A generalized lower bound theorem for balanced manifolds

TL;DR

This paper extends the generalized lower bound theorem to balanced triangulations of homology manifolds, establishing new inequalities and developing the theory of flag h''-vectors under certain topological conditions.

Contribution

It generalizes the lower bound theorem to a broader class of balanced manifolds and introduces the theory of flag h''-vectors for these complexes.

Findings

Proved a lower bound inequality for balanced triangulations of homology manifolds.

Verified a conjecture by Klee and Novik regarding h-vectors.

Developed the theory of flag h''-vectors for balanced complexes.

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 $\Delta$ is an arbitrary balanced triangulation of any closed homology manifold of dimension $d-1 \geq 3$, then $2h_2(\Delta) - (d-1)h_1(\Delta) \geq 4{d \choose 2}(\tilde{\beta}_1(\Delta)-\tilde{\beta}_0(\Delta))$, thus verifying a conjecture by Klee and Novik. To prove these results we develop the theory of flag $h''$-vectors.