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The homotopy types of moment-angle complexes for flag complexes


Authors: Jelena Grbić, Taras Panov, Stephen Theriault and Jie Wu
Journal: Trans. Amer. Math. Soc. 368 (2016), 6663-6682
MSC (2010): Primary 55P15; Secondary 13F55
DOI: https://doi.org/10.1090/tran/6578
Published electronically: November 12, 2015
MathSciNet review: 3461047
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Abstract: We study the homotopy types of moment-angle complexes, or equivalently, of complements of coordinate subspace arrangements. The overall aim is to identify the simplicial complexes $ K$ for which the corresponding moment-angle complex $ \mathcal Z_{K}$ has the homotopy type of a wedge of spheres or a connected sum of sphere products. When $ K$ is flag, we identify in algebraic and combinatorial terms those $ K$ for which $ \mathcal Z_{K}$ is homotopy equivalent to a wedge of spheres, and give a combinatorial formula for the number of spheres in the wedge. This extends results of Berglund and Jöllenbeck on Golod rings and homotopy theoretical results of the first and third authors. We also establish a connection between minimally non-Golod rings and moment-angle complexes $ \mathcal Z_{K}$ which are homotopy equivalent to a connected sum of sphere products. We go on to show that for any flag complex $ K$ the loop spaces $ \Omega \mathcal Z_{K}$ and $ \Omega$$ \mbox {$D{\mskip -1mu}J\/$}(K)$ are homotopy equivalent to a product of spheres and loops on spheres when localised rationally or at any prime $ p\neq 2$.


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Additional Information

Jelena Grbić
Affiliation: School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom
Email: J.Grbic@soton.ac.uk

Taras Panov
Affiliation: Department of Mathematics and Mechanics, Moscow State University, Leninskie Gory, 119991 Moscow, Russia – and – Institute for Theoretical and Experimental Physics, Moscow, Russia – and – Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia
Email: tpanov@mech.math.msu.su

Stephen Theriault
Affiliation: School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom
Email: S.D.Theriault@soton.ac.uk

Jie Wu
Affiliation: Department of Mathematics, National University of Singapore, Block S17 (SOC1), 06-02 10, Lower Kent Ridge Road, 119076 Singapore
Email: matwuj@nus.edu.sg

DOI: https://doi.org/10.1090/tran/6578
Received by editor(s): July 5, 2013
Received by editor(s) in revised form: July 7, 2014, and September 27, 2014
Published electronically: November 12, 2015
Additional Notes: The first author’s research was supported by the Leverhulme Trust (Research Project Grant RPG-2012-560).
The second author’s research was supported by the Russian Foundation for Basic Research (grants 14-01-00537 and 14-01-92612-KO), Council for Grants of the President of the Russian Federation (grants NSh-4833.2014.1 and MD-111.2013.1), and by Dmitri Zimin’s ‘Dynasty’ foundation.
The fourth author’s research was supported in part by the Singapore Ministry of Education research grant (AcRF Tier 1 WBS No. R-146-000-190-112) and a grant (No. 11329101) of NSFC of China.
Article copyright: © Copyright 2015 American Mathematical Society

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