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This article is cited in 7 scientific papers (total in 7 papers)
Toral Rank Conjecture for Moment-Angle Complexes
Yu. M. Ustinovskii M. V. Lomonosov Moscow State University
Abstract:
We consider an operation $K\mapsto L(K)$ on the set of simplicial complexes, which we call the “doubling operation.” This combinatorial operation was recently introduced in toric topology in an unpublished paper of Bahri, Bendersky, Cohen and Gitler on generalized moment-angle complexes (also known as $K$-powers). The main property of the doubling operation is that the moment-angle complex $\mathscr Z_K$ can be identified with the real moment-angle complex $\mathbb R\mathscr Z_{L(K)}$ for the double $L(K)$. By way of application, we prove the toral rank conjecture for the spaces $\mathscr{Z}_K$ by providing a lower bound for the rank of the cohomology ring of the real moment-angle complexes $\mathbb R\mathscr Z_K$. This paper can be viewed as a continuation of the author's previous paper, where the doubling operation for polytopes was used to prove the toral rank conjecture for moment-angle manifolds.
Keywords:
moment-angle manifold, moment-angle complex, simplicial complex, doubling, toral rank conjecture, cohomology rank, Mayer–Vietoris sequence
DOI:
https://doi.org/10.4213/mzm8732
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English version:
Mathematical Notes, 2011, 90:2, 279–283
Bibliographic databases:
UDC:
515.146.39 Received: 27.01.2010 Revised: 24.10.2010
Citation:
Yu. M. Ustinovskii, “Toral Rank Conjecture for Moment-Angle Complexes”, Mat. Zametki, 90:2 (2011), 300–305; Math. Notes, 90:2 (2011), 279–283
Citation in format AMSBIB
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Linking options:
http://mi.mathnet.ru/eng/mz8732https://doi.org/10.4213/mzm8732 http://mi.mathnet.ru/eng/mz/v90/i2/p300
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Yu. M. Ustinovskii, “O pochti svobodnykh deistviyakh tora i gipoteze Khorroksa”, Dalnevost. matem. zhurn., 12:1 (2012), 98–107
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Yu Li, “Small covers and the Halperin-Carlsson conjecture”, Pacific J. Math., 256:2 (2012), 489–507
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Bahri A., Bendersky M., Cohen F.R., Gitler S., “Operations on Polyhedral Products and a New Topological Construction of Infinite Families of Toric Manifolds”, Homol. Homotopy Appl., 17:2 (2015), 137–160
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Cho H.W., “Periodicity and the values of the real Buchstaber invariants”, J. Math. Soc. Jpn., 68:4 (2016), 1695–1723
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Park H., “Wedge Operations and Doubling Operations of Real Toric Manifolds”, Chin. Ann. Math. Ser. B, 38:6 (2017), 1321–1334
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Vidaurre E., “On Polyhedral Product Spaces Over Polyhedral Joins”, Homol. Homotopy Appl., 20:2 (2018), 259–280
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Yu L., “On Free Z(P)-Torus Actions in Dimensions Two and Three”, Sci. China-Math., 62:2 (2019), 391–410
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