Related papers: On Simplicial Complexes with Extremal Total Betti …

We find some general lower bounds of the sum of certain families of multigraded Betti numbers of any simplicial complex with a vertex coloring.

Let X be a k-dimensional simplicial complex such that the (k-j-2)-dimensional homology of the links of all j-dimensional simplices in X vanishes. An upper bound is given on the (k-1)-th Betti number of X. Examples based on sum complexes…

We study simplicial complexes with a given number of vertices whose Stanley-Reisner ring has the minimal possible Betti numbers. We find that these simplicial complexes have very special combinatorial and topological structures. For…

We study the problem of maximizing Betti numbers of simplicial complexes. We prove an upper bound of 1.32^n for the sum of Betti numbers of any n-vertex flag complex and 1.25^n for the independence complex of a triangle-free graph. These…

Let $S$ be a polynomial ring in $n$ variables over a field $K$ of characteristic $0$. A numerical characterization of all possible extremal Betti numbers of any graded submodule of a finitely generated graded free $S$-module is given.

We consider the multi-parameter random simplicial complex as a higher dimensional extension of the classical Erd\"os-R\'enyi graph. We investigate appearance of "unusual" topological structures in the complex from the point of view of large…

Upper bounds on the topological Betti numbers of Vietoris-Rips complexes are established, and examples of such complexes with high Betti numbers are given.

We consider the line graph of a pure simplicial complex. We prove that, as in the case of line graphs of simple graphs, one can compute the second graded Betti number of the facet ideal of a pure simplicial complex in terms of the…

We consider the multiparameter random simplicial complex on a vertex set $\{ 1,\dots,n \}$, which is parameterized by multiple connectivity probabilities. Our key results concern the topology of this complex of dimensions higher than the…

We show the minimal total Betti number of a closed almost complex manifold of dimension $2n\ge 8$ is four, thus confirming a conjecture of Sullivan except for dimension $6$. Along the way, we prove the only simply connected closed complex…

The bigraded Betti numbers b^{-i,2j}(P) of a simple polytope P are the dimensions of the bigraded components of the Tor groups of the face ring k[P]. The numbers b^{-i,2j}(P) reflect the combinatorial structure of P as well as the topology…

In this paper we discuss two general models of random simplicial complexes which we call the lower and the upper models. We show that these models are dual to each other with respect to combinatorial Alexander duality. The behaviour of the…

We initiate a classification of complex polynomials f of degree d having the top Betti number of the general fibre close to the maximum. We find a range in which the polynomial must have isolated singularities and another range where it may…

We prove that multigraded Betti numbers of a simplicial forest are always either 0 or 1. Moreover a nonzero multidegree appears exactly at one homological degree in the resolution. Our work generalizes work of Bouchat on edge ideals of…

Let $T$ be a perfect binary tree and $I$ be its edge ideal in the polynomial ring $S$. We determine the vertex cover number, independent number, and establish the recursive formula to compute the number of minimal vertex covers. As a…

We prove that the Betti numbers of simplicial complexes of bounded vertex degrees are testable in constant time.

We describe topology of random simplicial complexes in the lower and upper models in the medial regime, i.e. under the assumption that the probability parameters $p_\sigma$ approach neither $0$ nor $1$. We show that nontrivial Betti numbers…

We compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number.

Let $G$ be a chordal graph and $I(G)$ its edge ideal. Let $\beta (I(G)) = (\beta_0, \beta_1, ..., \beta_p)$ denote the Betti sequence of $I(G)$, where $\beta_i$ stands for the $i$th total Betti number of $I(G)$ and where $p$ is the…

We give some explicit bounds for the number of cobordism classes of real algebraic manifolds of real degree less than $d$, and for the size of the sum of $\mod 2$ Betti numbers for the real form of complex manifolds of complex degree less…