Classification of real Bott manifolds and acyclic digraphs
HTML articles powered by AMS MathViewer
- by Suyoung Choi, Mikiya Masuda and Sang-il Oum PDF
- Trans. Amer. Math. Soc. 369 (2017), 2987-3011 Request permission
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.
References
- C. Allday and V. Puppe, Cohomological methods in transformation groups, Cambridge Studies in Advanced Mathematics, vol. 32, Cambridge University Press, Cambridge, 1993. MR 1236839, DOI 10.1017/CBO9780511526275
- Jørgen Bang-Jensen and Gregory Gutin, Digraphs, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 2001. Theory, algorithms and applications. MR 1798170
- André Bouchet, Connectivity of isotropic systems, Combinatorial Mathematics: Proceedings of the Third International Conference (New York, 1985) Ann. New York Acad. Sci., vol. 555, New York Acad. Sci., New York, 1989, pp. 81–93. MR 1018611, DOI 10.1111/j.1749-6632.1989.tb22439.x
- André Bouchet, Digraph decompositions and Eulerian systems, SIAM J. Algebraic Discrete Methods 8 (1987), no. 3, 323–337. MR 897733, DOI 10.1137/0608028
- Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
- Leonard S. Charlap, Compact flat riemannian manifolds. I, Ann. of Math. (2) 81 (1965), 15–30. MR 170305, DOI 10.2307/1970379
- Suyoung Choi, The number of small covers over cubes, Algebr. Geom. Topol. 8 (2008), no. 4, 2391–2399. MR 2465745, DOI 10.2140/agt.2008.8.2391
- S. Choi and S. Oum, Real Bott manifolds and acyclic digraphs, arXiv:1002.4704.
- S. Choi and H. Park, On the cohomology and their torsion of real toric objects, to appear in Forum Math., arXiv:1311.7056.
- Suyoung Choi and Dong Youp Suh, Properties of Bott manifolds and cohomological rigidity, Algebr. Geom. Topol. 11 (2011), no. 2, 1053–1076. MR 2792373, DOI 10.2140/agt.2011.11.1053
- P. E. Conner and Frank Raymond, Injective operations of the toral groups, Topology 10 (1971), 283–296. MR 281218, DOI 10.1016/0040-9383(71)90021-8
- Lars Eirik Danielsen and Matthew G. Parker, Directed graph representation of half-rate additive codes over $\textrm {GF}(4)$, Des. Codes Cryptogr. 59 (2011), no. 1-3, 119–130. MR 2781603, DOI 10.1007/s10623-010-9469-6
- D. G. Fon-Der-Flaass, Local complementations of simple and oriented graphs, Sibirsk. Zh. Issled. Oper. 1 (1994), no. 1, 43–62, 87 (Russian, with Russian summary). MR 1292836
- Michael Grossberg and Yael Karshon, Bott towers, complete integrability, and the extended character of representations, Duke Math. J. 76 (1994), no. 1, 23–58. MR 1301185, DOI 10.1215/S0012-7094-94-07602-3
- Stephen Halperin, Rational homotopy and torus actions, Aspects of topology, London Math. Soc. Lecture Note Ser., vol. 93, Cambridge Univ. Press, Cambridge, 1985, pp. 293–306. MR 787835
- Hiroaki Ishida, Symplectic real Bott manifolds, Proc. Amer. Math. Soc. 139 (2011), no. 8, 3009–3014. MR 2801640, DOI 10.1090/S0002-9939-2011-10729-9
- Yoshinobu Kamishima and Mikiya Masuda, Cohomological rigidity of real Bott manifolds, Algebr. Geom. Topol. 9 (2009), no. 4, 2479–2502. MR 2576506, DOI 10.2140/agt.2009.9.2479
- Yoshinobu Kamishima and Admi Nazra, Seifert fibred structure and rigidity on real Bott towers, Discrete groups and geometric structures, Contemp. Math., vol. 501, Amer. Math. Soc., Providence, RI, 2009, pp. 103–122. MR 2581918, DOI 10.1090/conm/501/09843
- M. Masuda, Classification of real Bott manifolds, arXiv:0809.2178.
- M. Masuda and T. E. Panov, Semi-free circle actions, Bott towers, and quasitoric manifolds, Mat. Sb. 199 (2008), no. 8, 95–122 (Russian, with Russian summary); English transl., Sb. Math. 199 (2008), no. 7-8, 1201–1223 (2008). MR 2452268, DOI 10.1070/SM2008v199n08ABEH003959
- A. Nazra, Real Bott tower, Tokyo Metropolitan University, Master Thesis 2008.
- Admi Nazra, Diffeomorphism classes of real Bott manifolds, Tokyo J. Math. 34 (2011), no. 1, 229–260. MR 2866645, DOI 10.3836/tjm/1313074453
- Sang-il Oum, Rank-width and vertex-minors, J. Combin. Theory Ser. B 95 (2005), no. 1, 79–100. MR 2156341, DOI 10.1016/j.jctb.2005.03.003
- Sang-il Oum and Paul Seymour, Approximating clique-width and branch-width, J. Combin. Theory Ser. B 96 (2006), no. 4, 514–528. MR 2232389, DOI 10.1016/j.jctb.2005.10.006
- R. W. Robinson, Counting unlabeled acyclic digraphs, Combinatorial mathematics, V (Proc. Fifth Austral. Conf., Roy. Melbourne Inst. Tech., Melbourne, 1976) Lecture Notes in Math., Vol. 622, Springer, Berlin, 1977, pp. 28–43. MR 0476569
- A. Suciu and A. Trevisan, Real toric varieties and abelian covers of generalized Davis-Januszkiewicz spaces, preprint, 2012.
- A. Trevisan, Generalized Davis-Januszkiewicz spaces and their applications in algebra and topology, Ph.D. thesis, Vrije University Amsterdam, 2012; available at http://dspace.ubvu.vu.nl/handle/1871/32835.
- M. Van den Nest, J. Dehaene and B. De Moor, Graphical description of the action of local Clifford transformations on graph states, Phys. Rev. A 69(2) 022316 (2004).
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
- MR Author ID: 203919
- 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
- MR Author ID: 765385
- Email: sangil@kaist.edu
- 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). - © Copyright 2016 American Mathematical Society
- 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
- MathSciNet review: 3592535