The conifold is a cone over the space T , which is known to be topologically S × S. The coordinates used in the literature describe a sphere-bundle which can be proven to be topologically trivializable. We provide an explicit trivialization of this bundle, with simultaneous global coordinates for both spheres. Using this trivialization we are able to describe the topology of the base of several...

We consider two varieties associated to a web of quadrics W in P. One is the base locus and the second one is the double cover of P branched along the determinant surface of W . We show that small resolutions of these varieties are Calabi-Yau manifolds. We compute their Betti numbers and show that they are not birational in the generic case. The main result states that if the base locus of W co...

Let R be a standard graded K-algebra, that is, an algebra of the form R = K[x1, . . . , xn]/I where K[x1, . . . , xn] is a polynomial ring over the field K and I is a homogeneous ideal with respect to the grading deg(xi) = 1. Let M be a finitely generated graded R-module. Consider the (essentially unique) minimal graded Rfree resolution of M · · · → Ri → Ri−1 → · · · → R1 → R0 → M → 0 The rank ...

Let A be a semigroup whose only invertible element is 0. For an A-homogeneous ideal we discuss the notions of simple i-syzygies and simple minimal free resolutions of R/I. When I is a lattice ideal, the simple 0-syzygies of R/I are the binomials in I. We show that for an appropriate choice of bases every A-homogeneous minimal free resolution of R/I is simple. We introduce the gcd-complex ∆gcd(b...

for an $n$-gon with vertices at points $1,2,cdots,n$, the betti numbers of its suspension, the simplicial complex that involves two more vertices $n+1$ and $n+2$, is known. in this paper, with a constructive and simple proof, wegeneralize this result to find the minimal free resolution and betti numbers of the $s$-module $s/i$ where  $s=k[x_{1},cdots, x_{n}]$ and $i$ is the associated ideal to ...

0. Introduction 1. L-Betti numbers for CW -complexes of finite type 2. Basic conjectures 3. Low-dimensional manifolds 4. Aspherical manifolds and amenability 5. Approximating L-Betti numbers by ordinary Betti numbers 6. L-Betti numbers and groups 7. Kähler hyperbolic manifolds 8. Novikov-Shubin invariants 9. L-torsion 10. Algebraic dimension theory of finite von Neumann algebras 11. The zero-in...

The behaviour of graded Betti numbers under exterior and symmetric algebraic shifting is studied. It is shown that the extremal Betti numbers are stable under these operations. Moreover, the possible sequences of super extremal Betti numbers for a graded ideal with given Hilbert function are characterized. Finally it is shown that over a field of characteristic 0, the graded Betti numbers of a ...

An important application of the algebraic theory of L2-Betti numbers [10] (see Farber [8] for an alternative approach) is that the L2-Betti numbers β i (Γ ) of a group Γ vanish if it has a normal subgroup whose L2-Betti numbers vanish. With regard to the first L2-Betti number, one can significantly relax the normality condition to obtain similar vanishing results [14]. Peterson and Thom prove i...

Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector-valued functions, called filtering functions. As is well known, in the case of scalar-valued filtering functions, persistent homology groups can be studied through their persistent Betti numbers, i.e. the dimensions of the images of the homomorphisms induced by the inclusions o...

For an $n$-gon with vertices at points $1,2,cdots,n$, the Betti numbers of its suspension, the simplicial complex that involves two more vertices $n+1$ and $n+2$, is known. In this paper, with a constructive and simple proof, wegeneralize this result to find the minimal free resolution and Betti numbers of the $S$-module $S/I$ where  $S=K[x_{1},cdots, x_{n}]$ and $I$ is the associated ideal to ...