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Product decompositions of moment-angle manifolds and B-rigidity

Part of: Applied homological algebra and category theory Arithmetic rings and other special rings

Published online by Cambridge University Press:  15 May 2023

Steven Amelotte Open the ORCID record for Steven Amelotte [Opens in a new window]  and
Benjamin Briggs Open the ORCID record for Benjamin Briggs [Opens in a new window]
Steven Amelotte*
Affiliation:
Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, RI 02903, USA
Benjamin Briggs
Affiliation:
Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, CA 94720, USA e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

A simple polytope P is called B-rigid if its combinatorial type is determined by the cohomology ring of the moment-angle manifold $\mathcal {Z}_P$ over P. We show that any tensor product decomposition of this cohomology ring is geometrically realized by a product decomposition of the moment-angle manifold up to equivariant diffeomorphism. As an application, we find that B-rigid polytopes are closed under products, generalizing some recent results in the toric topology literature. Algebraically, our proof establishes that the Koszul homology of a Gorenstein Stanley–Reisner ring admits a nontrivial tensor product decomposition if and only if the underlying simplicial complex decomposes as a join of full subcomplexes.

Keywords

Moment-angle complexcohomological rigidityStanley–Reisner ringquasitoric manifold

MSC classification

Primary: 13F55: Stanley-Reisner face rings; simplicial complexes 55U10: Simplicial sets and complexes
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 4 , December 2023 , pp. 1313 - 1325
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

During this work, the first author was hosted by the Institute for Computational and Experimental Research in Mathematics in Providence, RI, supported by the National Science Foundation under Grant No. 1929284. The second author was hosted by the Mathematical Sciences Research Institute in Berkeley, CA, supported by the National Science Foundation under Grant No. 1928930.

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