arXiv research

We prove that a closed arithmetic hyperbolic 3-manifold with positive first betti number has virtually infinite first betti number.

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over $S^1$ with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over $S^1$. In a different direction, we prove that if the …

New proof shows RFRS groups virtually fibred if their $L^2$-Betti number vanishes.

Study $L^2$-Betti numbers of Dehn fillings for special groups.

Study virtual Betti numbers of 3-manifold mapping tori.

The virtual Betti number conjecture states that any hyperbolic three-manifold has a finite cover with positive first Betti number. We show that this would follow if it were known that the derived series of the fundamental group $G$ of a hyperbolic three-manifold satisfies a certain stability property. The stability pro…

We prove that every finitely presented group with positive first $\ell^2$-Betti number that virtually surjects onto $\mathbb Z$ is acylindrically hyperbolic. In particular, this implies acylindrical hyperbolicity of finitely presented residually finite groups with positive first $\ell^2$-Betti number as well as groups …

Lower bounds of betti numbers for homology groups of racks and quandles will be given using the quotient homomorphism to the orbit quandles. Exact sequences relating various types of homology groups are analyzed. Geometric methods of proving non-triviality of cohomology groups are also given, using virtual knots. The r…

Using the virtual fibering theorem of Agol we show that a sutured 3-manifold $(M, R_+,R_-,γ)$ is taut if and only if the $\ell^2$-Betti numbers of the pair $(M,R_-)$ are zero. As an application we can characterize Thurston norm minimizing surfaces in a 3-manifold $N$ with empty or toroidal boundary by the vanishing of …

We prove that if a closed oriented 4-manifold X fibers over a 2- or 3-dimensional manifold, in most cases all of its virtual Betti numbers are infinite. In turn, we show that a closed oriented 4-manifold X which is not a tower of torus bundles and fibering over a 2- or 3-dimensional manifold does not admit a torsion sy…

The paper proves a criterion for virtual Euler class one in hyperbolic 3-manifolds.

Study non-vanishing $\ell^2$-Betti numbers for specific groups.

Groups with specific properties have vanishing $\ell^2$-Betti numbers.

Simplified proof of cosmic singularity theorem using new mathematical techniques.

We show that closed, immersed, minimal hypersurfaces in a compact symmetric space satisfy a lower bound on the index plus nullity, which depends linearly on their first Betti number. Moreover, if either the minimal hypersurface satisfies a certain genericity condition, or if the ambient space is a product of two CROSSe…

We show that if M is a surface bundle over S^1 with fiber of genus 2, then for any integer n, M has a finite cover tilde(M) with b_1(tilde(M)) > n. A corollary is that M can be geometrized using only the `non-fiber' case of Thurston's Geometrization Theorem for Haken manifolds.

We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P^2-irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize whi…

Analyzes a finite set of metrics and functions to determine manifold torsion.

The purpose of this paper is to produce restrictions on fundamental groups of manifolds admitting good complexifications by proving the following Cheeger-Gromoll type splitting theorem: Any closed manifold $M$ admitting a good complexification has a finite-sheeted regular covering $M_1$ such that $M_1$ admits a fiber b…

Study on profinite rigidity of direct products of free and surface groups.

Upper bounds on Betti numbers via curvature constraints.

Study shows Betti numbers of curves and orbifolds relate to volume and genus.

Study L^2-Betti numbers in prime characteristic for a conjecture about 2-complex towers.

Research confirms a conjecture about complex manifolds with total Betti number three.

Lower Ricci curvature bound prevents first Betti number from dropping more than dimension in collapsing manifolds.

New relation found between embolic volume and Betti numbers.

This paper grew out of an attempt to find a suitable finite sheeted covering of an aspherical 3-manifold so that the cover either has infinite or trivial first homology group. With this motivation we define a new class of groups. These groups are in some sense eventually perfect. We prove results giving several classes…

Flat open manifolds with full first Betti number have zero curvature.

Estimates Betti numbers of loop spaces of compact manifolds.

Study on lens spaces bounding 4-manifolds with specific Betti numbers.

Study on normalized Betti numbers in non-positively curved manifolds.

Twists agrarian and $\ell^2$-Betti numbers for locally indicable groups.

Study computability of real numbers from group properties.

The paper calculates Betti numbers for special geometric manifolds with curvature constraints.

Found the smallest 4-manifold with a specific Betti number.

Study Euler and Betti numbers of homology groups for a specific type of superalgebra.

We provide a proof for an inequality between volume and L2-Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau's theory of L2-Betti numbers of R-simplicial complexes, and other …

Let G be an arithmetic Kleinian group, and let O be the associated hyperbolic 3-orbifold or 3-manifold. In this paper, we prove that, in many cases, G is large, which means that some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. This has many consequences, including that O has in…

Linear bound on Betti numbers of negatively curved orbifolds.

We present a rough classification of differential forms on a Riemannian manifold, we consider definitions and properties of conformal Killing forms on a compact Riemannian manifold and define Tachibana numbers as an analog of the well known Betti numbers. We state the conditions that characterize these numbers. In the …

In this paper we state and prove Morse type inequalities for Morse functions as well as for closed differential 1-forms. These inequalities involve delocalized Betti numbers. As an immediate consequence, we prove the vanishing of delocalized Betti numbers of manifolds fibering over the circle.

The study finds large Betti numbers in minimal hypersurfaces with positive Ricci curvature.

Study Bochner formula on metric measure spaces for vanishing Betti numbers.

Study estimates index of minimal hypersurfaces using Betti numbers.

The paper estimates Betti numbers for graphs with specific curvatures, proving bounds and characterizing rigidity.

It is shown that the Novikov inequalities for critical points of closed 1-forms hold with the von Neumann Betti numbers replacing the Novikov numbers. As a corollary we obtain a vanishing theorem for $L^2$ cohomology, generalizing a theorem of W. Lueck. We also prove that von Neumann Betti numbers coincide with the Nov…

Study rigidifies torus bundles under first Betti number constraints.

Suppose X is any finite complex with vanishing L^2 Betti number. We prove upper bounds on the Betti numbers for regular coverings of X, sublinear in the order of covering. The bounds are sensitive to the Novikov-Shubin invariants of X, and are improved in the presence of a spectral gap.