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 was proved by Bonamy, Charbit and Thomassé. We show an upper bound on the total Betti number in terms of the number of vertex disjoint cycles in a graph. Th...

We find a minimal generating set for the defining ideal of the schematic intersection of the set of diagonal matrices with the closure of the conjugacy class of a nilpotent matrix indexed by a hook partition. The structure of this ideal allows us to compute its minimal free resolution and give an explicit description of the graded Betti numbers, and study its Hilbert series and regularity.

We show that there exist non-formal compact oriented manifolds of dimension n and with first Betti number b1 = b ≥ 0 if and only if n ≥ 3 and b ≥ 2, or n ≥ (7 − 2b) and 0 ≤ b ≤ 2. Moreover, we present explicit examples for each one of these cases.

Copyright q 2010 Zisheng Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We extend the classical Bishop-Gromov volume comparison from constant Ricci curvature lower bound to radially symmetric Ricci curvature lower bound...

The purpose of this note is to determine the Picard group of the moduli space of vector bundles over an arbitrary algebraic surface. Since Donaldson’s pioneer work on using moduli of vector bundles to define smooth invariants of an algebraic surface, there has been a surge of interest in understanding the geometry of this moduli space. Among other things, the study of line bundles on this modul...

Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer cohomology group as well as a quilted Lagrangian Floer cohomology group. Each of these is equipped with a chain level grading. We show that the gradings agree.

The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is smooth. We give an explicit com...

We study the Betti numbers of binomial edge ideal associated to some classes of graphs with large Castelnuovo-Mumford regularity. As an application we give several lower bounds of the Castelnuovo-Mumford regularity of arbitrary graphs depending on induced subgraphs.