Related papers: Extremal Betti numbers of Rips complexes

In this paper, we shall provide explicit formulas for the extremal Betti numbers of $R/I$, where $I$ is the defining ideal of certain weighted hyperplanes in $\Bbb{P}^{n-1}$ and $R$ is the polynomial ring in $n$ indeterminates over a field.…

We prove a new bound for the number of connected components of a real regular elliptic surface with a real section and we show the sharpness of this bound. Furthermore, all possible values for the Betti numbers of such a surface are…

We survey results related to the magnitude of the Betti numbers of numerical semigroup rings and of their tangent cones.

We prove upper bounds on the sum of Betti numbers of tropical prevarieties in dense and sparse settings. In the dense setting the bound is in terms of the volume of Minkowski sum of Newton polytopes of defining tropical polynomials, or,…

We obtain sharp lower and upper bounds for the number of maximal (under inclusion) independent sets in trees with fixed number of vertices and diameter. All extremal trees are described up to isomorphism.

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 compute the Betti numbers of the geometric spaces associated to nonrational simple convex polytopes and find that they depend on the combinatorial type of the polytope exactly as in the rational case. This shows that the combinatorial…

We obtain strong upper bounds for the Betti numbers of compact complex-hyperbolic manifolds. We use the unitary holonomy to improve the results given by the most direct application of the techniques of [DS17]. We also provide effective…

We show that there exists a saturated graded ideal in a standard graded polynomial ring which has the largest total Betti numbers among all saturated graded ideals for a fixed Hilbert polynomial.

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

We establish upper bounds for the complexity of Seifert fibered manifolds with nonempty boundary. In particular, we obtain potentially sharp bounds on the complexity of torus knot complements.

We are interested in the maximal values of the Betti numbers b_i({\mathbb R}X_m^n) for fixed i,m,n; where {\mathbb R}X_m^n is the real part of a real nonsingular hypersurface of degree m in the complex projective space {\mathbb C}P^n, and…

In this paper we study the number of finite topologies on an $n$-element set subject to various restrictions.

We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by…

We bound from above the expected total Betti number of a high degree random real hypersurface in a smooth real projective manifold. This upper bound is deduced from the equirepartition of critical points of a real Lefschetz pencil…

We construct a family of examples of complete $(2+n)-$dimensional ($n\ge 2$) open manifolds with positive Ricci curvature, sectional curvature bounded from below and infinite Betti numbers $b_2,b_n$, moreover its volume growth can be…

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 derive a uniform bound for the total betti number of a closed manifold in terms of a Ricci curvature lower bound, a conjugate radius lower bound and a diameter upper bound. The result is based on an angle version of Toponogov comparison…

Using a method of F\'elix and Thomas we compute the Betti numbers of unordered configuration spaces of the torus.

Using basic properties of perverse sheaves, we give new upper bounds for compactly supported Betti numbers for arbitrary affine varieties in $\mathbb{A}^n$ defined by $r$ polynomial equations of degrees at most $d$. As arithmetic…