\ar[d]& & \ar[d] \pi_\beta^\ast(\eta_{\alpha}) \cong B_n \\
    B_{n-1}' = P({\mathbb{C}}\oplus \gamma^\alpha) \ar[dr]_{
         \pi_\alpha} & & \ar[dl]^{\pi_\beta
        }
    P({\mathbb{C}}\oplus \gamma^\beta) =B_{n-1} \\
    & B = B_{n-2} &
},$$ where $\pi_\beta^\ast(\eta_{\alpha})=\{ (x,y) \in P({\mathbb{C}}\oplus \gamma^\beta) \times P({\mathbb{C}}\oplus \gamma^\alpha) | \pi_\beta(x) = \pi_\alpha(y)\}$ and $\pi_\alpha^\ast(\eta_{\beta})=\{ (a,b) \in P({\mathbb{C}}\oplus \gamma^\alpha) \times P({\mathbb{C}}\oplus \gamma^\beta) | \pi_\alpha(a) = \pi_\beta(b)\}.$ Therefore $\pi_\beta^\ast(\eta_{\alpha}) \cong \pi_\alpha^\ast(\eta_{\beta})$.

\[corollary:last twist\] If a Bott manifold has a one-twist Bott tower structure, then it has another Bott tower structure whose last stage is nontrivial and all other stages are trivial.

By successive applications of Lemma \[lemma:2-1\], we can push the trivial fibration down to lower levels.

\[corollary:columns with zeros\] If $B_n$ is a Bott tower with the associated matrix $$\Lambda=
\left(\begin{array}{cccc}
  0 & \c{2}{1} & \cdots  & \c{n}{1}   \\
    & 0 &  & \vdots \\
     &  & \ddots & \c{n}{n-1}\\
  &   &   & 0
\end{array}
\right)$$ such that $\c{k+1}{k} = \c{k+2}{k} = \cdots = \c{n}{k} = 0$. Then $B_n$ is diffeomorphic to a Bott tower with the associated matrix $$\Lambda'=
    \left(\begin{array}{ccccccccc}
0 &\c{2}{1}& \cdots & \c{k-1}{1}& \c{k+1}{1}& \c{k+2}{1}& \cdots& \c{n}{1}& \c{k}{1} \\   & 0 & \cdots & \c{k-1}{2}&\c{k+1}{2}&\c{k+2}{2}&\cdots&\c{n}{2}&\c{k}{2} \\   &   & \ddots & \vdots& \vdots& \vdots & \vdots&\vdots &\vdots\\
  &   & & 0 