Lower bounds for Buchsbaum* complexes

Abstract

The class of (d − 1)-dimensional Buchsbaum* simplicial complexes is studied. It is shown that the rank-selected subcomplexes of a (completely) balanced Buchsbaum* simplicial complex are also Buchsbaum*. Using this result, lower bounds on the h-numbers of balanced Buchsbaum* simplicial complexes are established. In addition, sharp lower bounds on the h-numbers of flag m-Buchsbaum* simplicial complexes are derived, and the case of equality is treated.

Lower Bound Theorems and a Generalized Lower Bound Conjecture for Balanced Simplicial Complexes

“…• In Section 3, we show that the balanced analog of the Lower Bound Theorem (established in [19] for balanced spheres and in [13] for balanced manifolds) continues to hold for all balanced triangulations of normal pseudomanifolds, see Theorem 3.4.…”

“…For any pair of integers n and d with n divisible by d, a stacked cross-polytopal (d − 1)-sphere on n vertices, denoted ST × (n, d − 1), is defined as the balanced connected sum of n d − 1 copies of C * d with itself. Goff, Klee, and Novik [19] (for simplicial spheres and more generally for doubly Cohen-Macaulay complexes) and Browder and Klee [13] (for simplicial manifolds and more generally for Buchsbaum * simplicial complexes) showed that when n is divisible by d, ST × (n, d − 1) has the componentwise minimal f -vector among all balanced spheres/manifolds of dimension d − 1 with n vertices. Once again, this can be stated as a simple inequality on the level of h-numbers.…”

Minimal Balanced Triangulations of Sphere Bundles over the Circle

“…Since st∆ v is contractible, by (4.3) and (4.4) it implies thatH i (∆\{v i }; Q) =H i (∆; Q) =H i (∆; Q) for i = 1, 2.Iterating the argument on other vertices of V 1 , it follows thatH 2 (∆\V 1 ; Q) = 0 andH 1 (∆\V 1 ; Q) = 0. Hence by the proof of Theorem 3.6, we obtain that c(∆ i,j ) = s ≥ 2 and c(∆ 1,2,3 ) = 2s − 1 ≥ 3 for every {i, j} ⊆[3].Next, by Lemma 4.3, for every {i, j} ≤ 3 we also have…”

“…A (d − 1)-simplicial complex ∆ is Buchsbaum over k if ∆ is pure and for every nonempty face σ in ∆, and every i < d − 1 − dim σ, we haveH i (lk ∆ σ; k) = 0. A (d − 1)-dimensional simplicial complex ∆ is Buchsbaum* over k if it is Buchsbaum over k, and for every pair of faces σ ⊆ τ of ∆, the map i * : For more properties of balanced Buchsbaum* complexes, see [3] for a reference.…”

“…Further assume that for {i, j, k} = [3], every edge of G i is also an edge of either G j or G k , and that every G i ∩ G j has s connected components. Then there exist distinct vertices u 1 , u 2 , u 3 such that the graph G i \{u i } is disconnected for i = 1, 2, 3.…”