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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.
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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
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.
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.
In this paper, we attempt to develop the Quillen Suslin theory for the algebraic fundamental group of a ring. We give a surjective group homomorphism from the algebraic fundamental group of the field of the real numbers to the group of integers. At the end of the paper, we also propose some problems related to the algebraic fundamental group of some particular type of rings.
We define the principal divisor of a free noncommuatative function. We use these divisors to compare the determinantal singularity sets of free noncommutative functions. We show that the divisor of a noncommutative rational function is the difference of two polynomial divisors. We formulate a nontrivial theory of cohomology, fundamental groups and covering spaces for tracial free functions. We show that the natural fundamental group arising from analytic continuation for tracial free functions is a direct sum of copies of $mathbb{Q}$. Our results contrast the classical case, where the analogous groups may not be abelian, and the free case, where free universal monodromy implies such notions would be trivial.
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