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Classification of real Bott manifolds and acyclic digraphs


Authors: Suyoung Choi, Mikiya Masuda and Sang-il Oum
Journal: Trans. Amer. Math. Soc. 369 (2017), 2987-3011
MSC (2010): Primary 37F20, 57R91, 05C90; Secondary 53C25, 14M25
DOI: https://doi.org/10.1090/tran/6896
Published electronically: November 8, 2016
MathSciNet review: 3592535
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Abstract: We completely characterize real Bott manifolds up to affine diffeomorphism in terms of three simple matrix operations on square binary matrices obtained from strictly upper triangular matrices by permuting rows and columns simultaneously. We also prove that any graded ring isomorphism between the cohomology rings of real Bott manifolds with $ \mathbb{Z}/2$ coefficients is induced by an affine diffeomorphism between the real Bott manifolds.

Our characterization can also be described in terms of graph operations on directed acyclic graphs. Using this combinatorial interpretation, we prove that the decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors. Finally, we produce some numerical invariants of real Bott manifolds from the viewpoint of graph theory and discuss their topological meaning. As a by-product, we prove that the toral rank conjecture holds for real Bott manifolds.


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Additional Information

Suyoung Choi
Affiliation: Department of Mathematics, Ajou University, San 5, Woncheondong, Yeongtonggu, Suwon 16499, Republic of Korea
Email: schoi@ajou.ac.kr

Mikiya Masuda
Affiliation: Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan
Email: masuda@sci.osaka-cu.ac.jp

Sang-il Oum
Affiliation: Department of Mathematical Sciences, KAIST, 291 Daehakro, Yuseong-gu, Daejeon 34141, Republic of Korea
Email: sangil@kaist.edu

DOI: https://doi.org/10.1090/tran/6896
Keywords: Real toric manifold, real Bott manifold, real Bott tower, acyclic digraph, local complementation, flat Riemannian manifold, toral rank conjecture
Received by editor(s): July 6, 2013
Received by editor(s) in revised form: May 24, 2014, December 1, 2015, and January 3, 2016
Published electronically: November 8, 2016
Additional Notes: The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2011-0024975) and a TJ Park Science Fellowship.
The second author was partially supported by Grant-in-Aid for Scientific Research 19204007, 22540094, and 25400095.
The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2011-0011653).
Article copyright: © Copyright 2016 American Mathematical Society