A generalized lower bound theorem for balanced manifolds

Abstract

A simplicial complex of dimension d − 1 is said to be balanced if its graph is d-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a generalization of their result to balanced triangulations of closed homology manifolds and balanced triangulations of orientable homology manifolds with boundary under an additional assumption that all proper links of these triangulations have the weak Lefschetz property. As a corollary, we show that if ∆ is an arbitrary balanced triangulation of any closed homology manifold of dimension d − 1 ≥ 3, then 2h 2 (∆) − (d − 1)h 1 (∆) ≥ 4 d 2 (β 1 (∆) −β 0 (∆)), thus verifying a conjecture by Klee and Novik. To prove these results we develop the theory of flag h ′′ -vectors. d 1 +β 1 (∆).Equivalently, 2h 2 (∆) − (d − 1)h 1 (∆) ≥ 4 d 2 (β 1 (∆) −β 0 (∆)). Furthermore, if d ≥ 5, then this inequality holds as equality if and only if each connected component of ∆ is in the balanced Walkup class.

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.…”

“…Theorem 5.4. [JKMNS18] Let ∆ be a connected d-dimensional balanced F-homology manifold, with d ≥ 3. Then…”

“…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?…”

Octahedralizing 3-colorable 3-polytopes

“…Clearly since the graph of any (d − 1)-simplex is the complete graph on d vertices, d is the minimum chromatic number that the graph of any simplicial d-polytope can have. Balanced simplicial complexes were introduced by Stanley [Sta79] and recently they have gained attention from the point of view of face enumeration [KN16,JKM18,JKMNS18,Ven19]. For results of a more topological flavour regarding balancedness and colorings we refer to [Fis77,IJ03,IKN17,JKV18].…”

Positive Plücker tree certificates for non-realizability

“…Now we can say why Zheng's example is important -in fact, it is important in at least two ways. First, there has been a lot of work on analogies between combinatorial data in the balanced and the non-balanced settings [JM18,JMNS18,Ven19]. For example, one would like to have a balanced analogue of the celebrated Upper Bound Theorem by McMullen and Stanley.…”