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Mat. Zametki, 2001, Volume 69, Issue 3, Pages 375–382 (Mi mz511)  

This article is cited in 18 scientific papers (total in 18 papers)

Three-Dimensional Manifolds Defined by Coloring a Simple Polytope

I. V. Izmest'ev

M. V. Lomonosov Moscow State University

Abstract: In the present paper we introduce and study a class of three-dimensional manifolds endowed with the action of the group $\mathbb Z_2^3$ whose orbit space is a simple convex polytope. These manifolds originate from three-dimensional polytopes whose faces allow a coloring into three colors with the help of the construction used for studying quasitoric manifolds. For such manifolds we prove the existence of an equivariant embedding into Euclidean space $\mathbb R^4$. We also describe the action on the set of operations of the equivariant connected sum and the equivariant Dehn surgery. We prove that any such manifold can be obtained from a finitely many three-dimensional tori with the canonical action of the group $\mathbb Z_2^3$ by using these operations.


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English version:
Mathematical Notes, 2001, 69:3, 340–346

Bibliographic databases:

UDC: 515.162.3+515.164.8
Received: 19.06.2000

Citation: I. V. Izmest'ev, “Three-Dimensional Manifolds Defined by Coloring a Simple Polytope”, Mat. Zametki, 69:3 (2001), 375–382; Math. Notes, 69:3 (2001), 340–346

Citation in format AMSBIB
\by I.~V.~Izmest'ev
\paper Three-Dimensional Manifolds Defined by Coloring a Simple Polytope
\jour Mat. Zametki
\yr 2001
\vol 69
\issue 3
\pages 375--382
\jour Math. Notes
\yr 2001
\vol 69
\issue 3
\pages 340--346

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    This publication is cited in the following articles:
    1. V. M. Buchstaber, T. E. Panov, “Torus actions, combinatorial topology, and homological algebra”, Russian Math. Surveys, 55:5 (2000), 825–921  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. M. Joswig, “The group of projectivities and colouring of the facets of a simple polytope”, Russian Math. Surveys, 56:3 (2001), 584–585  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. Nakayama, H, “The orientability of Small Covers and coloring simple polytopes”, Osaka Journal of Mathematics, 42:1 (2005), 243  mathscinet  zmath  isi
    4. N. Yu. Erokhovets, “Buchstaber invariant of simple polytopes”, Russian Math. Surveys, 63:5 (2008), 962–964  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. Zivaljevic, RT, “Combinatorial Groupoids, Cubical Complexes, and the Lovasz Conjecture”, Discrete & Computational Geometry, 41:1 (2009), 135  crossref  mathscinet  zmath  isi  scopus
    6. Kuroki Sh., “Operations on 3-Dimensional Small Covers”, Chin. Ann. Math. Ser. B, 31:3 (2010), 393–410  crossref  mathscinet  zmath  isi  elib  scopus
    7. A. A. Aizenberg, “Svyaz invariantov Bukhshtabera i obobschennykh khromaticheskikh chisel”, Dalnevost. matem. zhurn., 11:2 (2011), 113–139  mathnet
    8. Cao X., Lue Zh., “Cohomological Rigidity and the Number of Homeomorphism Types for Small Covers Over Prisms”, Topology Appl., 158:6 (2011), 813–834  crossref  mathscinet  zmath  isi  elib  scopus
    9. Lue Zh., Yu L., “Topological Types of 3-Dimensional Small Covers”, Forum Math., 23:2 (2011), 245–284  crossref  mathscinet  zmath  isi  scopus
    10. Nishimura Ya., “Combinatorial Constructions of Three-Dimensional Small Covers”, Pac. J. Math., 256:1 (2012), 177–199  crossref  mathscinet  zmath  isi  scopus
    11. 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
    12. Kolpakov A., Martelli B., Tschantz S., “Some Hyperbolic Three-Manifolds That Bound Geometrically”, Proc. Amer. Math. Soc., 143:9 (2015), 4103–4111  crossref  mathscinet  zmath  isi  elib  scopus
    13. Kuroki Sh., “An Orlik-Raymond Type Classification of Simply Connected 6-Dimensional Torus Manifolds With Vanishing Odd-Degree Cohomology”, Pac. J. Math., 280:1 (2016), 89–114  crossref  mathscinet  zmath  isi  scopus
    14. Kuroki Sh., Lu Zh., “Projective bundles over small covers and the bundle triviality problem”, Forum Math., 28:4 (2016), 761–781  crossref  mathscinet  zmath  isi  elib  scopus
    15. Ayzenberg A., “Buchstaber Invariant, Minimal Non-Simplices and Related”, Osaka J. Math., 53:2 (2016), 377–395  mathscinet  zmath  isi  elib
    16. A. Yu. Vesnin, “Right-angled polyhedra and hyperbolic 3-manifolds”, Russian Math. Surveys, 72:2 (2017), 335–374  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    17. Chen B., Lu Zh., Yu L., “Self-Dual Binary Codes From Small Covers and Simple Polytopes”, Algebr. Geom. Topol., 18:5 (2018), 2729–2767  crossref  mathscinet  zmath  isi  scopus
    18. N. Yu. Erokhovets, “Three-Dimensional Right-Angled Polytopes of Finite Volume in the Lobachevsky Space: Combinatorics and Constructions”, Proc. Steklov Inst. Math., 305 (2019), 78–134  mathnet  crossref  crossref  isi  elib
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