Topological classification of generalized Bott towers

If $B$ is a toric manifold and $E$ is a Whitney sum of complex line bundles over $B$, then the projectivization $P(E)$ of $E$ is again a toric manifold. Starting with $B$ as a point and repeating this construction, we obtain a sequence of complex projective bundles which we call a generalized Bott tower. We prove that if the top manifold in the tower has the same cohomology ring as a product of complex projective spaces, then every fibration in the tower is trivial so that the top manifold is diffeomorphic to the product of complex projective spaces. This gives a supporting evidence to what we call cohomological rigidity problem for toric manifolds"Are toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic?"We provide two more results which support the cohomological rigidity problem.


Introduction
A toric variety X of dimension n is a normal complex algebraic variety with an action of an n-dimensional algebraic torus (C * ) n having a dense orbit. A fundamental result in the theory of toric varieties says that there is a one-to-one correspondence between toric varieties and fans. It follows that the classification of toric varieties is equivalent to the classification of fans up to isomorphism.
Among toric varieties, compact smooth toric varieties, which we call toric manifolds, are well studied. Recently the second author has shown in [7] that toric manifolds as varieties can be distinguished by their equivariant cohomology. So we are led to ask how much information ordinary cohomology contains for toric manifolds and posed the following problem in [9]. Throughout this paper, an isomorphism of cohomology rings is as graded rings unless otherwise stated.
Cohomological rigidity problem for toric manifolds. Are toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic?
If B is a toric manifold and E is a Whitney sum of complex line bundles over B, then the projectivization P (E) of E is again a toric manifold. Starting with B as a point and repeating this construction, say m times, we obtain a sequence of toric manifolds: where the fiber of π i : B i → B i−1 for i = 1, . . . , m is a complex projective space CP n i . We call the above sequence a generalized Bott tower of height m and omit "generalized" when n i = 1 for every i ( [4]). We also call B k in the tower a k-stage generalized Bott manifold and omit "generalized" as well when n i = 1 for every i. We note that H * (B m ) is isomorphic to H * ( m i=1 CP n i ) as a group but not necessarily as a graded ring. If every fibration in the tower (1.1) is trivial, then B m is diffeomorphic to m i=1 CP n i and H * (B m ) is isomorphic to H * ( m i=1 CP n i ) as a graded ring. The following theorem shows that the converse is true and generalizes Theorem 5.1 in [8] treating Bott towers. It also gives a supporting evidence to the cohomological rigidity problem above.
, then every fibration in the tower (1.1) is trivial, in particular, B m is diffeomorphic to m i=1 CP n i . Remark 1.2. It is shown in [2] that a toric manifold X whose cohomology ring is isomorphic to that of a generalized Bott manifold is a generalized Bott manifold. So we can conclude that if H * (X) is isomorphic to H * ( m i=1 CP n i ), then X is diffeomorphic to m i=1 CP n i 2-stage Bott manifolds are famous Hirzebruch surfaces and their diffeomorphism types can also be distinguished by their cohomology rings. The following theorem generalizes this fact and gives a partial affirmative solution to the cohomological rigidity problem. Actually we obtain a diffeomorphism classification result for those manifolds (see Theorem 6.1) and it would be interesting to compare it with the variety classification result in [6].
We also prove the following which gives another partial affirmative solution to the cohomological rigidity problem. This paper is organized as follows. In Section 2 we recall well-known facts on projective bundles and discuss their Pontrjagin classes. We prepare two lemmas on cohomology of generalized Bott manifolds in Section 3. Section 4 is devoted to the proof of Theorem 1.1. For the proof we need to show that a Whitney sum of complex line bundles over a product of complex projective spaces is trivial if its total Chern class is trivial. This result is of independent interest and proved in Section 5. We will discuss 2-stage generalized Bott manifolds in Section 6 and 3-stage Bott manifolds in Section 7. In Section 8 which is an appendix, we give a sufficient condition for an isomorphism of cohomology rings with Z/2 coefficients to preserve Stiefel-Whitney classes.

Projective bundles
Let B be a smooth manifold and let E be a complex vector bundle over B. We denote by P (E) the projectivization of E.
