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 thei
r 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.
We resolve a conjecture of Kalai asserting that the $g_2$-number of any simplicial complex $Delta$ that represents a connected normal pseudomanifold of dimension $dgeq 3$ is at least as large as ${d+2 choose 2}m(Delta)$, where $m(Delta)$ denotes the
minimum number of generators of the fundamental group of $Delta$. Furthermore, we prove that a weaker bound, $h_2(Delta)geq {d+1 choose 2}m(Delta)$, applies to any $d$-dimensional pure simplicial poset $Delta$ all of whose faces of co-dimension $geq 2$ have connected links. This generalizes a result of Klee. Finally, for a pure relative simplicial poset $Psi$ all of whose vertex links satisfy Serres condition $(S_r)$, we establish lower bounds on $h_1(Psi),ldots,h_r(Psi)$ in terms of the $mu$-numbers introduced by Bagchi and Datta.
Let $Delta$ be a triangulated homology ball whose boundary complex is $partialDelta$. A result of Hochster asserts that the canonical module of the Stanley--Reisner ring of $Delta$, $mathbb F[Delta]$, is isomorphic to the Stanley--Reisner module of t
he pair $(Delta, partialDelta)$, $mathbb F[Delta,partial Delta]$. This result implies that an Artinian reduction of $mathbb F[Delta,partial Delta]$ is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of $mathbb F[Delta]$. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the $h$-numbers of Buchsbaum complexes and use it to prove the monotonicity of $h$-numbers for pairs of Buchsbaum complexes as well as the unimodality of $h$-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold $g$-conjecture.
We study face numbers of simplicial complexes that triangulate manifolds (or even normal pseudomanifolds) with boundary. Specifically, we establish a sharp lower bound on the number of interior edges of a simplicial normal pseudomanifold with boundar
y in terms of the number of interior vertices and relative Betti numbers. Moreover, for triangulations of manifolds with boundary all of whose vertex links have the weak Lefschetz property, we extend this result to sharp lower bounds on the number of higher-dimensional interior faces. Along the way we develop a version of Bagchi and Dattas $sigma$- and $mu$-numbers for the case of relative simplicial complexes and prove stronge
