
where each fibration i W Bi ! Bi 1 for i D 1, : : : , n is the projectivization of a rank
2 decomposable complex vector bundle over Bi 1. A decomposable complex vector
bundle is the Whitney sum of line bundles. Each Bi is called a Bott manifold. As is
known, a Bott manifold Bn is a smooth projective toric variety. We denote the Bott
tower (1) of height n by B

D ({Bi }niD0, {i }niD1).
Two Bott towers B

D ({Bi }niD0,{i }niD1) and B 0 D ({B 0i }niD0,{ 0i }niD1) are isomorphic
if there is a collection of diffeomorphisms '

D {'i W Bi ! B 0i }
n
iD0 such that the follow-
ing diagram is commutative:
Bn
n
K
'n
K
Bn 1
n 1
K
'n 1
K
: : :
2
K B1
1
K
'1
K
B0
'0
K
B 0n

0
n
K B 0n 1

0
n 1
K   

0
2
K B 01

0
1
K B 00.
Since i W Bi ! Bi 1 has a cross section, the induced homomorphism i W H(Bi 1)!
H(Bi ) is injective for each i . Therefore the cohomology ring H(Bi 1) can be regarded
as a subring of H(Bi ) via i . We define a filtered graded Z-algebra FH(B) by
• For i  n, Fi H(B) WD H(Bn).
• For 0  j  n   1, F j H(B) WD n Æ n 1 Æ    Æ jC1(H(B j )).
2010 Mathematics Subject Classification. 57R19, 57R20, 57S25, 14M25.
516 H. ISHIDA
We call F

H(B

) the filtered cohomology ring of the Bott tower B

.
The following is our main result:
Theorem 1.1. Isomorphism classes of Bott towers are distinguished by their fil-
tered cohomology rings. Moreover, any filtered cohomology ring isomorphism between
two Bott towers are induced by an isomorphism between the Bott towers.
Our study is motivated by the so-called cohomological rigidity problem for toric
manifolds. A toric manifold is a smooth (complete) toric variety. Bott manifolds are ex-
amples of toric manifolds. The cohomological rigidity problem asks whether topological
types or diffeomorphism types of toric manifolds are distinguished by their cohomology
rings or not. This problem is open and we do not know any counterexamples.
This paper is organized as follows. In Section 2, we recall some preliminary facts
about Bott manifolds. In Section 3, we see that rank 2 decomposable vector bundles
over Bott manifolds are distinguished by their total Chern classes. In Section 4, we
study CP1-bundles over Bott manifolds by using the results obtained in Section 3. In
Section 5, we give a proof of Theorem 1.1.
Throughout this paper, all cohomology groups are taken with Z-coefficient.
2. Preliminaries
In this section, we recall some preliminary facts (see [1, Section 2] and [2, Sec-
tion 2] for more details). For a complex vector bundle  over a topological space, we
denote its projectivization by P().
Lemma 2.1 (Lemma 2.1, [2]). Let B be a smooth manifold,  be a complex line
bundle over B and  be a complex vector bundle over B. Then, the projectivizations
P() and P( 
 ) are isomorphic as bundles.
Let Bi 1 be a Bott manifold and i D i   0i be the Whitney sum of complex
line bundles i and  0i over Bi 1. Since i 
 i D C  (i 
  0i ), the projectivization
Bi D P(i ) is isomorphic to P(C  (i 
  0i )) as fiber bundles by Lemma 2.1, where
C denotes the trivial line bundle over Bi 1 and i denotes the dual line bundle of i .
Hence we can assume that i is the Whitney sum of C and a complex line bundle,
denoted i again, in the definition of Bott towers.
Let B

D ({Bi }niD0, {i }niD1) be a Bott tower of height n. We suppose that each fi-
bration Bi ! Bi 1 is the projectivization P(Ci ). Let i be the tautological line bun-
dle over P(C  i ) D Bi . By Borel–Hirzebruch formula, the cohomology ring H(Bi )
viewed as an H(Bi 1)-algebra via i is of the form
H(Bi )  H(Bi 1)[X ]=(X2   c1(i )X )
FILTERED COHOMOLOGICAL RIGIDITY OF BOTT TOWERS 517
where c1(i ) denotes the first Chern class of i and X represents the first Chern class
of the tautological line bundle i . Using this formula inductively on i , we see that the
cohomology ring H(Bn) is generated by
xi WD 

