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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2024, Volume 326, Pages 275–292
DOI: https://doi.org/10.4213/tm4401 (Mi tm4401) 		 
Bier Spheres and Toric Topology

Ivan Yu. Limonchenkoa, Matvey A. Sergeevb

a Mathematical Institute of the Serbian Academy of Sciences and Arts (SASA), Belgrade, Serbia
b National Research University Higher School of Economics, Moscow, Russia
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DOI: https://doi.org/10.4213/tm4401
Abstract: We compute the real and complex Buchstaber numbers of an arbitrary Bier sphere. In dimension two, we identify all the 13 different combinatorial types of Bier spheres and show that 12 of them are nerve complexes of nestohedra, while the remaining one is a nerve complex of a generalized permutohedron. As an application of our results, we construct a regular normal fan for each of those 13 Delzant polytopes, compute the cohomology rings of the corresponding nonsingular projective toric varieties, and examine the orientability of the corresponding small covers.
Keywords: Bier sphere, nestohedron, Delzant polytope, quasitoric manifold, small cover, Buchstaber number.
Funding agency Grant number
Ministry of Science, Technological Development and Innovation of the Republic of Serbia
HSE Basic Research Program
I. Yu. Limonchenko was supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia through the Mathematical Institute of the Serbian Academy of Sciences and Arts. M. A. Sergeev's work was carried out within the project “Mirror Laboratories” of HSE University, Russia.
Received: December 29, 2023
Revised: March 25, 2024
Accepted: April 28, 2024
English version:
Proceedings of the Steklov Institute of Mathematics, 2024, Volume 326, Pages 252–268
DOI: https://doi.org/10.1134/S0081543824040114
Document Type: Article
UDC: 515.145
Language: Russian
Citation: Ivan Yu. Limonchenko, Matvey A. Sergeev, “Bier Spheres and Toric Topology”, Topology, Geometry, Combinatorics, and Mathematical Physics, Collected papers. Dedicated to Victor Matveevich Buchstaber on the occasion of his 80th birthday, Trudy Mat. Inst. Steklova, 326, Steklov Math. Inst., Moscow, 2024, 275–292; Proc. Steklov Inst. Math., 326 (2024), 252–268
Citation in format AMSBIB
\Bibitem{LimSer24}
\by Ivan~Yu.~Limonchenko, Matvey~A.~Sergeev
\paper Bier Spheres and Toric Topology
\inbook Topology, Geometry, Combinatorics, and Mathematical Physics
\bookinfo Collected papers. Dedicated to Victor Matveevich Buchstaber on the occasion of his 80th birthday
\serial Trudy Mat. Inst. Steklova
\yr 2024
\vol 326
\pages 275--292
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4401}
\crossref{https://doi.org/10.4213/tm4401}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2024
\vol 326
\pages 252--268
\crossref{https://doi.org/10.1134/S0081543824040114}
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    		Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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