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Register a new userAlgebraic $h$-vectors of simplicial complexes through local cohomology, part 1
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Authors
Connor Sawaske
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Given an infinite field $mathbb{k}$ and a simplicial complex $Delta$, a common theme in studying the $f$- and $h$-vectors of $Delta$ has been the consideration of the Hilbert series of the Stanley--Reisner ring $mathbb{k}[Delta]$ modulo a generic linear system of parameters $Theta$. Historically, these computations have been restricted to special classes of complexes (most typically triangulations of spheres or manifolds). We provide a compact topological expression of $h_{d-1}^mathfrak{a}(Delta)$, the dimension over $mathbb{k}$ in degree $d-1$ of $mathbb{k}[Delta]/(Theta)$, for any complex $Delta$ of dimension $d-1$. In the process, we provide tools and techniques for the possible extension to other coefficients in the Hilbert series.
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We consider a generalised model of a random simplicial complex, which arises from a random hypergraph. Our model is generated by taking the downward-closure of a non-uniform binomial random hypergraph, in which for each $k$, each set of $k+1$ vertices forms an edge with some probability $p_k$ independently. As a special case, this contains an extensively studied model of a (uniform) random simplicial complex, introduced by Meshulam and Wallach [Random Structures & Algorithms 34 (2009), no. 3, pp. 408-417]. We consider a higher-dimensional notion of connectedness on this new model according to the vanishing of cohomology groups over an arbitrary abelian group $R$. We prove that this notion of connectedness displays a phase transition and determine the threshold. We also prove a hitting time result for a natural process interpretation, in which simplices and their downward-closure are added one by one. In addition, we determine the asymptotic behaviour of cohomology groups inside the critical window around the time of the phase transition.
A well-known conjecture of Richard Stanley posits that the $h$-vector of the independence complex of a matroid is a pure ${mathcal O}$-sequence. The conjecture has been established for various classes but is open for graphic matroids. A biconed graph is a graph with two specified `coning vertices, such that every vertex of the graph is connected to at least one coning vertex. The class of biconed graphs includes coned graphs, Ferrers graphs, and complete multipartite graphs. We study the $h$-vectors of graphic matroids arising from biconed graphs, providing a combinatorial interpretation of their entries in terms of `edge-rooted forests of the underlying graph. This generalizes constructions of Kook and Lee who studied the Mobius coinvariant (the last nonzero entry of the $h$-vector) of graphic matroids of complete bipartite graphs. We show that allowing for partially edge-rooted forests gives rise to a pure multicomplex whose face count recovers the $h$-vector, establishing Stanleys conjecture for this class of matroids.
We provide a random simplicial complex by applying standard constructions to a Poisson point process in Euclidean space. It is gigantic in the sense that - up to homotopy equivalence - it almost surely contains infinitely many copies of every compact topological manifold, both in isolation and in percolation.
Let $mathcal{H}$ be a hypergraph of rank $r$. We show that the simplicial complex whose simplices are the hypergraphs $mathcal{F}subsetmathcal{H}$ with covering number at most $p$ is $left(binom{r+p}{r}-1right)$-collapsible, and the simplicial complex whose simplices are the pairwise intersecting hypergraphs $mathcal{F}subsetmathcal{H}$ is $frac{1}{2}binom{2r}{r}$-collapsible.
We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these combinatorial Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their $f$-vectors. We also use characters to give a generalization of Stanleys $(-1)$-color theorem. A $q$-analog version of this family of characters is also studied.
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