Face rings of simplicial complexes with singularities

Miller, Ezra; Novik, Isabella; Swartz, Ed

doi:10.1007/s00208-010-0620-5

Cited by 15 publications

(24 citation statements)

References 20 publications

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“…The crucial part of the proof is the following property of the face rings of complexes with homologically isolated singularities that was observed in [18,Section 4]. …”

Section: Homologically Isolated Singularitiesmentioning

confidence: 98%

“…In this section we derive lower bounds on the face numbers of spaces with homologically isolated singularities -a certain subclass of spaces with isolated singularities introduced in [18,Section 4]. To achieve this goal we first compute the Hilbert series of Artinian reductions of such complexes generalizing Schenzel's formula for Buchsbaum complexes [25], which in turn is a generalization of Stanley's formula for CM complexes [28].…”

Section: Homologically Isolated Singularitiesmentioning

confidence: 99%

“…Many other examples of such complexes are described in [18,Section 4]. On the other hand, suspensions of manifolds have isolated singularities that are not homologically isolated.…”

Section: Homologically Isolated Singularitiesmentioning

confidence: 99%

“…A natural question raised in [18] is whether for an arbitrary (d − 1)-dimensional complex , h i ( )−h i ( ) is determined by the homeomorphism type of . As noted earlier, for i = d, the answer is yes as…”

Section: Pl Homeomorphic Pseudomanifoldsmentioning

confidence: 99%

“…This includes triangulations of manifolds (with and without boundary). In [18] algebraic aspects of face rings of spaces with more complicated singularities were studied.…”

Section: Introductionmentioning

confidence: 99%

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Face numbers of pseudomanifolds with isolated singularities

Novik¹,

Swartz²

2012

MATH. SCAND.

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We investigate the face numbers of simplicial complexes with Buchsbaum vertex links, especially pseudomanifolds with isolated singularities. This includes deriving Dehn-Sommerville relations for pseudomanifolds with isolated singularities and establishing lower and upper bound theorems when the singularities are also homologically isolated. We give formulas for the Hilbert function of a generic Artinian reduction of the face ring when the singularities are homologically isolated and for any pure two-dimensional complex. Some examples of spaces where the f -vector can be completely characterized are described. We also show that the Hilbert function of a generic Artinian reduction of the face ring of a simplicial complex with isolated singularities minus the h-vector of is a PL-topological invariant.

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“…The crucial part of the proof is the following property of the face rings of complexes with homologically isolated singularities that was observed in [18,Section 4]. …”

Section: Homologically Isolated Singularitiesmentioning

confidence: 98%

Section: Homologically Isolated Singularitiesmentioning

confidence: 99%

“…Many other examples of such complexes are described in [18,Section 4]. On the other hand, suspensions of manifolds have isolated singularities that are not homologically isolated.…”

Section: Homologically Isolated Singularitiesmentioning

confidence: 99%

Section: Pl Homeomorphic Pseudomanifoldsmentioning

confidence: 99%

“…This includes triangulations of manifolds (with and without boundary). In [18] algebraic aspects of face rings of spaces with more complicated singularities were studied.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Face numbers of pseudomanifolds with isolated singularities

Novik¹,

Swartz²

2012

MATH. SCAND.

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Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums

Adiprasito

Sanyal

2016

Publ.math.IHES

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Abstract. In this paper we settle two long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed facets of Minkowski sums. This has a wide range of applications and generalizes the classical Upper Bound Theorems of McMullen and Stanley.Our main observation is that within (relative) Stanley-Reisner theory, it is possible to encode topological as well as combinatorial/geometric restrictions in an algebraic setup. We illustrate the technology by providing several simplicial isoperimetric and reverse isoperimetric inequalities in addition to our treatment of Minkowski sums.The Upper Bound Theorem (UBT) for polytopes is one of the cornerstones of discrete geometry. The UBT gives precise bounds on the 'combinatorial complexity' of a convex polytope P as measured by the number of k-dimensional faces f k (P ) in terms of its dimension and the number of vertices. Upper Bound Theorem for polytopes. For a d-dimensional polytope P on n vertices and 0where Polytopes attaining the upper bound are called (simplicial) neighborly polytopes and are characterized by the fact that all non-faces are of dimension at least d 2 . Cyclic polytopes are a particularly interesting class of neighborly polytopes whose combinatorial structure allows for an elementary and explicit calculation of f k (Cyc d (n)) in terms of d and n; cf. [Zie95, Section 0]. The UBT was conjectured by Motzkin [Mot57] and proved by McMullen [McM70]. One of the salient features to note is that for given d and n there is a polytope that maximizes f k for all k simultaneously -a priori, this is not to be expected.In this paper we will address more general upper bound problems for polytopes and polytopal complexes. To state the main applications of the theory to be developed, recall that the Minkowski sum of polytopes and game theory. Our methods also apply to the study of mixed faces and we establish strong upper bounds and in particular characterize the case of equality in the most important case. Upper Bound Theorem for mixed facets. The number of mixed facets of a Minkowski sum is maximized by Minkowski neighborly families.From discrete geometry to combinatorial topology to commutative algebra. An intriguing feature of the UBT is that its validity extends beyond the realm of convex polytopes and into combinatorial topology. Let ∆ be a triangulation of the (d − 1)-sphere and, as before, let us write f k (∆) for the number of k-dimensional faces. For example, boundaries of simplicial d-dimensional polytopes yield simplicial spheres, but these are by far not all. The UBTM too will be the consequence of a statement in the topological domain that we derive using algebra, though we will also briefly comment on a geometric approach to the problem. The appropriate combinatorial/topological setup for the UBPM is that of relative simplicial complexes: A re...

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Level algebras through Buchsbaum* manifolds

Nagel

2010

Collect. Math.

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Stanley-Reisner rings of Buchsbaum* complexes are studied by means of their quotients modulo a linear system of parameters. The socle of these quotients is computed. Extending a recent result by Novik and Swartz for orientable homology manifolds without boundary, it is shown that modulo a part of their socle these quotients are level algebras. This provides new restrictions on the face vectors of Buchsbaum* complexes.

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Face rings of simplicial complexes with singularities

Cited by 15 publications

References 20 publications

Face numbers of pseudomanifolds with isolated singularities

Face numbers of pseudomanifolds with isolated singularities

Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums

Level algebras through Buchsbaum* manifolds

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