arXiv research

Upper bounds on Betti numbers via curvature constraints.

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

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

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.

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.

The paper studies Betti numbers of HOMFLYPT homology modules for links.

Sutured manifolds have zero $L^2$-Betti numbers if and only if they are taut.

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

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

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 …

Proves inequality between Betti numbers and total curvature of complex projective manifolds.

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 …

Study Price inequalities and Betti number growth on specific manifolds.

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.

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

Study extends Elkalla's work on subnormal subgroups to $PD_3$-groups, but $L^2$-Betti numbers need verification.

New bounds on manifold Betti numbers derived from semigroup norms.

Arithmetic spaces' thin parts are negligible, impacting Betti numbers.

The paper extends $L^2$-Betti numbers to groups with finite subgroups and introduces approximation theorems.

Study connects lens spaces' fundamental group to their symplectic fillings' second Betti numbers.

Study of twisted $L^2$-Betti numbers in manifolds.

Constructs non-Kähler Calabi-Yau 3-folds with large Betti numbers.

Gromov-Thurston covers have Betti numbers as expected.

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 …

Study $f$-Laplace bounds on gradient Ricci shrinkers, applying to Betti numbers.

The study bounds growth of Hodge numbers and computes $L^2$-Betti numbers for irregular varieties.

Study topological invariants of complexes for Riemannian manifolds.

We study Kahler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold $K$ has $l\ge2$ boundary components (possibly $l=\infty$), then it has first betti number at least $l-1$, and the Levi form of any boundary component is zero. If $K…

Positive braids have a signature bound by their Betti number.

New method to decompose 4-manifolds with positive scalar curvature.

The study shows conditions for almost complex structures on rational homology spheres and limits on their Betti numbers.