We show that, for any k ≥ 1, there exist non-formal compact orientable (k−1)-connected n-manifolds with k-th Betti number bk = b ≥ 0 if and only if n ≥ max{4k − 1, 4k + 3− 2b}.

We show that for any discrete finitely-generated group G and any self-adjoint n-tuple X1, . . . , Xn of generators of the group algebra CG, Voiculescu’s non-microstates free entropy dimension δ(X1, . . . , Xn) is exactly equal to β1(G) − β0(G) + 1, where βi are the L Betti numbers of G.

We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space RPn (e.g. X is the intersection of two real quadrics). We give explicit formulae for its Betti numbers and for those of its double cover in the sphere Sn; we also give similar formulae for level sets of homogeneous quadratic maps to the plane. We discuss some applications o...

Let X = (xij) and Y = (yij) be generic n by n matrices and Z = XY − Y X. Let S = k[x11, . . . , xnn, y11, . . . , ynn], where k is a field, let I be the ideal generated by the entries of Z and let R = S/I . We give a conjecture on the first syzygies of I , show how these can be used to give a conjecture on the canonical module of R. Using this and the Hilbert series of I we give a conjecture on...

Here we are concerned with series involving generalized Fibonacci numbers Un(p,q) and generalized Lucas numbers Vn(p,q). The aim of this paper is to find triples (p,q,r) for which the seriesUn(p,q)/rn and Vn(p,q)/rn (for r running from 0 to infinity) are unconcerned at the introduction of the factor n. The results established in this paper generalize the known fact that the series Fn/2n (Fn the...

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w ∈ Sn is at most the number of elements below w in the Bruhat order, and (B) that equality holds if and only if w avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of thi...

We characterize all pairs of graphs (G1, G2), for which the binomial edge ideal JG1,G2 has linear relations. We show that JG1,G2 has a linear resolution if and only if G1 and G2 are complete and one of them is just an edge. We also compute some of the graded Betti numbers of the binomial edge ideal of a pair of graphs with respect to some graphical terms. In particular, we show that for every p...

Let M be a compact locally conformal hyperkähler manifold. We prove a version of Kodaira-Nakano vanishing theorem for M . This is used to show that M admits no holomorphic differential forms, and the cohomology of the structure sheaf H(OM ) vanishes for i > 1. We also prove that the first Betti number of M is 1. This leads to a structure theorem for locally conformally hyperkähler manifolds, de...

Over the course of this article, we will discuss irrational numbers and several different ways to prove their existence. As is commonly known, the real numbers can be partitioned into rational numbers and irrational numbers. Rational numbers are those which can be represented as a ratio of two integers — i.e., the set { b : a, b ∈ Z, b 6= 0} — and the irrational numbers are those which cannot b...

We show that every closed, oriented, topologically PSC 4-manifold can be obtained via 0 and 1-surgeries from a 4-orbifold with vanishing first Betti number second at most as large the original one.