with $\deg x_j=2$. Here the numbers $\c{j}{i}$ can be determined by a Bott tower structure of $M$. Indeed, $\c{j}{i}$’s are the entries of the matrix (\[eqn:lambda\]). Hence if the fibration of the $i$-th stage of a Bott tower structure on $M$ is trivial, then we may assume that $f_j=0$. Therefore the number of nonzero $f_j$’s may depend not only on the choices of generators of the cohomology ring $H^\ast(M)$ but also the Bott tower structures of $M$. The [*cohomological complexity of $M$*]{} is the minimal number of nonzero $f_j$’s among all possible such choices.

In the following theorem we show that the twist number of any Bott tower structure of a Bott manifold $M$ is equal to the cohomological complexity of $M$. This, in particular, shows that the twist number of a Bott manifold is well-defined.

\[theorem:twist number = complexity\] Let $M$ be a Bott manifold. Then the twist number of any Bott tower structure of $M$ is equal to the cohomological complexity of $M$.

Let $$B_n \rightarrow B_{n-1} \rightarrow \cdots \rightarrow B_1 \rightarrow B_0 = \{\text{a point}\}$$ be a Bott tower structure of $M$ whose twist number is equal to $t$. By Corollary \[corollary:last twist\], we may assume that 