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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.


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MSC: 55N22, 52B20, 14M25
Received: September 15, 2006

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
\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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    This publication is cited in the following articles:
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    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
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    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
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    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
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    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
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    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
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