ON A GENERALIZATION OF HIRZEBRUCHS THEOREM TO BOTT TOWERS

Title & Authors
ON A GENERALIZATION OF HIRZEBRUCHS THEOREM TO BOTT TOWERS
Kim, Jin Hong;

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 $\small{B_n({\alpha}_1,{\ldots},{\alpha}_{n-1},{\alpha}_n)}$ and $\small{B_n({\alpha}_1,{\ldots},{\alpha}_{n-1},{\alpha}_n^{\prime})}$ are isomorphic to each other, as Bott towers if and only if both $\small{{\alpha}_n{\equiv}{\alpha}_n^{\prime}}$ mod 2 and ${\alpha}_n^2 Keywords Bott towers;Bott manifolds;Hirzebruch surfaces;toric varieties;Petrie`s conjecture;strong cohomological rigidity conjecture; Language English Cited by References 1. A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458-538. 2. R. Bott and H. Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964-1029. 3. R. Bott and L. W. Tu, Differential Forms in Algebraic Topology, Grad. Texts Math. 82, Springer, 1982. 4. V. Buchstaber and T. Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series, Vol. 24, Amer. Math. Soc., Providence, Rhode Island, 2002. 5. S. Choi and M. Masuda, Classification of${\mathbb{Q}}\$-trivial Bott towers, J. Symp. Geom. 10 (2012), no. 3, 447-462.

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