Lemma 2.1. Let B and E be as above and let L be a complex line bundle over B. We denote by E * the complex vector bundle dual to E. Then P (E ⊗ L), P (E) and P (E * ) are isomorphic as fiber bundles over B, in particular they are diffeomorphic.
Proof. For each x ∈ B, we choose a non-zero vector v x from the fiber of L over x and define a map Ψ : where u x is an element of the fiber of E over x. The map Ψ depends on the choice of v x 's but the induced map from P (E) to P (E ⊗ L) does not because L is a line bundle. It is easy to check that the induced map gives an isomorphism of P (E) and P (E ⊗ L) as fiber bundles over B.
Choose a hermitian metric , on E, which is anti-C-linear on the first entry and C-linear on the second entry, and define a map Φ : E → E * by Φ(u) := u, . This map is not C-linear but anti-C-linear, so it induces a map from P (E) to P (E * ), which gives an isomorphism as fiber bundles.
Let y ∈ H 2 (P (E)) be minus * the first Chern class of the tautological line bundle over P (E) where vectors in a line ℓ of E form the fiber over ℓ ∈ P (E). H * (P (E)) can be viewed as an algebra over H * (B) via π * : H * (B) → H * (P (E)) where π : P (E) → B denotes the projection. When H * (B) is finitely generated and torsion free (this is the case when B is a toric manifold), π * is injective and H * (P (E)) as an algebra over H * (B) is known to be described as where n denotes the complex fiber dimension of E. If we formally express then the relation in (2.1) is written as (y + u i ), and the total Chern class of the tangent bundle along the fibers T f (P (E)) of P (E) is given by see [1, (2) in p.515]. It follows that the total Pontrjagin class of T f (P (E)) is given by Proposition 2.2. Let E ′ → B ′ be another complex vector bundle over a smooth manifold B ′ with the same fiber dimension as E. Suppose that ϕ : H * (P (E ′ )) → H * (P (E)) is an isomorphism such that ϕ(H * (B ′ )) = H * (B). Then ϕ(p(T f P (E ′ ))) = p(T f P (E)). If ϕ satisfies ϕ(p(B ′ )) = p(B) in addition, then ϕ(p(P (E ′ ))) = p(P (E)).
As in (2.2) we formally express c(E ′ ) = n i=0 (1 + u ′ i ). Then we have the relation (2.3) and the formula (2.4) for E ′ → B ′ with prime.
Since ϕ( n i=0 (y ′ + u ′ i )) = n i=0 (ǫy + w + ϕ(u ′ i )) is zero in H * (P (E)), we have an identity Replace y with √ −1 + y and − √ −1 + y in the identity above and multiply the resulting two identities at each side. Then we obtain an identity in the ring H * (B)[y], in particular, in H * (P (E)). It follows from (2.4), (2.5) and (2.6) that This proves the former part of the proposition. Since the tangent bundle T P (E) of P (E) decomposes into a Whitney sum of π * (T B) and T f (P (E)), we obtain the latter part of the proposition.
We conclude this section with an observation on Pontrjagin classes of generalized Bott manifolds in (1.1). Since π * j : for any j. Proof. It follows from the assumption that the fiber dimensions of B j → B j−1 and B ′ j → B ′ j−1 must agree for each j. If ϕ(p(B ′ j−1 )) = p(B j−1 ), then Proposition 2.2 implies that ϕ(p(B ′ j )) = p(B j ). Therefore, the theorem follows by induction on j.

Cohomology of generalized Bott manifolds
Complex vector bundles involved in a generalized Bott tower (1.1) are Whitney sums of complex line bundles. Since P (E ⊗ L) and P (E) are isomorphic as fiber bundles by Lemma 2.1, we may assume that at least one of the complex line bundles is trivial at each stage of the tower, that is, where C denotes the trivial complex line bundle and ξ i a Whitney sum of complex line bundles over B i−1 . We set n i = dim ξ i .
