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 Mat. Zametki, 2011, Volume 90, Issue 2, Pages 300–305 (Mi mz8732)

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 This combinatorial operation was recently introduced in toric topology in an unpublished paper of Bahri, Bendersky, Cohen and Gitler on generalized moment-angle complexes (also known as $K$-powers).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
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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• http://mi.mathnet.ru/eng/mz8732
• https://doi.org/10.4213/mzm8732
• http://mi.mathnet.ru/eng/mz/v90/i2/p300

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This publication is cited in the following articles:
1. Yu. M. Ustinovskii, “O pochti svobodnykh deistviyakh tora i gipoteze Khorroksa”, Dalnevost. matem. zhurn., 12:1 (2012), 98–107
2. Yu Li, “Small covers and the Halperin-Carlsson conjecture”, Pacific J. Math., 256:2 (2012), 489–507
3. 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
4. Cho H.W., “Periodicity and the values of the real Buchstaber invariants”, J. Math. Soc. Jpn., 68:4 (2016), 1695–1723
5. Park H., “Wedge Operations and Doubling Operations of Real Toric Manifolds”, Chin. Ann. Math. Ser. B, 38:6 (2017), 1321–1334
6. Vidaurre E., “On Polyhedral Product Spaces Over Polyhedral Joins”, Homol. Homotopy Appl., 20:2 (2018), 259–280
7. 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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