the cohomology of projectivised bundles, we can see that the cohomology of $B_j$ is a free module over $H^\ast(B_{j-1})$ on generators $1$ and $x_j$ of dimension $0$ and $2$ respectively. The ring structure of $H^\ast(B_j)$ is determined by a single relation $$x_j^2=c_1(\xi_{j-1})x_j$$ where $x_j$ is the first Chern class of the line bundle $\gamma_j$ which is the pull-back bundle of the tautological line bundle of $P(\mathbb C\oplus \xi_{j-1})=B_j$ via the projection $B_n\to B_j$. Since $c_1(\xi_{j-1})\in H^2(B_{j-1})$, we can write $$f_j:= c_1 (\xi_{j-1}) = \sum_{i=1}^{j-1}\c{j}{i}x_i.$$ Since complex line bundles are distinguished by their first Chern classes, Bott manifold $B_n$ is determined by the above list of integers ($\c{j}{i} : 1 \leq i < j \leq
n$).

It is convenient to organize the integers $\c{j}{i}$ into an $n
\times n$ upper triangular matrix, $$\label{eqn:lambda}
\Lambda = \left(\begin{array}{cccc}
  0 & \c{2}{1} & \cdots & \c{n}{1} \\
    & 0 & \cdots & \vdots \\
    &  & \ddots & \c{n}{n-1} \\
    &  &  & 0 \\
\end{array}
\right).$$ We call it the *associated matrix* of the Bott tower.

One of the basic questions in vector bundle theory is to determine when two bundles with equal characteristic classes are isomorphic. In particular, we would like to 