Strong cohomological rigidity of Hirzebruch surface bundles in Bott towers

A Bott tower of height $n$ is an iterated $\mathbb{C}P^{1}$-bundle

where each fibration $\pi_{j}\thinspace\colon B_{j}\to B_{j-1}$ is the projectivization of a Whitney sum of two complex line bundles over $B_{j-1}$. Each $B_{j}$ is called a Bott manifold. The notion of the Bott tower is introduced in [GK1994]. By definition, $B_{j}$ is a closed manifold of dimension $2j$ and $B_{1}$ is nothing but the projective line $\mathbb{C}P^{1}$. Bott manifolds of dimension $4$ are known as Hirzebruch surfaces and introduced in [Hirzebruch1951].

By composing the first $k$ fibrations $\pi_{n},\dots,\pi_{n-k+1}$, we have a fiber bundle $B_{n}\to B_{n-k}$ whose fibers are Bott manifold of dimension $2k$. Moreover the cohomology $H^{*}(B_{n})$ of $B_{n}$ has a structure of $H^{*}(B_{n-k})$-algebra. In this paper, we clarify the relation between the topology of Hirzebruch surface bundles over a Bott manifold $B_{n}$ and the structure of the cohomology as $H^{*}(B_{n})$-algebras. The main theorem of this paper is the following.

This study is motivated by the strong cohomological rigidity conjecure posed in [Choi2015].

So far, no counterexamples are known to Conjecture 1.2 and some partial affirmative results are known. Conjecture 1.2 is known to be true for Bott manifolds of dimension up to $6$ (see [Choi2015, Theorem A]). Conjecture 1.2 is also true if we restrict Bott manifolds to $\mathbb{Q}$-trivial (see [CM2012, Corollary 5.1]) or $\mathbb{Z}/2\mathbb{Z}$-trivial ([CMM2015, Theorem 1.3]) ones. A Bott manifold $B_{n}$ is said to be $\mathbb{Q}$-trivial (respectively, $\mathbb{Z}/2\mathbb{Z}$-trivial) if $H^{*}(B_{n})\otimes\mathbb{Q}\cong H^{*}((\mathbb{C}P^{1})^{n})\otimes\mathbb{Q}$ (respectively, $H^{*}(B_{n})\otimes\mathbb{Z}/2\mathbb{Z}\cong H^{*}((\mathbb{C}P^{1})^{n})\otimes\mathbb{Z}/2\mathbb{Z}$). In [Choi2015, Problem 3.5], it is pointed out that Theorem 1.1 implies that Conjecture 1.2 is also true for Bott manifolds of dimension $8$.

This paper is organized as follows. In Section 2, we recall some preliminary facts about Bott manifolds that we need. In Section 3, we study the topology of Hirzebruch surfaces. Especially, we show an $S^{1}$-equivariant version of the cohomological rigidity of a Hirzebruch surface. Sections 4, 5, 6 are devoted to prove Theorem 1.1. In Section 4, we study the automorphism of $H^{*}(B_{n+2})$ as an $H^{*}(B_{n})$-algebra. In Section 5, we show that any automorphism of $H^{*}(B_{n+2})$ as an $H^{*}(B_{n})$-algebra is induced by a bundle automorphism. In Section 6, we show that isomorphism classes of Hirzebruch surface bundles over a Bott manifold $B_{n}$ are distinguished by their cohomologies as $H^{*}(B_{n})$-algebras. These results obtained in Sections 5 and 6 imply Theorem 1.1 immediately.

Throughout this paper, all cohomologies are taken with $\mathbb{Z}$-coefficient.

In this section we recall elementary facts about Bott manifolds for later use.

Thanks to Lemma 2.1 we may assume that one of line bundles over $B_{j}$ is the product line bundle $\underline{\mathbb{C}}$ in (1.1) without loss of generality. Throughout this paper, we assume that each fibration $B_{j}\to B_{j-1}$ is a projectivization of $P(\underline{\mathbb{C}}\oplus\xi_{j})\to B_{j-1}$, where $\underline{\mathbb{C}}$ is a product line bundle and $\xi_{j}$ is a complex line bundle over $B_{j-1}$.

