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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 boundary 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
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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.
The $q,t$-Catalan numbers can be defined using rational functions, geometry related to Hilbert schemes, symmetric functions, representation theory, Dyck paths, partition statistics, or Dyck words. After decades of intensive study, it was eventually proved that all these definitions are equivalent. In this paper, we study the similar situation for higher $q,t$-Catalan numbers, where the equivalence of the algebraic and combinatorial definitions is still conjectural. We compute the limits of sever
We show that J_n, the Stanley-Reisner ideal of the n-cycle, has a free resolution supported on the (n-3)-dimensional simplicial associahedron A_n. This resolution is not minimal for n > 5; in this case the Betti numbers of J_n are strictly smaller than the f-vector of A_n. We show that in fact the Betti numbers of J_n are in bijection with the number of standard Young tableaux of shape (d+1, 2, 1^{n-d-3}). This complements the fact that the number of (d-1)-dimensional faces of A_n are given by the number of standard Young tableaux of (super)shape (d+1, d+1, 1^{n-d-3}); a bijective proof of this result was first provided by Stanley. An application of discrete Morse theory yields a cellular resolution of J_n that we show is minimal at the first syzygy. We furthermore exhibit a simple involution on the set of associahedron tableaux with fixed points given by the Betti tableaux, suggesting a Morse matching and in particular a poset structure on these objects.
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.
This paper has two related parts. The first generalizes Hochsters formula on resolutions of Stanley-Reisner rings to a colorful version, applicable to any proper vertex-coloring of a simplicial complex. The second part examines a universal system of parameters for Stanley-Reisner rings of simplicial complexes, and more generally, face rings of simplicial posets. These parameters have good properties, including being fixed under symmetries, and detecting depth of the face ring. Moreover, when resolving the face ring over these parameters, the shape is predicted, conjecturally, by the colorful Hochster formula.
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