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Transactions of the Moscow Mathematical Society

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Substitutions of polytopes and of simplicial complexes, and multigraded betti numbers

Author: A. A. Ayzenberg
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 74 (2013), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2013, 175-202
MSC (2010): Primary 05E45; Secondary 52B11, 52B05, 55U10, 13F55
Published electronically: April 9, 2014
MathSciNet review: 3235795
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Abstract: For a simplicial complex $ K$ on $ m$ vertices and simplicial complexes $ K_1,\ldots ,K_m$, we introduce a new simplicial complex $ K(K_1,\ldots ,K_m)$, called a substitution complex. This construction is a generalization of the iterated simplicial wedge studied by A. Bari, M. Bendersky, F. R. Cohen, and S. Gitler. In a number of cases it allows us to describe the combinatorics of generalized joins of polytopes $ P(P_1,\ldots ,P_m)$, as introduced by G. Agnarsson. The substitution gives rise to an operad structure on the set of finite simplicial complexes in which a simplicial complex on $ m$ vertices is considered as an $ m$-ary operation. We prove the following main results: (1) the complex $ K(K_1,\ldots ,K_m)$ is a simplicial sphere if and only if $ K$ is a simplicial sphere and the $ K_i$ are the boundaries of simplices, (2) the class of spherical nerve-complexes is closed under substitution, (3) multigraded betti numbers of $ K(K_1,\ldots ,K_m)$ are expressed in terms of those of the original complexes $ K, K_1,\ldots ,K_m$. We also describe connections between the obtained results and the known results of other authors.

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Additional Information

A. A. Ayzenberg
Affiliation: Moscow, MSU, Department of Mechanics and Mathematics, Chair of Geometric Methods of Mathematical Physics

Keywords: Generalized polyhedral join, simplicial wedge, simplicial complex operad, polyhedral product, polyhedral join, graded betti numbers, enumerating polynomials, polarization of a homogeneous ideal.
Published electronically: April 9, 2014
Additional Notes: Supported by the RFFI Grant 12-01-92104–YaFa and the RF Government Grant 2010-220-01-077.
Article copyright: © Copyright 2014 American Mathematical Society

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