n Æ    Æ 

iC1(c1(i ))
for i D 1, : : : , n. More precisely, H(Bn) is of the form
H(Bn) D Z[x1, : : : , xn]=(x2i   i xi I i D 1, : : : , n)
as rings, where i WD n Æ    Æ iC1(c1(i )) for all i .
With this understood, we may regard H(Bk) as a subring of H(Bn) generated by
x j ’s with j D 1, : : : , k. Namely, we have
FkH(B) D Z[x1, : : : , xk]=(x2j    j x j I j D 1, : : : , k).
A vanishing pair is an ordered pair (z, Nz) of elements in H2(Bn) such that z Nz D 0.
We say that a vanishing pair is primitive if both elements are primitive. The following
lemma is very helpful for our purpose.
Lemma 2.2 (Lemma 2.3, [1]). A primitive vanishing pair is of the form
(ax j C u, (a(x j    j )   u))
for some j , where a is a non-zero integer, u is a linear combination of xi ’s with i < j ,
and u(u C a j ) D 0.
3. Rank 2 decomposable vector bundles over Bott manifolds
For an element  2 H2(Bn), we denote by   the complex line bundle over a Bott
manifold Bn whose first Chern class is . The purpose of this section is to prove
the following.
Theorem 3.1. Rank 2 decomposable vector bundles over a Bott manifold are dis-
tinguished by their total Chern classes.
We prepare two lemmas.
Lemma 3.2.  x j    x jC j  C    j .
Proof. Since  x j is the pull-back bundle of the tautological line bundle over P(C
 j ) D B j ,  x j is a subbundle of C    j where  j D c1( j ). It follows that the first
Chern class of the orthogonal complement of  x j in C    j (with an Hermitian metric)
518 H. ISHIDA
is  x j C  j . Since the orthogonal complement is a complex line bundle and complex
line bundles are classified by their first Chern classes, it must be isomorphic to   x jC j ,
proving the lemma.
Lemma 3.3. If       C   C , then  a   b  C   aCb for any
integers a and b.
Proof. There is a nowhere zero cross section f of      since      con-
tains a trivial complex line bundle as a subbundle by assumption. Let f

and f

be the
projections of f on   and   , which are cross sections of   and   respectively.
Their zero loci do not intersect. If a (respectively, b) is positive, we define fa WD
f a

D f


    
 f

(respectively, fb WD f b

) which is a cross section of  a (respect-
ively,  b). Note that there is a map 'a W  a !   a (respectively, 'b W  b !   b)
which is an isomorphism as real 2-plane bundles but anti-C-linear on fibers. If a (re-
spectively, b) is negative, we define a cross section fa of  a (respectively, fb of

b) to be the pull-back of f  a

(respectively, f  b

) by 'a W  a !   a (respectively,
'b W 
b
! 
 b). The pair ( fa , fb) determines a nowhere zero cross section of

a
 
b
. Thus  a   b contains a trivial complex line bundle as a subbundle. By
computing the first Chern class of the orthogonal complement of the trivial subbundle
in  a   b , we get the lemma.
Proposition 3.4. Let  and  0 be complex line bundles over Bn . Then, the second
Chern class c2(   0) is zero if and only if    0 is isomorphic to C  ( 
  0).
Proof. The “if” part is clear. So we show the “only if” part.
We assume that c2(   0) D 0. Then c1( )c1( 0) D 0. Note that if either c1( ) D 0
or c1( 0) D 0, then  or  0 is trivial and Proposition 3.4 holds. Thus, by Lemma 3.3, it
suffices to show that    0  C ( 
  0) only when the vanishing pair (c1(i ), c1( 0i ))
is primitive.
By Lemmas 2.2 and 3.3, we may assume that
(c1( ), c1( 0)) D (ax j C u, a(x j    j )   u)
for some j , where a is a non-zero integer, u is a linear combination of xi ’s with i < j ,
and u(u C a j ) D 0. Then, j is the minimal integer such that c1( ) and c1( 0) can
be written as linear combinations of xk’s with k  j . We will show that    0 
C  ( 
  0) by induction on j . If j D 1, then (c1( ), c1( 0)) D (x1, x1) and we are
done by Lemma 3.2 and 3.3. Now we assume that    0  C ( 
  0 ) for any
vanishing pair (, 0) such that  and 0 can be written as linear combinations of xk’s
FILTERED COHOMOLOGICAL RIGIDITY OF BOTT TOWERS 519
with k < j . Then,

ax jCu
 
a(x j  j ) u
 
ax j

 ( u    (uCa j ))
 
ax j

 (C    a j ) (by hypothesis of induction)
 
ax j
 
a(x j  j )
 C  
a(2x j  j ) (by Lemma 3.2 and 3.3).
This completes the induction step, proving the proposition.
Proof of Theorem 3.1. Let , , 0 and  0 be elements in H2(Bn) such that
c(    ) D c( 0    0),
that is, (1C )(1C ) D (1C 0)(1C  0). Then, we have
    D (0   )C ( 0   ) and (0   )( 0   ) D 0.
Therefore, it follows from Proposition 3.4 that
C  
 
 

0
 
 

