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Mosc. Math. J., 2007, Volume 7, Number 2, Pages 219–242 (Mi mmj280)  

This article is cited in 34 scientific papers (total in 35 papers)

Spaces of polytopes and cobordism of quasitoric manifolds

V. M. Buchstabera, T. E. Panovb, N. Rayc

a Steklov Mathematical Institute, Russian Academy of Sciences
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
c University of Manchester, Department of Mathematics

Abstract: Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular projective toric varieties (otherwise known as toric manifolds). By expressing the first and third authors' approach to the representability of cobordism classes in these terms, we simplify and correct two of their original proofs concerning quotient polytopes; the first relates to framed embeddings in the positive cone, and the second involves modifying the operation of connected sum to take account of orientations. Analogous polytopes provide an informative setting for several of the details.

Key words and phrases: Analogous polytopes, complex cobordism, connected sum, framing, omniorientation, quasitoric manifold, stable tangent bundle.

DOI: https://doi.org/10.17323/1609-4514-2007-7-2-219-242

Full text: http://www.ams.org/.../abst7-2-2007.html
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MSC: 55N22, 52B20, 14M25
Received: September 15, 2006
Language:

Citation: V. M. Buchstaber, T. E. Panov, N. Ray, “Spaces of polytopes and cobordism of quasitoric manifolds”, Mosc. Math. J., 7:2 (2007), 219–242

