On a generalization of Hirzebruch's theorem to Bott towers (Q2813355)

A complex Bott tower \(\{ (B_j(\alpha_1,\ldots, \alpha_j),\pi_j)\}_{j=1}^n\) of height \(n\) is a sequence of \({\mathbb{C}}P^1\)-bundles NEWLINE\[NEWLINE {{B_n(\alpha_1,\ldots, \alpha_n)}}{\overset{\pi_n} \longrightarrow } {B_{n-1}(\alpha_1,\ldots, \alpha_{n-1})} \overset{\pi_{n-1}}\longrightarrow {\cdots} \overset{\pi_2}\longrightarrow{B_1(\alpha_1)}\overset{\pi_1}\longrightarrow B_0, NEWLINE\]NEWLINE where \(B_0\) is a point, \(B_j(\alpha_1,\ldots, \alpha_j)\) is the projectivization \( {\mathbb{P}}({\underline{\mathbb{C}}}\oplus {\mathbb{L}}_j)\) of the trivial complex line bundle \({\underline{\mathbb{C}}}\) and a complex line bundle \({\mathbb{L}}_j\) over \(B_{j-1}(\alpha_1,\ldots, \alpha_{j-1})\) with the first Chern class \(\alpha_j\). The \(\pi_j\) denote the projections. The manifold \(B_j(\alpha_1,\ldots, \alpha_j)\) is a toric manifold called a Bott manifold of height \(j\), or a \(j\)-step Bott manifold. An isomorphism between two Bott towers \(\{ (B_j(\alpha_1,\ldots, \alpha_j),\pi_j)\}_{j=1}^n\) and \(\{ (B_j(\alpha'_1,\ldots, \alpha'_j),\pi'_j)\}_{j=1}^n\) is a collection \(\{ F_j\}_{j=1}^n\) of diffeomorphisms NEWLINE\[NEWLINE F_j: B_j(\alpha_1,\ldots, \alpha_j)\to B_j(\alpha'_1,\ldots, \alpha'_j) NEWLINE\]NEWLINE that commute with the projection maps \(\pi_j\) and \(\pi'_j\).NEWLINENEWLINEIt follows from a result in [\textit{F. Hirzebruch}, Math. Ann. 124, 77--86 (1951; Zbl 0043.30302)] that \(B_2(\alpha_1,\alpha_2)\) is isomorphic (or diffeomorphic) to \(B_2(\alpha_1, \alpha'_2)\) if and only if \(\alpha_2 \equiv \alpha'_2 \mod 2\). The author generalizes this result to Bott towers of height \(n\). He first shows that all complex vector bundles of rank \(2\) over a Bott manifold are classified by their total Chern classes. As a consequence, he shows that two Bott manifolds \(B_n(\alpha_1,\ldots, \alpha_{n-1}, \alpha_n)\) and \(B_n(\alpha_1,\ldots, \alpha_{n-1},\alpha'_n)\) are isomorphic as Bott towers if and only if both \(\alpha_n\equiv \alpha'_n\mod 2\) and \(\alpha_n^2=(\alpha'_n)^2\) hold in the integral cohomology ring of \(B_{n-1}(\alpha_1,\ldots,\alpha_{n-1})\).