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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
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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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http://mi.mathnet.ru/eng/mmj280 http://mi.mathnet.ru/eng/mmj/v7/i2/p219
Citing articles on Google Scholar:
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This publication is cited in the following articles:
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V. M. Buchstaber, N. Ray, “Universal equivariant genus and Krichever's formula”, Russian Math. Surveys, 62:1 (2007), 178–180
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M. Masuda, T. E. Panov, “Semifree circle actions, Bott towers and quasitoric manifolds”, Sb. Math., 199:8 (2008), 1201–1223
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T. E. Panov, “Toric Kempf–Ness Sets”, Proc. Steklov Inst. Math., 263 (2008), 150–162
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Masuda M., Suh D.Y., “Classification problems of toric manifolds via topology”, Toric Topology, Contemporary Mathematics Series, 460, 2008, 273–286
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A. A. Kustarev, “Equivariant almost complex structures on quasi-toric manifolds”, Russian Math. Surveys, 64:1 (2009), 156–158
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A. A. Kustarev, “Equivariant Almost Complex Structures on Quasitoric Manifolds”, Proc. Steklov Inst. Math., 266 (2009), 133–141
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Yu. M. Ustinovskii, “Doubling operation for polytopes and torus actions”, Russian Math. Surveys, 64:5 (2009), 952–954
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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
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Buchstaber V., Panov T., Ray N., “Toric Genera”, Int. Math. Res. Not. IMRN, 2010, no. 16, 3207–3262
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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
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Choi Suyoung, Masuda Mikiya, Suh Dong Youp, “Quasitoric manifolds over a product of simplices”, Osaka J. Math., 47:1 (2010), 109–129
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V. M. Buchstaber, E. Yu. Bun'kova, “Krichever Formal Groups”, Funct. Anal. Appl., 45:2 (2011), 99–116
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N. Yu. Erokhovets, “Moment-angle manifolds of simple $n$-polytopes with $n+3$ facets”, Russian Math. Surveys, 66:5 (2011), 1006–1008
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A. A. Aizenberg, V. M. Buchstaber, “Nerve complexes and moment–angle spaces of convex polytopes”, Proc. Steklov Inst. Math., 275 (2011), 15–46
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Taras Panov, Yuri Ustinovsky, “Complex-analytic structures on moment-angle manifolds”, Mosc. Math. J., 12:1 (2012), 149–172
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V. M. Buchstaber, “Complex cobordism and formal groups”, Russian Math. Surveys, 67:5 (2012), 891–950
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A. A. Aizenberg, “Topological applications of Stanley-Reisner rings of simplicial complexes”, Trans. Moscow Math. Soc., 73 (2012), 37–65
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T. E. Panov, “Geometric structures on moment-angle manifolds”, Russian Math. Surveys, 68:3 (2013), 503–568
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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
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Hiroaki Ishida, Yukiko Fukukawa, Mikiya Masuda, “Topological toric manifolds”, Mosc. Math. J., 13:1 (2013), 57–98
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A. A. Aizenberg, “Substitutions of polytopes and of simplicial complexes, and multigraded Betti numbers”, Trans. Moscow Math. Soc., 74 (2013), 175–202
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Trans. Moscow Math. Soc., 74 (2013), 203–216
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N. Yu. Erokhovets, “Buchstaber invariant theory of simplicial complexes and convex polytopes”, Proc. Steklov Inst. Math., 286 (2014), 128–187
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A. A. Aizenberg, M. Masuda, Seonjeong Park, Haozhi Zeng, “Toric origami structures on quasitoric manifolds”, Proc. Steklov Inst. Math., 288 (2015), 10–28
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V. M. Buchstaber, A. A. Kustarev, “Embedding theorems for quasi-toric manifolds given by combinatorial data”, Izv. Math., 79:6 (2015), 1157–1183
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Kuroki Sh., Masuda M., Yu L., “Small Covers, Infra-Solvmanifolds and Curvature”, Forum Math., 27:5 (2015), 2981–3004
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Darby A., “Torus Manifolds in Equivariant Complex Bordism”, Topology Appl., 189 (2015), 31–64
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G. D. Solomadin, Yu. M. Ustinovskiy, “Projective toric polynomial generators in the unitary cobordism ring”, Sb. Math., 207:11 (2016), 1601–1624
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Lu Zh., Wang W., “Examples of Quasitoric Manifolds as Special Unitary Manifolds”, Math. Res. Lett., 23:5 (2016), 1453–1468
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Lu Zh., Panov T., “on Toric Generators in the Unitary and Special Unitary Bordism Rings”, Algebr. Geom. Topol., 16:5 (2016), 2865–2893
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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
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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
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G. D. Solomadin, “Quasitoric Totally Normally Split Representatives in the Unitary Cobordism Ring”, Math. Notes, 105:5 (2019), 763–780
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I. Yu. Limonchenko, T. E. Panov, G. Chernykh, “$SU$-bordism: structure results and geometric representatives”, Russian Math. Surveys, 74:3 (2019), 461–524
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Li X., Wang G., “a Moment-Angle Manifold Whose Cohomology Has Torsion”, Homol. Homotopy Appl., 21:2 (2019), 199–212
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