is to determine whether the class of toric (or quasitoric) manifolds is cohomologically rigid. A quasitoric manifold is a topological analogue of a toric manifold, which was first introduced by Davis and Januszkiewicz in [@DJ], see also [@BP].

Since the class of toric or quasitoric manifolds is too large to handle it is reasonable to restrict our attention to a smaller but an interesting subclass of manifolds. Namely, we would like to restrict our focus on Bott manifolds or cohomology Bott manifolds.

A *(complex) Bott tower* $\{B_j \mid j=0,\ldots, n\}$ of height $n$ (or *$n$-stage Bott tower*) is a sequence $$B_n\stackrel{\pi_n}\longrightarrow B_{n-1} \stackrel{\pi_{n-1}}\longrightarrow
\dots \stackrel{\pi_2}\longrightarrow B_1 \stackrel{\pi_1}\longrightarrow
B_0=\{\text{a point}\},$$ of manifolds $B_j = P( \underline{{\mathbb{C}}} \oplus \xi_{j-1} )$ where $\xi_{j-1}$ is a complex line bundle over $B_{j-1}$ for each $j=1,\ldots,n$. In this case we call $B_j$ the [*$j$-th stage Bott manifold*]{} of the Bott tower. A smooth manifold $M$ diffeomorphic to the top stage $B_n$ of a Bott tower is also called a Bott manifold, and in this case $\{B_j\mid j=0,\ldots n\}$ is callded a [*Bott tower structure* ]{} of $M$.

A Bott tower was first introduced by Bott and Samelson in [@BS], and later named as Bott tower in 