may have two Bott tower structures $\{B_j \mid j=0,\ldots, n\}$ and $\{B_j' \mid j=0,\ldots, n\}$. The question we are interested in here is whether the twist number (i.e., the number of nontrivial fibrations) of the two Bott tower structures are equal? If so, the twist number of a Bott manifold is well-defined.

On the other hand the cohomology ring of an $n$-stage Bott manifold $M$ is a truncated polynomial ring $$\label{eqn:cohomology}
    H^\ast(M) \cong {\mathbb{Z}}[x_1, \ldots, x_n]/I,$$ where $I=<x_j(x_j - f_j) : j=1, \ldots, n>$ and $f_j = \sum_{i=1}^{j-1}\c{j}{i}x_i$ with $\deg x_i=2$. If the fibration of the $j$-th stage of a Bott tower structure on $M$ is trivial, then we may assume that $f_j=0$. The number of nonzero $f_j$’s may depend on the choices of both generators of the cohomology ring $H^\ast(M)$ and Bott tower structures of $M$. The [*cohomological complexity of $M$*]{} is the minimal number of nonzero $f_j$’s among all possible such choices. It is obvious that cohomological complexity of $H^\ast(M)$ is less than or equal to the twist number of any Bott tower structure of $M$. In Theorem \[theorem:twist number = complexity\], we show that the twist number of any Bott tower structure of $M$ is equal to 