Characteristic-independence of Betti numbers of graph ideals

In this paper, we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first 6 Betti numbers do not depend on the characteristic of the ground field. We also show that, if the number of variables n is at most 10, all Betti numbers are independent of the ground field. For n=11, there exists precisely 4 examples in which the Betti numbers depend on the ground field. This is equivalent to the statement that the homology of flag complexes with at most 10 vertices is torsion free and that there exists precisely 4 non-isomorphic flag complexes with 11 vertices whose homology has torsion.In each of the 4 examples mentioned above the 8th Betti numbers depend on the ground field and so we conclude that the highest Betti number which is always independent of the ground field is either 6 or 7; if the former is true then we show that there must exist a graph with 12 vertices whose 7th Betti number depends on the ground field.     

The analytic torsion of a disc

In this article, we study the Reidemeister torsion and the analytic torsion of the m dimensional disc, with the Ray and Singer homology basis (Adv Math 7:145–210, 1971). We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger–Müller theorem. We use a formula proved by Brüning and Ma (GAFA 16:767–873, 2006) that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary (Lück, J Diff Geom 37:263–322, 1993). Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the two sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang (Asian J Math 4:695–714, 2000), and we compare the results. The results of these work were announced in the study of Hartmann et al. (BUMI 2:529–533, 2009).     

Computation of betti numbers of monomial ideals associated with cyclic polytopes

We give a combinatorial formula for the Betti numbers which appear in a minimal free resolution of the Stanley-Reisner ringk[Δ(P)]=A/I _Δ(P) of the boundary complex Δ(P) of an odd-dimensional cyclic polytopePover a fieldk. A corollary to the formula is that the Betti number sequence ofk[Δ(P)] is unimodal and does not depend on the base fieldk.     

Intersection Poincaré Polynomial for Nagaraj–Seshadri moduli space

We compute Betti numbers of both the components of the moduli space of rank 2 semi‐stable torsion‐free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.     

Applications of Klee’s Dehn–Sommerville Relations

We use Klee’s Dehn–Sommerville relations and other results on face numbers of homology manifolds without boundary to (i) prove Kalai’s conjecture providing lower bounds on the f-vectors of an even-dimensional manifold with all but the middle Betti number vanishing, (ii) verify Kühnel’s conjecture that gives an upper bound on the middle Betti number of a 2k-dimensional manifold in terms of k and the number of vertices, and (iii) partially prove Kühnel’s conjecture providing upper bounds on other Betti numbers of odd- and even-dimensional manifolds. For manifolds with boundary, we derive an extension of Klee’s Dehn–Sommerville relations and strengthen Kalai’s result on the number of their edges. I. Novik research partially supported by Alfred P. Sloan Research Fellowship and NSF grant DMS-0500748. E. Swartz research partially supported by NSF grant DMS-0600502.     

Extremal Betti Numbers of Vietoris–Rips Complexes

Upper bounds on the Betti numbers over an arbitrary field of Vietoris–Rips complexes are established, and examples of such complexes with large Betti numbers are given.     

The geometry of the moduli space of one-dimensional sheaves

Let Md be the moduli space of stable sheaves on P2with Hilbert polynomial dm+1.In this paper,we determine the effective and the nef cone of the space Md by natural geometric divisors.Main idea is to use the wall-crossing on the space of Bridgeland stability conditions and to compute the intersection numbers of divisors with curves by using the Grothendieck-Riemann-Roch theorem.We also present the stable base locus decomposition of the space M6.As a byproduct,we obtain the Betti numbers of the moduli spaces,which confirm the prediction in physics.     

Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph ℋ appears within the resolution of its edge ideal ℐ(ℋ). We discuss when recursive formulas to compute the graded Betti numbers of ℐ(ℋ) in terms of its sub-hypergraphs can be obtained; these results generalize our previous work (Hà, H.T., Van Tuyl, A. in J. Algebra 309:405–425, 2007) on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are “well behaved.” For such a hypergraph ℋ (and thus, for any simple graph), we give a lower bound for the regularity of ℐ(ℋ) via combinatorial information describing ℋ and an upper bound for the regularity when ℋ=G is a simple graph. We also introduce triangulated hypergraphs that are properly-connected hypergraphs generalizing chordal graphs. When ℋ is a triangulated hypergraph, we explicitly compute the regularity of ℐ(ℋ) and show that the graded Betti numbers of ℐ(ℋ) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs. Dedicated to Anthony V. Geramita on the occasion of his 65th birthday.     