Citation in format AMSBIB
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\by V.~M.~Buchstaber, T.~E.~Panov, N.~Ray
\paper Spaces of polytopes and cobordism of quasitoric manifolds
\jour Mosc. Math.~J.
\yr 2007
\vol 7
\issue 2
\pages 219--242
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\crossref{https://doi.org/10.17323/1609-4514-2007-7-2-219-242}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. V. M. Buchstaber, N. Ray, “Universal equivariant genus and Krichever's formula”, Russian Math. Surveys, 62:1 (2007), 178–180  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. M. Masuda, T. E. Panov, “Semifree circle actions, Bott towers and quasitoric manifolds”, Sb. Math., 199:8 (2008), 1201–1223  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. T. E. Panov, “Toric Kempf–Ness Sets”, Proc. Steklov Inst. Math., 263 (2008), 150–162  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    4. Masuda M., Suh D.Y., “Classification problems of toric manifolds via topology”, Toric Topology, Contemporary Mathematics Series, 460, 2008, 273–286  crossref  mathscinet  zmath  isi
    5. A. A. Kustarev, “Equivariant almost complex structures on quasi-toric manifolds”, Russian Math. Surveys, 64:1 (2009), 156–158  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. A. A. Kustarev, “Equivariant Almost Complex Structures on Quasitoric Manifolds”, Proc. Steklov Inst. Math., 266 (2009), 133–141  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    7. Yu. M. Ustinovskii, “Doubling operation for polytopes and torus actions”, Russian Math. Surveys, 64:5 (2009), 952–954  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    8. Bahri A., Bendersky M., Cohen F.R., Gitler S., “Decompositions of the polyhedral product functor with applications to moment-angle complexes and related spaces”, Proc. Natl. Acad. Sci. USA, 106:30 (2009), 12241–12244  crossref  mathscinet  zmath  adsnasa  isi  scopus
    9. Buchstaber V., Panov T., Ray N., “Toric Genera”, Int. Math. Res. Not. IMRN, 2010, no. 16, 3207–3262  mathscinet  zmath  isi
    10. Bahri A., Bendersky M., Cohen F.R., Gitler S., “The polyhedral product functor: A method of decomposition for moment-angle complexes, arrangements and related spaces”, Adv. Math., 225:3 (2010), 1634–1668  crossref  mathscinet  zmath  isi  elib  scopus
    11. Choi Suyoung, Masuda Mikiya, Suh Dong Youp, “Quasitoric manifolds over a product of simplices”, Osaka J. Math., 47:1 (2010), 109–129  mathscinet  zmath  isi  elib
    12. V. M. Buchstaber, E. Yu. Bun'kova, “Krichever Formal Groups”, Funct. Anal. Appl., 45:2 (2011), 99–116  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. N. Yu. Erokhovets, “Moment-angle manifolds of simple $n$-polytopes with $n+3$ facets”, Russian Math. Surveys, 66:5 (2011), 1006–1008  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    14. A. A. Aizenberg, V. M. Buchstaber, “Nerve complexes and moment–angle spaces of convex polytopes”, Proc. Steklov Inst. Math., 275 (2011), 15–46  mathnet  crossref  mathscinet  isi  elib  elib
    15. Taras Panov, Yuri Ustinovsky, “Complex-analytic structures on moment-angle manifolds”, Mosc. Math. J., 12:1 (2012), 149–172  mathnet  crossref  mathscinet
    16. V. M. Buchstaber, “Complex cobordism and formal groups”, Russian Math. Surveys, 67:5 (2012), 891–950  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    17. A. A. Aizenberg, “Topological applications of Stanley-Reisner rings of simplicial complexes”, Trans. Moscow Math. Soc., 73 (2012), 37–65  mathnet  crossref  zmath  elib
    18. T. E. Panov, “Geometric structures on moment-angle manifolds”, Russian Math. Surveys, 68:3 (2013), 503–568  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    19. A. M. Vershik, A. P. Veselov, A. A. Gaifullin, B. A. Dubrovin, A. B. Zhizhchenko, I. M. Krichever, A. A. Mal'tsev, D. V. Millionshchikov, S. P. Novikov, T. E. Panov, A. G. Sergeev, I. A. Taimanov, “Viktor Matveevich Buchstaber (on his 70th birthday)”, Russian Math. Surveys, 68:3 (2013), 581–590  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    20. Hiroaki Ishida, Yukiko Fukukawa, Mikiya Masuda, “Topological toric manifolds”, Mosc. Math. J., 13:1 (2013), 57–98  mathnet  crossref  mathscinet
    21. A. A. Aizenberg, “Substitutions of polytopes and of simplicial complexes, and multigraded Betti numbers”, Trans. Moscow Math. Soc., 74 (2013), 175–202  mathnet  crossref  mathscinet  zmath  elib
    22. Trans. Moscow Math. Soc., 74 (2013), 203–216  mathnet  crossref  mathscinet  zmath  elib
    23. N. Yu. Erokhovets, “Buchstaber invariant theory of simplicial complexes and convex polytopes”, Proc. Steklov Inst. Math., 286 (2014), 128–187  mathnet  crossref  crossref  isi  elib  elib
    24. A. A. Aizenberg, M. Masuda, Seonjeong Park, Haozhi Zeng, “Toric origami structures on quasitoric manifolds”, Proc. Steklov Inst. Math., 288 (2015), 10–28  mathnet  crossref  crossref  isi  elib
    25. V. M. Buchstaber, A. A. Kustarev, “Embedding theorems for quasi-toric manifolds given by combinatorial data”, Izv. Math., 79:6 (2015), 1157–1183  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    26. Kuroki Sh., Masuda M., Yu L., “Small Covers, Infra-Solvmanifolds and Curvature”, Forum Math., 27:5 (2015), 2981–3004  crossref  mathscinet  zmath  isi  scopus
    27. Darby A., “Torus Manifolds in Equivariant Complex Bordism”, Topology Appl., 189 (2015), 31–64  crossref  mathscinet  zmath  isi  elib  scopus
    28. G. D. Solomadin, Yu. M. Ustinovskiy, “Projective toric polynomial generators in the unitary cobordism ring”, Sb. Math., 207:11 (2016), 1601–1624  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    29. Lu Zh., Wang W., “Examples of Quasitoric Manifolds as Special Unitary Manifolds”, Math. Res. Lett., 23:5 (2016), 1453–1468  crossref  mathscinet  zmath  isi
    30. Lu Zh., Panov T., “on Toric Generators in the Unitary and Special Unitary Bordism Rings”, Algebr. Geom. Topol., 16:5 (2016), 2865–2893  crossref  mathscinet  zmath  isi  scopus
    31. 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
    32. Buchstaber V.M., Erokhovets N.Yu., “Fullerenes, Polytopes and Toric Topology”, Combinatorial and Toric Homotopy: Introductory Lectures, Lecture Notes Series Institute For Mathematical Sciences National University of Singapore, 35, eds. Darby A., Grbic J., Lu Z., Wu J., World Scientific Publ Co Pte Ltd, 2018, 67–178  crossref  mathscinet  zmath  isi
    33. G. D. Solomadin, “Quasitoric Totally Normally Split Representatives in the Unitary Cobordism Ring”, Math. Notes, 105:5 (2019), 763–780  mathnet  crossref  crossref  mathscinet  isi  elib
    34. I. Yu. Limonchenko, T. E. Panov, G. Chernykh, “$SU$-bordism: structure results and geometric representatives”, Russian Math. Surveys, 74:3 (2019), 461–524  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    35. Li X., Wang G., “a Moment-Angle Manifold Whose Cohomology Has Torsion”, Homol. Homotopy Appl., 21:2 (2019), 199–212  crossref  mathscinet  zmath  isi  scopus
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