Let y i ∈ H 2 (B i ) denote minus the first Chern class of the tautological line bundle over B i = P (C ⊕ ξ i ). We may think of y i as an element of Then the repeated use of (2.1) shows that the ring structure of H * (B m ) can be described as We prepare two lemmas used later.
where (3.2) is used at the second identity. If b = 0 and (by m +w) nm+1 = 0, then we see bc 1 (ξ m ) = (n m + 1)w by looking at the coefficients of y nm m at the identity above and hence b and w must be proportional, proving the lemma.
Proof. Suppose x n j = 0 on the contrary. Then ( m j=1 b j y j ) n j must be in the ideal generated by the polynomials in (3.2) while a non-zero scalar multiple of y n j j appears in ( m j=1 b j y j ) n j when we expand it because b j = 0. However, it follows from (3.2) that if a non-zero scalar multiple of a power of y j appears in the ideal, then the exponent must be at least n j + 1, which is a contradiction.

Cohomologically product generalized Bott manifolds
The purpose of this section is to prove Theorem 1.1 in the Introduction.
We continue to use the notation of the previous section and from now until this section ends, we assume that and one has an expression Then S is the image of µ. Let J ℓ = µ −1 (N ℓ ) for ℓ = 1, . . . , k and let C J ℓ and D J ℓ be the matrices formed from c ij and d ij with i, j ∈ J ℓ respectively. Since if n 1 ≥ n 2 ≥ · · · ≥ n m , and in general conjugate to the above by a permutation matrix. Since C = D −1 , C is also conjugate to a block upper triangular matrix     by a permutation matrix. Then a similar argument to Case 1 above can be applied to C J ℓ and D J ℓ (for J ℓ containing m) instead of C and D, and the lemma follows.
We may further assume that c mm = d mm = 1 if necessary by taking −x m instead of x m , so that we may assume On the other hand H * (B m ) has an expression (4.1) by assumption. It follows from (4.3) that H * (B m−1 ) agrees with the right hand side of (4.1) with the relation x m + m−1 j=1 c mj x j = 0 added. Therefore, we can eliminate x m using the added relation, so that we obtain a surjective homomorphism But the both sides above are torsion free and have the same rank, so the homomorphism above is an isomorphism, proving the lemma.
We need one more result for the proof of Theorem 1.1. a ij y j with a ij ∈ Z.
Therefore, it suffices to find a complex line bundle L such that the total Chern class of (C ⊕ ξ m ) ⊗ L is trivial because the triviality of the bundle follows from the triviality of the Chern class by Theorem 4.3.
We take L = m−1 j=1 γ −d mj j with d mj in (4.3). Then because a natural homomorphism from the right hand side above to H * (B m ) is surjective by (3.1) and (4.3), and hence isomorphic since both are torsion free and have the same rank. Therefore, when we expand the right hand side of (4.7), the coefficient of x k m must be zero for any k = 1, . . . , n m . This implies that the right hand side of (4.5) is equal to 1, proving the theorem.
Combining Theorem 1.1 with Theorem 8.1 in [3], we obtain the following corollary which generalizes Theorem 5.1 in [8] treating the case where n i = 1 for any i. Corollary 4.4. If the cohomology ring of a quasitoric manifold over a product of simplices is isomorphic to that of m i=1 CP n i , then it is homeomorphic to m i=1 CP n i . Remark 4.5. Similarly to Remark 1.2 the assumption "over a product of simplices" in the corollary above can be dropped by a result in [2].

Proof of Theorem 4.3
This section is devoted to the proof of Theorem 4.3. We recall a general fact. A more refined result can be found in [11].
Lemma 5.1. Let X be a finite CW-complex such that H odd (X) = 0 and H * (X) has no torsion. Then complex n-dimensional vector bundles over X with 2n ≥ dim X are isomorphic if and only if their total Chern classes are same.
Proof. The assumption on H * (X) implies that K(X) is torsion free, so Chern character gives a monomorphism from K(X) to H * (X; Q). On the other hand, if dim X ≤ 2n, then the homotopy set [X, BU(n)], where BU(n) denotes the classifying space of a unitary group U(n), agrees with K(X). This implies the lemma.