0
 
.
By tensoring the both sides above with   , we obtain the theorem.
4. CP1-bundles over Bott manifolds
In this section, we study CP1-bundles over Bott manifolds.
Theorem 4.1. Let Bn be a Bott manifold. Let   and   be complex line bundles
over Bn . If H(P(C   )) and H(P(C   )) are isomorphic as H(Bn)-algebras,
then P(C   ) and P(C   ) are isomorphic as bundles.
Proof. By Borel–Hirzebruch formula, H(P(C  )) and H(P(C  )) are of
the forms
H(P(C   )) D H(Bk)[X ]=(X2   X )
and
H(P(C   )) D H(Bk)[Y ]=(Y 2   Y ).
Let 8W H(P(C ))! H(P(C )) be an H(Bn)-algebra isomorphism. We write
8(Y ) DW s XC0, s D1, 0 2 H2(Bk). Since 8(Y 2 Y ) D 0 and X2 D X , we have
0 D (s X C 0)2   (s X C 0)
D ( C 2s0   s)X C 0(0   ).
520 H. ISHIDA
Therefore  D s( 20 C ) and 0(0   ) D 0. Thus we obtain
P(C   ) D P(C   s( 20C))
 P(C    20C) (by taking complex conjugation when s D  1)
 P( 0    0C) (by Lemma 2.1)
 P(C   ) (by Theorem 3.1).
This proves the theorem.
Lemma 4.2. The group of H(Bn)-algebra automorphisms of H(P(C   )) is
of order 2 and generated by the automorphism induced by the bundle map p W P(C 

) ! P(C  ) which assigns a line l in C    (with an Hermitian metric) to its
orthogonal complement l?.
Proof. Let 8 W H(P(C   )) ! H(P(C   )) be an H(Bn)-algebra auto-
morphism. The same argument as in the proof of Theorem 4.1 shows that 8(X ) D X
or  X C . Thus the group of H(Bn)-algebra automorphisms of H(P(C   )) is
of order 2. We have to show that p(X ) D  X C . Note that the total space of the
tautological line bundle  X over P(C   ) is the set
{(l, v) 2 P(C   )  E(C   )I l 3 v}
and the total space of the pull-back p X of  X by p is given by
{(l, v) 2 P(C   )  E(C   )I l? 3 v},
so p X   X  (C  ), where  W P(C  ) ! Bn is the projection. Therefore
we obtain p(X ) D  X C .
The following corollary follows from Theorem 4.1 and Lemma 4.2.
Corollary 4.3. Let Bn be a Bott manifold. Let  and 0 be rank 2 decomposable
vector bundles over Bn . Then, their projectivizations P() and P(0) are isomorphic
as bundles if and only if their cohomology rings are isomorphic as H(Bn)-algebras.
Moreover, any H(Bn)-algebra isomorphism between H(P()) and H(P(0)) is in-
duced by a bundle isomorphism.
5. Proof of Theorem 1.1
Let B

D ({Bi }niD0, {i }niD1) and B 0 D ({B 0i }niD0, { 0i }niD1) be Bott towers of height
n. Let 8

W F

H(B 0

) ! F

H(B

) be an isomorphism. We will show Theorem 1.1 by
induction on the height. In the case when the height k D 0, Theorem 1.1 clearly holds.
FILTERED COHOMOLOGICAL RIGIDITY OF BOTT TOWERS 521
Assume that Theorem 1.1 holds for Bott towers of height k. For 8k W FkH(B 0

) D
H(B 0k) ! H(Bk) D FkH(B), there is a diffeomorphism 'k W Bk ! B 0k such that 'k D
8k by the hypothesis of induction. Then, the pull-back bundle 'k B 0kC1 ! Bk is iso-
morphic to BkC1 ! Bk . In fact, there is a commutative diagram
H(BkC1) H(B 0kC1)
8kC1
K
Q'

k
K
H('k B 0kC1)
H(Bk)


kC1
K
('k  0kC1) K
H(B 0k)
'

k D8k
K

0
kC1
K
where Q'k is the homomorphism induced from the bundle map Q'k W 'k B 0kC1 ! B 0kC1 and
('k  0kC1) is the homomorphism induced from the projection ' 0kC1 W 'k B 0kC1 ! Bk .
Since Q'k is a diffeomorphism, the induced homomorphism Q'k is a ring isomorphism.
The composition 8kC1 Æ ( Q'k ) 1 W H('k B 0kC1) ! H(BkC1) is an H(Bk)-algebra iso-
morphism. By Corollary 4.3, there is a bundle isomorphism b W BkC1 ! 'k B 0kC1 such
that b D 8kC1 Æ ( Q'k ) 1.
By the construction of b, the bundle map 'kC1 WD Q'k Æ b satisfies 'kC1 D 8kC1.
ACKNOWLEDGEMENT. The author would like to thank Professor Mikiya Masuda
for stimulating discussion about toric topology.
References
[1] S. Choi and M. Masuda: Classification of Q-trivial Bott manifolds, preprint, arXiv:
math.AT/0912.5000.
[2] S. Choi, M. Masuda and D.Y. Suh: Topological classification of generalized Bott towers, Trans.
Amer. Math. Soc. 362 (2010), 1097–1112.
522 H. ISHIDA
Department of Mathematics
Graduate School of Science
Osaka City University
Sugimoto, Sumiyoshi-ku, Osaka 558-8585
Japan
Current address:
Osaka City University Advanced Mathematical Institute
Sugimoto, Sumiyoshi-ku, Osaka 558-8585
Japan
e-mail: ishida@sci.osaka-cu.ac.jp


Filtered cohomological rigidity of bott towers

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Filtered cohomological rigidity of bott towers

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