Different Bounds on the Different Betti Numbers of Semi-Algebraic Sets

Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers. We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined by quadratic inequalities.     

Computing the Homology of Semialgebraic Sets. II: General Formulas

Foundations of Computational Mathematics - We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean...     

Different Bounds on the Different Betti Numbers of Semi-Algebraic Sets

   Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers. We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined by quadratic inequalities.     

A Grothendieck module with applications to Poincaré rationality

In this paper, we define a Grothendieck module associated to a Noetherian ring A. This structure is designed to encode relations between A-modules which can be responsible for the relations among Betti numbers and therefore rationality of the Poincaré series. We will define the Grothendieck module, demonstrate that the condition of being torsion in the Grothendieck module implies rationality of the Poincaré series, and provide examples. The paper concludes with an example which demonstrates that the condition of being torsion in the Grothendieck module is strictly stronger than having rational Poincaré series.     

Approximating L 2-Invariants and Homology Growth

In this paper we consider the asymptotic behavior of invariants such as Betti numbers, minimal numbers of generators of singular homology, the order of the torsion subgroup of singular homology, and torsion invariants. We will show that all these vanish in the limit if the CW-complex under consideration fibers in a specific way. In particular we will show that all these vanish in the limit if one considers an aspherical closed manifold which admits a non-trivial S ¹-action or whose fundamental group contains an infinite normal elementary amenable subgroup. By considering classifying spaces we also get results for groups.     

Rationally smooth algebraic monoids

A reductive monoid M is called rationally smooth if it has sufficiently mild singularities as a topological space. We characterize this class of monoids in combinatorial terms. We then use our results to calculate the Betti numbers of certain projective, rationally smooth group embeddings using the “monoid BB-decomposition”.     

On 0-dimensional schemes with all permissible conductor degrees

IfX is a set of distinct points in ℙ² with given graded Betti numbers, we produce a new set of pointsY with the same graded Betti numbers asX which admits all possible conductor degrees according to the graded Betti numbers. Moreover, for such schemes we can compute the conductor degree for each point. We conclude by generalizing the construction of these schemes, obtaining again the same results.     

Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture

We give conjectures on the possible graded Betti numbers ofCohen–Macaulay modules up to multiplication by positiverational numbers. The idea is that the Betti diagrams shouldbe non-negative linear combinations of pure diagrams. The conjecturesare verified in the cases where the structure of resolutionsis known, that is: for modules of codimension two, for Gorensteinalgebras of codimension three and for complete intersections.The motivation for proposing the conjectures comes from theMultiplicity conjecture of Herzog, Huneke and Srinivasan.     

The analytic torsion of a cone over a sphere

We compute the analytic torsion of a cone over a sphere of dimensions 1, 2, and 3, and we conjecture a general formula for the cone over an odd dimensional sphere.     

On the reducibility of the postulation Hilbert scheme

We give a condition in terms of the possible graded Betti numbers compatible with a given Hilbert functionH of 0-dimensional subschemes of ℙ ⁿ which implies the reducibility of the postulation Hilbert scheme and of its subscheme which parametrizes reduced subschemes with Hilbert functionH.     

Empty Simplices of Polytopes and Graded Betti Numbers

The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope and applying a result of Migliore and the author. This approach allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function.     

Iterated homology and decompositions of simplicial complexes

Kalai has conjectured that a simplicial complex can be partitioned into Boolean algebras at least as roughly, as a shifting-preserving collapse sequence of its algebraically shifted complex. In particular, then, a simplicial complex could (conjecturally) be partitioned into Boolean intervals whose sizes are indexed by its iterated Betti numbers, a generalization of ordinary homology Betti numbers. This would imply a long-standing conjecture made (separately) by Garsia and Stanley concerning partitions of Cohen-Macaulay complexes into Boolean intervals. We prove a relaxation of Kalai’s conjecture, showing that a simplicial complex can be partitioned into recursively defined spanning trees of Boolean intervals indexed by its iterated Betti numbers.