Let B = k j=1 CP n j be a product of complex projective spaces and let E = n i=1 η i be a Whitney sum of complex line bundles η i over B. Suppose that c(E) = 1. Then since H odd (B) = 0 and H * (B) has no torsion, E is trivial by Lemma 5.1 when n ≥ k j=1 n j . So we assume that n < k j=1 n j in the following. By assumption where we can take x j as the first Chern class of the pullback γ j of the tautological line bundle over CP n j via the projection k j=1 CP n j → CP n j . Then we may assume that η i = k j=1 γ a ij j with a ij ∈ Z and It follows that Since x j 's are linearly independent, the identity above implies that a ij = 0 for each j = 1, . . . , k.
Moreover it follows from (5.1) that We need to consider two cases. Case I n j ≥ 2 for some j = 1, . . . , k. Since Hence n i=1 a 2 ij = 0, which implies that a 1j = · · · = a nj = 0.
In this case n < k as n < k j=1 n j . Set v j = (a 1j , . . . , a nj ) ∈ Z n for j = 1, . . . , k. We claim that v j = 0 for some j = 1, . . . , k. Since This means that k vectors v 1 , . . . , v k in Z n ⊂ R n are mutually orthogonal. But since k > n, v j = 0 for some j = 1, . . . , k.
We have shown that in either cases there exists some j such that (a 1j , . . . , a nj ) = 0. For simplicity, assume j = k. Then η i is of the form k−1 j=1 γ a ij j . Let γ j be the pull-back bundle of the tautological line bundle of CP n j via the projection k−1 j=1 CP n j → CP n j . Then Hence the problem reduces to the bundle on k−1 j=1 CP n j . The argument above shows that the proof of the theorem reduces to the case k = 1, so the theorem follows from the following lemma. Proof. Let γ u denote a complex line bundle over CP n whose first Chern class is u ∈ H 2 (CP n ).
In case m ≥ n, the lemma follows from Lemma 5.1. In case m < n, c(E) = c(E ′ ) implies that {u 0 , . . . , u m } = {u ′ 0 , . . . , u ′ m } and hence E and E ′ are isomorphic. 2-stage generalized Bott manifolds can be thought of as a higher dimensional generalization of Hirzebruch surfaces and their classification as varieties is completed in [6]. In this section we complete the diffeomorphism classification of those manifolds.

2-stage generalized Bott manifolds
Let B 1 = CP n 1 and where u 0 = 0 and γ u i denotes the complex line bundle over B 1 whose first Chern class is u i ∈ H 2 (B 1 ) as before. Similarly let be another 2-stage generalized Bott manifold with B 1 = CP n 1 as 1stage, where u ′ 0 = 0. Theorem 6.1. Let B 2 and B ′ 2 be as above. Then the following are equivalent.
2 ) are isomorphic. Proof. Condition (1) means that ( n 2 i=0 γ u ′ i ) ⊗ γ w or its dual has the same total Chern class as n 2 i=0 γ u i , so that they are isomorphic as vector bundles by Lemma 5.2. This together with Lemma 2.1 implies (2). The implication (2)⇒(3) is obvious, so it suffices to prove the implication (3)⇒(1).
Therefore, (6.3) r(s + a ′ q) + s(r + a ′ p) = s(s + a ′ q)a and moreover since n 1 ≥ 2, we have r(r + a ′ p) = 0 and hence r = 0 or r = −a ′ p. If r = 0, then p = ±1 and s = ±1 from (6.2) and hence ±a ′ = (s + a ′ q)a from (6.3), which implies that a|a ′ . If r = −a ′ p, then from (6.2) we have ±1 = ps − qr = ps + a ′ pq = p(s + a ′ q). Thus p = ±1 and s + a ′ q = ±1. From (6.3) we have ±a ′ = sa and hence a|a ′ . In any case we have shown that a ′ is divisible by a. By the symmetry, a is divisible by a ′ . Thus a = ±a ′ , and hence the identity in (1) is satisfied with w = 0.
Case 2. The case where n 1 = n 2 = 1. We write u 1 = ax and u ′ 1 = a ′ x as in Case 1 above. The identity in (1) is equivalent to a ≡ a ′ mod 2.
