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Mat. Sb., 2008, Volume 199, Number 8, Pages 95–122 (Mi msb4110)  

This article is cited in 32 scientific papers (total in 32 papers)

Semifree circle actions, Bott towers and quasitoric manifolds

M. Masudaa, T. E. Panovbc*

a Osaka City University
b Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)
c M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: A Bott tower is the total space of a tower of fibre bundles with base $\mathbb C P^1$ and fibres $\mathbb C P^1$. Every Bott tower of height $n$ is a smooth projective toric variety whose moment polytope is combinatorially equivalent to an $n$-cube. A circle action is semifree if it is free on the complement to the fixed points. We show that a quasitoric manifold over a combinatorial $n$-cube admitting a semifree action of a 1-dimensional subtorus with isolated fixed points is a Bott tower. Then we show that every Bott tower obtained in this way is topologically trivial, that is, homeomorphic to a product of 2-spheres. This extends a recent result of Il'inskiǐ, who showed that a smooth compact toric variety admitting a semifree action of a 1-dimensional subtorus with isolated fixed points is homeomorphic to a product of 2-spheres, and makes a further step towards our understanding of Hattori's problem of semifree circle actions. Finally, we show that if the cohomology ring of a quasitoric manifold is isomorphic to that of a product of 2-spheres, then the manifold is homeomorphic to this product. In the case of Bott towers the homeomorphism is actually a diffeomorphism.
Bibliography: 18 titles.
* Author to whom correspondence should be addressed


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English version:
Sbornik: Mathematics, 2008, 199:8, 1201–1223

Bibliographic databases:

UDC: 515.14+515.16
MSC: Primary 57S15; Secondary 14M25
Received: 20.11.2007 and 04.03.2008

Citation: M. Masuda, T. E. Panov, “Semifree circle actions, Bott towers and quasitoric manifolds”, Mat. Sb., 199:8 (2008), 95–122; Sb. Math., 199:8 (2008), 1201–1223

Citation in format AMSBIB
\by M.~Masuda, T.~E.~Panov
\paper Semifree circle actions, Bott towers and quasitoric manifolds
\jour Mat. Sb.
\yr 2008
\vol 199
\issue 8
\pages 95--122
\jour Sb. Math.
\yr 2008
\vol 199
\issue 8
\pages 1201--1223

