Applications of Klee’s Dehn–Sommerville Relations

Novik, Isabella; Swartz, Ed

doi:10.1007/s00454-009-9187-x

Cited by 32 publications

(58 citation statements)

References 20 publications

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“…For the case of closed manifolds, the last decade of research led to tremendous progress on this question, see, for instance, [2,3,19,22,23,25,27,28,29,30,37,38]. On the other hand, face numbers of manifolds with boundary remained a big mystery, and very few papers even touched on this subject, see [9,14,25,28]. At present there is not even a conjecture for characterizing the set of f -vectors of balls of dimension six and above, see [14].…”

Section: Introductionmentioning

confidence: 99%

Face Numbers of Manifolds with Boundary

Murai

Novik

2016

Int Math Res Notices

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We study face numbers of simplicial complexes that triangulate manifolds (or even normal pseudomanifolds) with boundary. Specifically, we establish a sharp lower bound on the number of interior edges of a simplicial normal pseudomanifold with boundary in terms of the number of interior vertices and relative Betti numbers. Moreover, for triangulations of manifolds with boundary all of whose vertex links have the weak Lefschetz property, we extend this result to sharp lower bounds on the number of higher-dimensional interior faces. Along the way we develop a version of Bagchi and Datta's σ-and µ-numbers for the case of relative simplicial complexes and prove stronger versions of the above statements with the Betti numbers replaced by the µ-numbers. Our results provide natural generalizations of known theorems and conjectures for closed manifolds and appear to be new even for the case of a ball.

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Section: Introductionmentioning

confidence: 99%

Face Numbers of Manifolds with Boundary

Murai

Novik

2016

Int Math Res Notices

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“…In particular, face enumeration on manifolds with boundary is closely related to face enumeration on spaces with isolated singularities. For a specific example see [21,Theorem 5.1].…”

Section: Introductionmentioning

confidence: 99%

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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“…It seems, the version with the boundary goes back to Macdonald [26] and was rediscovered a number of times. We refer to [30] for the most general version of the equations, various applications, and further references.…”

Section: 8mentioning

confidence: 99%

Acute triangulations of polyhedra and ℝ N

Kopczyński

Pak²,

Przytycki

2012

Combinatorica

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We study the problem of acute triangulations of convex polyhedra and the space R n . Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the n-cube do not exist for n ≥ 4. Further, we prove that acute triangulations of the space R n do not exist for n ≥ 5. In the opposite direction, in R 3 , we present a construction of an acute triangulation of the cube, the regular octahedron and a non-trivial acute triangulation of the regular tetrahedron. We also prove nonexistence of an acute triangulation of R 4 if all dihedral angles are bounded away from π/2.

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Applications of Klee’s Dehn–Sommerville Relations

Cited by 32 publications

References 20 publications

Face Numbers of Manifolds with Boundary

Face Numbers of Manifolds with Boundary

Face numbers of pseudomanifolds with isolated singularities

Acute triangulations of polyhedra and ℝ N

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