Algebraic shifting of strongly edge decomposable spheres

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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“…Comparing the right-hand-sides of (13) and (14) and using dim I 1 + dβ 0 ≤ dim I 2 , implies the result.…”

Section: It Follows From Proposition 55 Below That If ∆ Ismentioning

confidence: 52%

“…The proof is by induction on d. Any ∆ homeomorphic to S 2 is k-rigid. This follows from [14,Cor. 3.5].…”

Section: It Follows From Proposition 55 Below That If ∆ Ismentioning

confidence: 92%

Applications of Klee’s Dehn–Sommerville Relations

Novik

Swartz

2009

Discrete Comput Geom

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“…Hence this approach is only valid in characteristic zero. However, Murai's recent paper [23,Corollary 3.5], combined with Whiteley's proof that two-dimensional spheres are strongly edge decomposable [42] (see [25] for the definition of strongly edge decomposable), provide an alternative proof which is valid in nonzero characteristics. 2 Problem 5.3.…”

Section: Page 50]) As Multiplication By a Generic One-form From (K[lmentioning

confidence: 99%

Socles of Buchsbaum modules, complexes and posets

Novik

Swartz

2009

Advances in Mathematics

119

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The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel's conjecture for the maximum value of the Euler characteristic of a 2k-dimensional simplicial manifold on n vertices as well as Kalai's conjecture providing a lower bound on the number of edges of a simplicial manifold in terms of its dimension, number of vertices, and the first Betti number.

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“…He proved that the g-vector of an edge decomposable sphere is non-negative in [21]. Later, it was proved in [1,20] that the g-vector of an edge decomposable complex is the f -vector of a multicomplex.…”

Section: Edge Decomposabilitymentioning

confidence: 99%

“…Kalai's squeezed spheres also arise from finite multicomplexes by certain operations (see [12, p. 6] and [19,Proposition 4.1]), and give many shellable edge decomposable spheres which are not realizable as polytopes [12,13,20]. It might be of interest to find a general construction of shellable spheres which includes both Bier spheres and Kalai's squeezed spheres.…”

Section: Introductionmentioning

confidence: 99%

Spheres arising from multicomplexes

Murai

2011

Journal of Combinatorial Theory, Series A

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In 1992, Thomas Bier introduced a surprisingly simple way to construct a large number of simplicial spheres. He proved that, for any simplicial complex $\Delta$ on the vertex set $V$ with $\Delta \ne 2^V$, the deleted join of $\Delta$ with its Alexander dual $\Delta^\vee$ is a combinatorial sphere. In this paper, we extend Bier's construction to multicomplexes, and study their combinatorial and algebraic properties. We show that all these spheres are shellable and edge decomposable, which yields a new class of many shellable edge decomposable spheres that are not realizable as polytopes. It is also shown that these spheres are related to polarizations and Alexander duality for monomial ideals which appear in commutative algebra theory.Comment: 20 pages. Improve presentation. To appear in Journal of Combinatorial Theory, Series

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Algebraic shifting of strongly edge decomposable spheres

Cited by 18 publications

References 26 publications

Applications of Klee’s Dehn–Sommerville Relations

Applications of Klee’s Dehn–Sommerville Relations

Socles of Buchsbaum modules, complexes and posets

Spheres arising from multicomplexes

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