On face numbers of manifolds with symmetry
type: Item
instance of: scholarly article
title: On face numbers of manifolds with symmetry
zbMATH Open document ID: 1092.52009
author: Isabella Novik
published in: Advances in Mathematics
publication date: 2005-04-07T00:00:00Z
review text: The author considers symmetric simplicial complexes~\(\Gamma\), specifically those that admit a proper action of the cyclic group~\(\mathbb Z_p\) of prime order~\(p\). Throughout, we follow the author and write the number of vertices of~\(\Gamma\) as \(n=l+pm\), where \(l\)~is the number of \(\mathbb Z_p\)-invariant vertices and \(m\)~the number of \(\mathbb Z_p\)-orbits of non-invariant vertices. The paper under review is roughly divided into two parts. The first part is dedicated to the case that \(\Gamma\)~is Cohen-Macaulay (e.g.,~a simplicial sphere) or Buchsbaum (e.g.,~a simplicial manifold); in these situations, the author gives necessary conditions on the \(f\)-vector of~\(\Gamma\). The second part treats the Upper Bound Theorem for numbers of faces, as well as bounds on the reduced Euler characteristic of a \(\mathbb Z_p\)-symmetric simplicial manifold, and on a certain weighted sum of its Betti numbers. The special case of \(p=2\), i.e., centrally symmetric manifolds, also receives attention. For each class of complexes considered in the first part of this paper, i.e., Cohen-Macaulay and Buchsbaum complexes, the necessary conditions on the \(f\)-vector just alluded to take two forms. The first one (treated in Theorems~1.1 and~3.3) says that the \(h\)-vector \(h(\Gamma)=(h_0(\Gamma),\dots, h_d(\Gamma))\) of any simplicial \((d-1)\)-dimensional Cohen-Macaulay complex~\(\Gamma\) must be the face vector~\(F(\Gamma)=(F_0(\Gamma),F_1(\Gamma),\dots)\) of a suitable multicomplex \[ M \;\subset \;S\big((p-1)^m, \infty^{n-d-m}\big); \] here \(S(a^r, b^s)\)~denotes the multicomplex of all monomials \(x_1^{a_1}\cdots x_{r+s}^{a_{r+s}}\) with \(0\leq a_i\leq a\) for \(1\leq i\leq r\) and \(0\leq a_{r+i}\leq b\) for \(1\leq i\leq s\), ordered by divisibility, and \(F_i(\Gamma)=| \{\mu\in\Gamma:\deg\mu=i\}| \). This strengthens \textit{R. P. Stanley}'s [Higher Comb., Proc. NATO Adv. Study Inst., Berlin (West) 1976, 51--62 (1977; Zbl 0376.55007)] earlier characterization o…
reviewed by: Julian Pfeifle
Mathematics Subject Classification ID: 52B70
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 57Q15
Mathematics Subject Classification ID: 52B12
zbMATH DE Number: 2153794
zbMATH Keywords: Cohen-Macaulay and Buchsbaum complexes
zbMATH Keywords: initial ideals
zbMATH Keywords: Clemens-Lindström theorem
zbMATH Keywords: normalized matching property
zbMATH Keywords: Kind-Kleinschmidt condition
zbMATH Keywords: Kühnel's conjecture
zbMATH Keywords: Upper Bound Theorem for manifolds
MaRDI profile type: Publication
full work available at URL: j.aim.2004.04.003
OpenAlex ID: W2091599323
cites work: On generalized \(h\)-vectors of rational polytopes with a symmetry of prime order
cites work: On face numbers of rational simplicial polytopes with symmetry
cites work: A simple proof of the upper bound theorem
cites work: An upper bound theorem for rational polytopes
cites work: An extended Euler-Poincaré theorem
cites work: The Poincaré duality in generalized manifolds
cites work: The non-neighbourliness of centrally symmetric convex polytopes having many vertices
cites work: A generalization of a combinatorial theorem of macaulay
cites work: Q4317713
cites work: Q4196436
cites work: A short simplicial \(h\)-vector and the upper bound theorem
cites work: Smith-type inequalities for a polytope with a solvable group of symmetries
cites work: Combinatorics of polytopes with a group of linear symmetries of prime power order
cites work: Q3974981
cites work: On separating systems of a finite set
cites work: Schälbare Cohen-Macaulay-Komplexe und ihre Parametrisierung. (Shellable Cohen-Macaulay complexes and their parametrization)
cites work: A Combinatorial Analogue of Poincaré's Duality Theorem
cites work: On the Number of Vertices of a Convex Polytope
cites work: Q5726070
cites work: Tight polyhedral submanifolds and tight triangulations
cites work: Triangulated manifolds with few vertices and vertex-transitive group actions
cites work: The maximum numbers of faces of a convex polytope
cites work: Diagrams for centrally symmetric polytopes
cites work: Topological results in combinatorics
cites work: Upper bound theorems for homology manifolds
cites work: Remarks on the upper bound theorem.
cites work: On the number of faces of simplicial complexes and the purity of Frobenius
cites work: An upper and a lower bound theorem for combinatorial 4-manifolds
cites work: A new lower bound theorem for combinatorial \(2k\)-manifolds
cites work: The Upper Bound Conjecture and Cohen-Macaulay Rings
cites work: Q4153339
cites work: Balanced Cohen-Macaulay Complexes
cites work: On the number of faces of centrally-symmetric simplicial polytopes
cites work: A monotonicity property of \(h\)-vectors and \(h^*\)-vectors
cites work: Combinatorics and commutative algebra.
cites work: Q3744271
cites work: Lectures on Polytopes
DOI: 10.1016/J.AIM.2004.04.003
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description: scientific article; zbMATH DE number 2153794