Face enumeration—from spheres to manifolds

Swartz, Ed

doi:10.4171/jems/156

Cited by 50 publications

(81 citation statements)

References 45 publications

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“…(In characteristic 0 this result was proved in [29]; for a field of arbitrary characteristic it follows from Murai's paper [19] combined with Whiteley's proof that 2-dimensional spheres are strongly edge decomposable [36].) The proof of Theorem 4.26 in [33] …”

Section: ( ) This Is Immediate From the Following Series Of Canonicamentioning

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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Section: ( ) This Is Immediate From the Following Series Of Canonicamentioning

confidence: 99%

“…For the left-hand map, first we observe that I 3 is generated by all monomials that contain at least one nonsingular vertex. Now we follow the proof of [33,Theorem 4.26] to see that each of these monomials is in the image.…”

Section: Examplesmentioning

confidence: 99%

Face numbers of pseudomanifolds with isolated singularities

Novik¹,

Swartz²

2012

MATH. SCAND.

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“…Proof: Since all vertex links of ∆ have the hard Lefschetz property, Theorem 4.26 of [21] implies that for a sufficiently generic choice of θ 1 , . .…”

Section: Eq (1) Then Impliesmentioning

confidence: 99%

Applications of Klee’s Dehn–Sommerville Relations

Novik

Swartz

2009

Discrete Comput Geom

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We use Klee's Dehn-Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai's conjecture providing lower bounds on the f -vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel's conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel's conjecture providing upper bounds on other Betti numbers of odd-and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee's Dehn-Sommerville relations and strengthen Kalai's result on the number of their edges.

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“…In the special case when is an orientable homology manifold, Lemma 3.1 essentially reduces to [20,Proposition 4.24].…”

Section: Lemma 31 Let Be a (D − 1)-dimensional Buchsbaum Complex Anmentioning

confidence: 99%

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 enumeration—from spheres to manifolds

Cited by 50 publications

References 45 publications

Face numbers of pseudomanifolds with isolated singularities

Face numbers of pseudomanifolds with isolated singularities

Applications of Klee’s Dehn–Sommerville Relations

Level algebras through Buchsbaum* manifolds

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