Classification of Bott towers by matrix

Abstract

A criterion for the classification of Bott towers is presented, i.e., two ABott towers B_∗(A) and B_∗(A') are isomorphic if and only if the matrices A and A' are equivalent. The equivalence relation is defined by two operations on matrices. And it is based on the observation that any Bott tower B_∗(A) is uniquely determined by its structure matrix A, which is a strictly upper triangular integer matrix. The classification of Bott towers is closely related to the cohomological rigidity problem for both Bott towers and Bott manifolds.

Keywords

Bott manifold / Bott tower / cohomological rigidity / toric manifold

References

[1] Bott R, Samelson H. Applications of the theory of Morse to symmetric spaces. Amer J Math, 1958, 80: 964–1029

[2] Choi S. Classification of Bott manifolds up to dimension eight. arXiv: 1112.2321

[3] Choi S, Masuda M. Classification of Q-trivial Bott manifolds. J Symplectic Geom, 2012, 10(3): 447–461

[4] Choi S, Masuda M, Oum S. Classification of real Bott manifolds and acyclic digraphs. arXiv: 1006.4658

[5] Choi S, Masuda M, Suh D. Topological classification of generalized Bott towers. Trans Amer Math Soc, 2010, 362: 1097–1112

[6] Choi S, Masuda M, Suh D. Rigidity problems in toric topology: a survey. Proc Steklov Inst Math, 2011, 275(1): 177–190

[7] Choi S, Suh D. Properties of Bott manifolds and cohomological rigidity. Algebra Geom Topol, 2011, 11(2): 1053–1076

[8] Duan H. The degree of a Schubert variety. Adv Math, 2003, 180(1): 112–133

[9] Grossberg M, Karshon Y. Bott towers, complete integrability, and the extended character of representations. Duke Math J, 1994, 76: 23–58

[10] Ishida H. Filtered cohomological rigidity of Bott towers. Osaka J Math, 2012, 49: 515–522

[11] Lü Z. Lectures on elements of transformation groups and orbifolds. In: Ji L Z, Yau S T, eds. Transformation Groups and Moduli Spaces of Curves. Adv Lectures in Mathematics, Vol 16. Boston/Beijing: International Press/Higher Education Press, 2011, 239–276

[12] Masuda M, Panov T E. Semi-free circle actions, Bott towers, and quasitoric manifolds. Mat Sb, 2008, 199(8): 95–122

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Higher Education Press and Springer-Verlag Berlin Heidelberg