In the following all congruence relations are taken modulo 2 unless stated otherwise. It follows from (6.3) and (6.2) that On the other hand, since x 2 = 0, the identity ϕ(x) 2 = 0 implies that 0 = (px + qy) 2 ≡ q 2 y 2 ≡ q 2 axy, so that If a ≡ 0, then so is a ′ from (6.4). If a ≡ 1, then q ≡ 0 from (6.5), so that a ′ ≡ s 2 a from (6.4) and ps ≡ 1 (and hence s ≡ 1) from (6.2). Therefore, a ≡ a ′ in any case. Case 3. The case where n 2 ≥ 2. If u i = u ′ i = 0 for all i's, then the identity in (1) holds with w = 0, so we may assume either u i or u ′ i is non-zero for some i and moreover u i = 0 for some i without loss of generality. Then, since 0 = ϕ(x n 1 +1 ) = (px + qy) n 1 +1 , q = 0 by Lemma 6.2 below. This means that ϕ preserves the subring H * (B 1 ), so that we have ϕ(y ′ ) = ǫy + w for some w ∈ H 2 (B 1 ), where ǫ = ±1. Therefore ϕ( n 2 i=0 (y ′ + u ′ i )) = n 2 i=0 (ǫy + w + ϕ(u ′ i )). Since this element vanishes in H * (B 2 ) and is a polynomial of degree n 2 + 1 in y, we have an identity as polynomials in y. Then, plugging y = 1, we obtain the identity in (1) in the theorem.
Here is the lemma used above. We shall use the same notation as above.
Proof. Since (αx+βy) n 1 +1 = 0 in the ring Z[x, y]/(x n 1 +1 , n 2 i=0 (y +u i )), there are a homogeneous polynomial g(x, y) in x, y of total degree n 1 − n 2 and an integer c such that (6.6) (αx + βy) n 1 +1 − cx n 1 +1 = g(x, y) as polynomials in x and y. In fact, c = α n 1 +1 as u 0 = 0. Suppose g(x, y) = 0 (so that n 1 ≥ n 2 ). When we split the left hand side into a product of linear polynomials in x and y, it has at most two linear polynomials over Z as factors while the right hand side has at least three linear polynomials over Z as n 2 ≥ 2 by assumption. This is a contradiction. Therefore g(x, y) = 0. But then β must be zero, proving the lemma. Proof. Let B 2 → B 1 = CP n 1 be a generalized Bott tower of height 2 where the fiber is CP n 2 and let B ′ 2 → B ′ 1 = CP n ′ 1 be another generalized Bott tower of height 2 where the fiber is CP n ′ 2 . Suppose that H * (B 2 ) is isomorphic to H * (B ′ 2 ). Then {n 1 , n 2 } = {n ′ 1 , n ′ 2 } which we can see from their Betti numbers. If n i = n ′ i for i = 1, 2, then the corollary follows from Theorem 6.1. Therefore, we may assume that n 1 = n ′ 2 , n 2 = n ′ 1 and they are different. If both B 2 and B ′ 2 are cohomologically product, then they are diffeomorphic to CP n 1 × CP n 2 by Theorem 1.1.
In the sequel it suffices to prove that H * (B 2 ) and H * (B ′ 2 ) are not isomorphic when they are not cohomologically product and n 1 = n ′ 2 = n 2 = n ′ 1 . We may assume n 1 > n 2 without loss of generality. Since B ′ 2 is a CP n 1 -bundle over CP n 2 , there is a non-zero element in H 2 (B ′ 2 ) whose n 1 -th power vanishes, in fact, a non-zero element in H 2 (B ′ 2 ) coming from the base space CP n 2 is such an element because n 1 > n 2 . On the other hand, it is not difficult to see that there is no such a non-zero element in H 2 (B 2 ) since H * (B 2 ) = Z[x, y]/(x n 1 +1 , where u 0 = 0 and u i = 0 for some 1 ≤ i ≤ n 2 . (It also follows from Lemma 6.2 when n 2 ≥ 2.)
Lemma 7.2. The following are equivalent.
Proof. (1) ⇔ (2). This equivalence follows from the observation made in the paragraph just before the lemma.