Related papers: Growth of Betti Numbers

The Bounded Negativity Conjecture predicts that for any smooth complex surface $X$ there exists a lower bound for the selfintersection of reduced divisors on $X$. This conjecture is open. It is also not known if the existence of such a…

For every integer \(n\ge 3\), every \(1\le \ell\le n-2\), and every sufficiently large integer \(m\), we construct harmonic functions \(u_{m,\ell}\) on the unit ball \(B_1(0)\subset\mathbb{R}^n\) such that the frequency is bounded…

We show that the lower bounds for Betti numbers given in math.GT/9909161 are equalities for a class of racks that includes dihedral and Alexander racks. We confirm a conjecture from the same paper by defining a splitting for the short exact…

The first $\ell^2$ Betti number of a group is non-decreasing under various embeddings arising from first order logic. Strict inequality is proved for elementary embeddings of non-abelian proper subgroups within torsion free hyperbolic…

Starting with an infinite set of non linear Equations for the Li-Keiper coefficients, we first specify a lower bound emerging from the infinite set and give a characterization of it. Then, we propose a possible new upper and lower bound for…

Assuming projective determinacy, we extend Spector's strong version of the Spector-Gandy Theorem to all odd levels of the projective hierarchy: Theorem. For every space $X$ which is a finite product of the natural numbers $N$ and Baire…

Cut ideals, introduced by Sturmfels and Sullivant, are used in phylogenetics and algebraic statistics. We study the minimal free resolutions of cut ideals of tree graphs. By employing basic methods from topological combinatorics, we obtain…

We propose a refinement of the Betti numbers and of the homology with coefficients in a field of a compact ANR in the presence of a continuous real valued function. The refinement of Betti numbers consists of finite configurations of points…

A bound for Betti numbers of sets definable in o-minimal structures is presented. An axiomatic complexity measure is defined, allowing various concrete complexity measures for definable functions to be covered. This includes common concrete…

The present work is concerned with characterizing some algebraic invariants of edge ideals of hypergraphs. To this aim, firstly, we introduce some kinds of combinatorial invariants similar to matching numbers for hypergraphs. Then we…

We prove an analogue of the Approximation Theorem of L^2-Betti numbers by Betti numbers for arbitrary coefficient fields and virtually torsionfree amenable groups. The limit of Betti numbers is identified as the dimension of some module…

We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical…

The total Betti number of the independence complex of a graph is an intriguing graph invariant. Kalai and Meshulam have raised the question on its relation to cycles and the chromatic number of a graph, and a recent conjecture on that theme…

We use Boij-S\"oderberg theory to provide some order of magnitude bounds on algebraic Betti numbers.

We compare the asymptotic grows of the number of rational points on modular varieties of D-elliptic sheaves over finite fields to the grows of their Betti numbers as the degree of the level tends to infinity. This is a generalization to…

Let X be a Riemannian manifold endowed with a co-compact isometric action of an infinite discrete group. We consider L2 spaces of harmonic vector-valued forms on the product manifold X^N, which are invariant with respect to an action of the…

Let $X$ be an irreducible smooth geometrically integral projective surface over a field. In this paper we give an effective bound in terms of the Neron--Severi rank $\rho(X)$ of $X$ for the number of irreducible curves $C$ on $X$ with…

We analyze the vanishing and non-vanishing behavior of the graded Betti numbers for Segre embeddings of products of projective spaces. We give lower bounds for when each of the rows of the Betti table becomes non-zero, and prove that our…

A significant theorem of L\"uck says that the first $L^2$-Betti number of the total space of a fibration vanishes under some conditions on the fundamental groups. The proof is based on constructions on chain complexes. In the present paper,…

Let $G$ be a group with a finite subgroup $H$. We define the $L^2$-multiplicity of an irreducible representation of $H$ in the $L^2$-homology of a proper $G$-CW-complex. These invariants generalize the $L^2$-Betti numbers. Our main results…