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    This publication is cited in the following articles:
    1. Choi Suyoung, “The number of small covers over cubes”, Algebr. Geom. Topol., 8:4 (2008), 2391–2399  crossref  mathscinet  zmath  isi  elib  scopus
    2. Kamishima Y., Masuda M., “Cohomological rigidity of real Bott manifolds”, Algebr. Geom. Topol., 9:4 (2009), 2479–2502  crossref  mathscinet  zmath  isi  scopus  scopus
    3. Choi Suyoung, Masuda M., Suh Dong Youp, “Topological classification of generalized Bott towers”, Trans. Amer. Math. Soc., 362:2 (2010), 1097–1112  crossref  mathscinet  zmath  isi  elib  scopus
    4. Choi S., Panov T., Suh D.Y., “Toric cohomological rigidity of simple convex polytopes”, J. Lond. Math. Soc. (2), 82:2 (2010), 343–360  crossref  mathscinet  zmath  isi  elib  scopus
    5. Buchstaber V., Panov T., Ray N., “Toric genera”, Int. Math. Res. Not. IMRN, 2010, no. 16, 3207–3262  mathscinet  zmath  isi  elib
    6. Choi Suyoung, Masuda M., Suh Dong Youp, “Quasitoric manifolds over a product of simplices”, Osaka J. Math., 47:1 (2010), 109–129  mathscinet  zmath  isi  elib
    7. Choi Suyoung, Suh Dong Youp, “Properties of Bott manifolds and cohomological rigidity”, Algebr. Geom. Topol, 11:2 (2011), 1053–1076  crossref  mathscinet  zmath  isi  elib  scopus
    8. Choi Suyoung, Kuroki Shintarô, “Topological classification of torus manifolds which have codimension one extended actions”, Algebr. Geom. Topol., 11:5 (2011), 2655–2679  crossref  mathscinet  zmath  isi  elib  scopus
    9. Proc. Steklov Inst. Math., 275 (2011), 177–190  mathnet  crossref  mathscinet  isi  elib  elib
    10. Yu L., “Discrete group actions and generalized real Bott manifolds”, Math. Res. Lett., 18:6 (2011), 1289–1303  crossref  mathscinet  zmath  isi  elib  scopus
    11. Wiemeler M., “Remarks on the classification of quasitoric manifolds up to equivariant homeomorphism”, Arch Math. (Basel), 98:1 (2012), 71–85  crossref  mathscinet  zmath  isi  scopus
    12. S. Theriault, “A homotopy-theoretic rigidity property of Bott manifolds”, Dalnevost. matem. zhurn., 12:1 (2012), 89–97  mathnet
    13. Choi S., Masuda M., “Classification of $\mathbb Q$-trivial Bott manifolds”, J. Symplectic Geom., 10:3 (2012), 447–461  crossref  mathscinet  zmath  isi  scopus
    14. Choi S., Suh D.Y., “Strong cohomological rigidity of a product of projective spaces”, Bull. Korean. Math. Soc., 49:4 (2012), 761–765  crossref  mathscinet  zmath  isi
    15. Choi S., Park S., Suh D.Y., “Topological classification of quasitoric manifolds with second Betti number 2”, Pacific J. Math., 256:1 (2012), 19–49  crossref  mathscinet  zmath  isi  elib  scopus
    16. Wiemeler M., “Quasitoric Manifolds Homeomorphic to Homogeneous Spaces”, Osaka J. Math., 50:1 (2013), 153–160  mathscinet  zmath  isi
    17. Proc. Steklov Inst. Math., 286 (2014), 285–307  mathnet  crossref  crossref  isi  elib
    18. Suyoung Choi, Mikiya Masuda, Satoshi Murai, “Invariance of Pontrjagin classes for Bott manifolds”, Algebr. Geom. Topol, 15:2 (2015), 965  crossref  mathscinet  zmath
    19. Choi S., “Classification of Bott Manifolds Up To Dimension 8”, Proc. Edinb. Math. Soc., 58:3 (2015), 653–659  crossref  mathscinet  zmath  isi  scopus
    20. Bai Q., Li F., “Classification of Bott towers by matrix”, Front. Math. China, 11:2 (2016), 255–268  crossref  mathscinet  zmath  isi  scopus
    21. Choi S., Park H., “Wedge operations and torus symmetries”, Tohoku Math. J., 68:1 (2016), 91–138  crossref  mathscinet  zmath  isi
    22. Kim J.H., “on a Generalization of Hirzebruch'S Theorem To Bott Towers”, J. Korean. Math. Soc., 53:2 (2016), 331–346  crossref  mathscinet  zmath  isi  scopus
    23. Choi S., Masuda M., Oum S.-i., “Classification of real Bott manifolds and acyclic digraphs”, Trans. Am. Math. Soc., 369:4 (2017), 2987–3011  crossref  mathscinet  zmath  isi  scopus
    24. V. M. Buchstaber, N. Yu. Erokhovets, M. Masuda, T. E. Panov, S. Park, “Cohomological rigidity of manifolds defined by 3-dimensional polytopes”, Russian Math. Surveys, 72:2 (2017), 199–256  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    25. Dsouza R., “On the Topology of Real Bott Manifolds”, Indian J. Pure Appl. Math., 49:4 (2018), 743–763  crossref  mathscinet  zmath  isi  scopus
    26. Lafont J.-F., Sorcar G., Zheng F., “Some Kahler Structures on Products of 2-Spheres”, Enseign. Math., 64:1-2 (2018), 127–142  crossref  mathscinet  zmath  isi
    27. Boyer Ch.P., Calderbank D.M.J., Tonnesen-Friedman Ch.W., “The Kahler Geometry of Bott Manifolds”, Adv. Math., 350 (2019), 1–62  crossref  mathscinet  zmath  isi
    28. Megumi Harada, Tatsuya Horiguchi, Mikiya Masuda, Seonjeong Park, “The Volume Polynomial of Regular Semisimple Hessenberg Varieties and the Gelfand–Zetlin Polytope”, Proc. Steklov Inst. Math., 305 (2019), 318–344  mathnet  crossref  crossref  mathscinet  isi  elib
    29. Lee E., Masuda M., “Generic Torus Orbit Closures in Schubert Varieties”, J. Comb. Theory Ser. A, 170 (2020), UNSP 105143  crossref  mathscinet  isi
    30. Hasui Sh., Kuwata H., Masuda M., Park S., “Classification of Toric Manifolds Over An N-Cube With One Vertex Cut”, Int. Math. Res. Notices, 2020:16 (2020), 4890–4941  crossref  mathscinet  isi
    31. Apostolov V., Calderbank D.M.J., Gauduchon P., Legendre E., “Levi-Kahler Reduction of Cr Structures, Products of Spheres, and Toric Geometry”, Math. Res. Lett., 27:6 (2020), 1565–1629  crossref  mathscinet  isi
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