Given a knot in the 3-sphere, non-orientable 4-genus or 4-dimensional crosscap number of is minimal first Betti surfaces, smoothly and properly embedded 4-ball, with boundary knot. In this paper, we calculate knots crossing 10.

We study L-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes in [4]. We give a definition of Lcohomology and show how the study of the first L-Betti number can be related to the study of derivations with values in a bi-module of affiliated operators. We show several results about the possibility of extending derivations from sub-algebras and about uniqueness of...

Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first l-Betti number of the group. Our main aim, however, is to present the basic elements of a higher-dimensional analogue on finite and infinite CW-complexes, which relate to the higher l-Betti numbers. One consequence is a ...

We show that Jn, the Stanley-Reisner ideal of the n-cycle, has a free resolution supported on the (n−3)-dimensional simplicial associahedron An. This resolution is not minimal for n > 6; in this case the Betti numbers of Jn are strictly smaller than the f -vector of An. We show that in fact the Betti numbers βd of Jn are in bijection with the number of standard Young tableaux of shape (d + 1, 2...

We show the non-vanishing of cohomology groups of sufficiently small congruence lattices in SL(1, D), where D is a quaternion division algebras defined over a number field E contained inside a solvable extension of a totally real number field. As a corollary, we obtain new examples of compact, arithmetic, hyperbolic three manifolds, with non-torsion first homology group, confirming a conjecture...

We prove the following two conjectures of Gromov. Firstly, all L2-Betti numbers of a manifold fibered over S1 are trivial. Secondly, the first L2-Betti number of a finitely presented group Γ vanishes provided that Γ is an extension {1} −→ ∆ −→ Γ −→ π −→ {1} of finitely presented groups such that ∆ is infinite and π contains Z as a subgroup. We conclude for such a group Γ that its deficiency is ...

The theme of this paper is the connection between topological properties of a closed orientable hyperbolic 3-manifold M and the maximal injectivity radius of M . In [4] we showed that if the first Betti number of M is at least 3 then the maximal injectivity radius of M is at least log 3. By contrast, the best known lower bound for the maximal injectivity radius of M with no topological restrict...

We prove the existence of rigid compact complex surfaces general type whose Chern slopes are arbitrarily close to Bogomolov--Miyaoka--Yau bound $3$. In addition, each these has first Betti number equal $4$.

(i) Y is simply-connected (ii) The first (integral) homology group of Y is trivial (iii) The first (integral) Betti number of Y is zero (iv) The first (integral) cohomology group of Y is trivial (v) The Euler Characteristic of Y is equal to the number of components of S\ Y (vi) Y has one (and hence all six) of the Phragmen-Brouwer properties [6] (The Phragmen-Brouwer properties are listed in th...