Let $B$ be a smooth manifold and $V$ a complex vector bundle over $B$. The tautological line bundle $\gamma$ of $P(V)$ is a line bundle over $P(V)$ given by

here we think of $P(V)$ as a set of all lines passing $0$ in $V$. The first Chern classes of tautological bundles play a role to describe the cohomology ring of Bott manifolds. Lemma 2.2 and Theorem 2.3 below are well known facts. For leader’s convenience, we give brief proofs.

Let

be a Bott tower of height $n$. Let $\gamma_{j}$ be the tautological line bundle over $B_{j}=P(\underline{\mathbb{C}}\oplus\xi_{j})$. For integers $j,k$ with $1\leq j\leq k\leq n$, we define

By applying Lemma 2.2 eventually, we have the following.

Put $x_{j}:=x_{j}^{(n)}$. It follows from Theorem 2.3 that $H^{*}(B_{n})$ is generated by degree $2$ elements $x_{1},\dots,x_{n}$ and they form a basis of $H^{2}(B_{n})$. We call them the standard generators of $H^{*}(B_{n})$ with respect to $\xi_{1},\dots,\xi_{n}$.

Suppose that

Let $\gamma_{j}^{(k)}$ be the line bundle over $B_{k}$ given by

Since $c_{1}(\gamma_{j}^{(k)})=x_{j}^{(k)}$ and isomorphism classes of line bundles are distinguished by their first Chern classes, we have that the line bundle $\xi_{j}$ over $B_{j-1}$ is isomorphic to $(\gamma_{1}^{(j-1)})^{\otimes a_{j}^{1}}\otimes\dots\otimes(\gamma_{j-1}^{(j-1)})^{\otimes a_{j}^{j-1}}$. For simplicity, we assume that $\xi_{j}=(\gamma_{1}^{(j-1)})^{\otimes a_{j}^{1}}\otimes\dots\otimes(\gamma_{j-1}^{(j-1)})^{\otimes a_{j}^{j-1}}$. Let $\rho_{j}\thinspace\colon(S^{1})^{k}\to S^{1}$ be the homomorphism given by $\rho_{j}(t)=\prod_{\ell=1}^{j-1}t_{\ell}^{a_{j}^{\ell}}$ for $t=(t_{1},\dots,t_{k})\in(S^{1})^{k}$. We use the same symbol even if $k$ is different. Let $M_{k}$ be the quotient manifold $(S^{3})^{k}/(S^{1})^{k}$ by the action of $(S^{1})^{k}$ on $(S^{3})^{k}$ given by

Let $p_{k}\thinspace\colon M_{k}\to M_{k-1}$ be the map induced by the projection $(S^{3})^{k}\to(S^{3})^{k-1}$. Then $p_{k}\thinspace\colon M_{k}\to M_{k-1}$ is a $\mathbb{C}P^{1}$-bundle. Therefore we have the iterated $\mathbb{C}P^{1}$-bundle

Let $L_{k}$ be the quotient $((S^{3})^{k}\times\mathbb{C})/(S^{1})^{k}$ by the action of $(S^{1})^{k}$ on $(S^{3})^{k}\times\mathbb{C}$ given by

For a homomorphism $\rho\thinspace\colon(S^{1})^{k}\to S^{1}$, let $L_{\rho}$ be the $((S^{3})^{k}\times\mathbb{C})/(S^{1})^{k}$ by the action of $(S^{1})^{k}$ on $(S^{3})^{k}\times\mathbb{C}$ given by

$L_{k}$ and $L_{\rho}$ are line bundles over $M_{k}$.

