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

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.

DOI: https://doi.org/10.4213/mzm511

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

Bibliographic databases:

UDC: 515.162.3+515.164.8

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
\Bibitem{Izm01} \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 \mathnet{http://mi.mathnet.ru/mz511} \crossref{https://doi.org/10.4213/mzm511} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1846836} \zmath{https://zbmath.org/?q=an:0991.57016} \transl \jour Math. Notes \yr 2001 \vol 69 \issue 3 \pages 340--346 \crossref{https://doi.org/10.1023/A:1010231424507} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000169324200007} 

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• https://doi.org/10.4213/mzm511
• http://mi.mathnet.ru/eng/mz/v69/i3/p375

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
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5. Zivaljevic, RT, “Combinatorial Groupoids, Cubical Complexes, and the Lovasz Conjecture”, Discrete & Computational Geometry, 41:1 (2009), 135
6. Kuroki Sh., “Operations on 3-Dimensional Small Covers”, Chin. Ann. Math. Ser. B, 31:3 (2010), 393–410
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11. N. Yu. Erokhovets, “Buchstaber invariant theory of simplicial complexes and convex polytopes”, Proc. Steklov Inst. Math., 286 (2014), 128–187
12. Kolpakov A., Martelli B., Tschantz S., “Some Hyperbolic Three-Manifolds That Bound Geometrically”, Proc. Amer. Math. Soc., 143:9 (2015), 4103–4111
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
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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
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