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

Abstract

A (d − 1)-dimensional simplicial complex is called balanced if its underlying graph admits a proper d-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs). Specifically, we prove the balanced analog of the celebrated Lower Bound Theorem for normal pseudomanifolds and characterize the case of equality; we introduce and characterize the balanced analog of the Walkup class; we propose the balanced analog of the Generalized Lower Bound Conjecture and establish some related results. We close with constructions of balanced manifolds with few vertices.

Balanced Triangulations on few Vertices and an Implementation of Cross-Flips

“…It is an interesting refinement of the lower bound theorem for homology spheres, obtained from the study of algebraic invariants of Buchsbaum graded rings. Juhnke-Kubitzke, Murai, Novik and Sawaske proved a balanced analog of this bound (see [JKMNS18]), and established a conjecture of Klee and Novik [KN16,Conjecture 4.14] for the characterization of the case of equality, when the dimension is greater or equal to 4. Let ∆ and Γ be pure balanced simplicial complexes of the same dimension on disjoint vertex sets, let F, G be two facets of ∆ and Γ respectively and let ϕ ∶ F → G be a bijection.…”

“…We point out that some of these triangulations were previously known, and they are referenced through this section. For instance Klee and Novik [KN16] proved the existence of a d-dimensional simplicial complex on 3d + 3 vertices which provides a vertex minimal balanced triangulation of S d−1 × S 1 when d is even, and of the twisted bundle S d−1 S 1 when d is odd. Moreover they construct balanced triangulations of both S d−1 × S 1 and S d−1 S 1 on 3d + 5 vertices.…”

“…In this article we focus on the family of balanced simplicial complexes, i.e., d-dimensional complexes whose underlying graph is (d + 1)-colorable, in the classical graph theoretic sense. Many questions and results in face enumeration have balanced analogs (see for instance [IKN17,JKM18,JKMNS18,KN16]. In particular we can ask the following question: what is the minimum number of vertices that a balanced triangulation of a manifold M can have?…”

“…The description of the construction for (S 2 S 1 ) #2 mostly relies on the simplicial complex ∆ S 2 S 1 12 . Since in [KN16] the authors showed that the analog construction in dimension d provides a triangulation on 3d vertices of S d−1 S 1 if d is even and of S d−1 × S 1 if d is odd, the extra handle that we add provides triangulations on 4d vertices of (S d−1 S 1 ) #2 if d is odd and of (S d−1 × S 1 ) #2 if d is even.…”

Centrally symmetric and balanced triangulations of S2×Sd−3Wang1, Zheng2 2020European Journal of Combinatorics1000View full textAdd to dashboardCite

“…In much of the same spirit as Kühnel's construction, Klee and Novik [8] provided a balanced triangulation of S 1 × S d−2 with 3d vertices for odd d and with 3d+2 vertices otherwise. Furthermore, Zheng [16] showed that the number of vertices for a minimal triangulation is indeed 3d for odd d and 3d + 2 otherwise.…”