Let

be another Bott tower of height $n$ such that $\pi_{j}^{\prime}\thinspace\colon B_{j}^{\prime}\to B_{j-1}^{\prime}$ is a projectivization $P(\underline{\mathbb{C}}\oplus\xi^{\prime})\to B_{j-1}^{\prime}$, where $\xi_{j}^{\prime}$ is a complex line bundle over $B_{j-1}^{\prime}$. Let $x_{1}^{\prime},\dots,x_{n}^{\prime}$ be the standard generators of $H^{*}(B_{n}^{\prime})$ with respect to $\xi_{1}^{\prime},\dots,\xi_{n}^{\prime}$.

Let $a\in\mathbb{Z}$. The Hirzebruch surface $\Sigma_{a}$ is the total space of the projectivization of the rank $2$ vector bundle over $\mathbb{C}P^{1}$:

In particular, $\Sigma_{a}$ is a Bott manifold of dimension $4$. By Theorem 2.3, the cohomology algebra $H^{*}(\Sigma_{a})$ is of the form

The strong cohomological rigidity for Hirzebruch surfaces is already shown in [CM2012]. In this section, we review it briefly and study the certain diffeomorphism of $\Sigma_{a}$.

First we shall review the classification of the diffeomorphism types of Hirzebruch surfaces given in [Hirzebruch1951] briefly. Let $a,b\in\mathbb{Z}$. By direct computation the isomorphism type of the cohomology algebra $H^{*}(\Sigma_{a})$ is determined by the parity of $a$. In case when $a$ is even, say $a=2k$, then we have that

In case when $a$ is odd, say $a=2k+1$, then we have that

Thus the diffeomorphism types of Hirzebruch surfaces are determined by the parity of $a$, and hence they are determined by the isomorphism types of the cohomologies.

We shall see the group of automorphisms of $H^{*}(\Sigma_{a})$. Since $x_{1}$ and $x_{2}$ form a basis of $H^{2}(\Sigma_{a})$ and $H^{*}(\Sigma_{a})$ is generated by $x_{1}$ and $x_{2}$, the automorphism of $H^{*}(\Sigma_{a})$ is determined by the images of $x_{1}$ and $x_{2}$. By direct computations one can see that there are $8$ automorphisms of $H^{*}(\Sigma_{a})$. Among them, there are $4$ diffeomorphisms whose representation matrices with respect to $x_{1},x_{2}$ are the following upper triangular matrices

Remaining $4$ automorphisms have different forms by the parity of $a$:

• •

  In case when $a$ is even, representation matrices are

  $$\pm\begin{pmatrix}\frac{a}{2}&\frac{a^{2}}{4}-1\\ -1&-\frac{a}{2}\end{pmatrix},\pm\begin{pmatrix}\frac{a}{2}&\frac{a^{2}}{4}+1\\ -1&-\frac{a}{2}\end{pmatrix}.$$

• •

  In case when $a$ is odd, representation matrices are

  $$\pm\begin{pmatrix}a&\frac{a^{2}-1}{2}\\ -2&-a\end{pmatrix},\pm\begin{pmatrix}a&\frac{a^{2}+1}{2}\\ -2&-a\end{pmatrix}.$$

To show the strong cohomological rigidity for Hirzebruch surfaces, we need the following: for any automorphism $\varphi$ of $H^{*}(\Sigma_{0})$ (respectively, $H^{*}(\Sigma_{1})$), there exists a diffeomorphism $f$ of $\Sigma_{0}$ (respectively, $\Sigma_{1}$) such that $f^{*}=\varphi$.

