On a generalization of Hirzebruch's theorem to Bott towers
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The primary aim of this paper is to generalize a theorem of Hirzebruch for the complex $2$-dimensional Bott manifolds, usually called Hirzebruch surfaces, to more general Bott towers of height $n$. To do so, we first show that all complex vector bundles of rank $2$ over a Bott manifold are classified by their total Chern classes. As a consequence, in this paper we show 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 to each other, 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 cohomology ring of $B_{n-1}(\alpha_1,\ldots, \alpha_{n-1})$ over integer coefficients. This result will complete a circle of ideas initiated in \cite{Ishida} by Ishida. We also give some partial affirmative remarks toward the assertion that under certain condition our main result still holds to be true for two Bott manifolds just diffeomorphic, but not necessarily isomorphic, to each other.
Abstract
The primary aim of this paper is to generalize a theorem of Hirzebruch for the complex $2$-dimensional Bott manifolds, usually called Hirzebruch surfaces, to more general Bott towers of height $n$. To do so, we first show that all complex vector bundles of rank $2$ over a Bott manifold are classified by their total Chern classes. As a consequence, in this paper we show 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 to each other, 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 cohomology ring of $B_{n-1}(\alpha_1,\ldots, \alpha_{n-1})$ over integer coefficients. This result will complete a circle of ideas initiated in \cite{Ishida} by Ishida. We also give some partial affirmative remarks toward the assertion that under certain condition our main result still holds to be true for two Bott manifolds just diffeomorphic, but not necessarily isomorphic, to each other.
참고문헌 (19)
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