For $\ell\in\mathbb{C}P^{1}$, we denote by $\ell^{\perp}\in\mathbb{C}P^{1}$ the orthogonal complement of $\ell$ in $\mathbb{C}^{2}$. Let $\varphi\thinspace\colon H^{*}(\Sigma_{0})\to H^{*}(\Sigma_{0})$ be an isomorphism. The representation matrix of $\varphi$ with respect to $x_{1}$, $x_{2}$ is a signed permutation matrix and vice versa. A signed permutation matrix of size $2$ is the identity matrix or a multiplication of several $\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$ and $\begin{pmatrix}-1&0\\ 0&1\end{pmatrix}$. Therefore $\varphi$ is induced by the identity map or a composition of diffeomorphisms $(\ell_{1},\ell_{2})\mapsto(\ell_{2},\ell_{1})$ and $(\ell_{1},\ell_{2})\mapsto(\ell_{1}^{\perp},\ell_{2})$ for $(\ell_{1},\ell_{2})\in\Sigma_{0}=\mathbb{C}P^{1}\times\mathbb{C}P^{1}$.

Let $a=\pm 1$. We will construct a diffeomorphism of $\Sigma_{a}$ explicitly for all automorphism of $H^{*}(\Sigma_{a})$. For a moment, we use column vector notation to represent elements in $\mathbb{C}^{2}$. By Proposition 2.4, $\Sigma_{a}$ is diffeomorphic to the quotient $(S^{3})^{2}/(S^{1})^{2}$ by the action of $(S^{1})^{2}$ on $(S^{3})^{2}$ given by

We identify $\Sigma_{a}$ with the quotient $(S^{3})^{2}/(S^{1})^{2}$ via the diffeomorphism. We denote by $\begin{bmatrix}\begin{pmatrix}z_{1}\\ w_{1}\end{pmatrix},\begin{pmatrix}z_{2}\\ w_{2}\end{pmatrix}\end{bmatrix}\in\Sigma_{a}$ the equivalence class of $(\begin{pmatrix}z_{1}\\ w_{1}\end{pmatrix},\begin{pmatrix}z_{2}\\ w_{2}\end{pmatrix})\in(S^{3})^{2}$. For a complex number $z$ and $k\in\mathbb{Z}$, we define

Let $f\thinspace\colon\Sigma_{a}\to\Sigma_{a}$ be the map given by

By direct computation we have that $f$ is well-defined. Moreover, $f$ admits a smooth inverse, because

is the first column vector of the special unitary matrix

We claim that $f^{*}(x_{1})=-2ax_{2}+x_{1}$. To see this, we consider the pull-back $L$ of the tautological line bundle of $\mathbb{C}P^{1}$ by $\pi_{2}\thinspace\colon\Sigma_{a}\to\mathbb{C}P^{1}$, that is,

Then the pull-back $f^{*}(L)$ is

On the other hand, $f^{*}(L)$ is isomorphic to $L_{1}^{\prime}\otimes L_{2}^{\otimes(-2a)}$, where $L_{1}^{\prime}$ and $L_{2}$ are defined as in the proof of Proposition 2.4. In fact, the map $L_{1}^{\prime}\otimes L_{2}^{\otimes(-2a)}\to f^{*}(L)$ given by

is a line bundle isomorphism. By definition, $c_{1}(L)=x_{1}$. The argument above shows that $c_{1}(f^{*}(L))=-2ax_{2}+x_{1}$. Therefore $f^{*}(x_{1})=-2ax_{2}+x_{1}$. By the same argument we have that $f^{*}(x_{2})=-x_{2}$. Let $g_{1}\thinspace\colon\Sigma_{a}\to\Sigma_{a}$ be the diffeomorphism given by

Then the representation matrix of $g_{1}^{*}$ is $\begin{pmatrix}1&a\\ 0&-1\end{pmatrix}$. Let $g_{2}\thinspace\colon\Sigma_{a}\to\Sigma_{a}$ be the diffeomorphism given by

Then the representation matrix of $g_{2}^{*}$ is $\begin{pmatrix}-1&-a\\ 0&1\end{pmatrix}$. These computations show that the diffeomorphism $\varphi$ is induced by one of the identity map or a composition of several $f$, $g_{1}$ and $g_{2}$.

Finally, we remark that $f$, $g_{1}$ and $g_{2}$ are equivariant with respect to the following $S^{1}$-action on $\Sigma_{a}$. We define the $S^{1}$-action on $\Sigma_{a}$ via