arXiv research

Upper bounds on Betti numbers via curvature constraints.

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.

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.

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 …

Linear bound on Betti numbers of negatively curved orbifolds.

We show that asymptotically the first Betti number, or the arithmetic genus, of a Shimura curve satisfies the Gauss--Bonnet equality. We also show that the first Betti number of a congruence hyperbolic 3--orbifold asymptotically vanishes relatively to hyperbolic volume.

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.

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.

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

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

In arXiv:math/0508510, Rasmussen observed that the Khovanov-Rozansky homology of a link is a finitely generated module over the polynomial ring generated by the components of this link. In the current paper, we study the module structure of the middle HOMFLYPT homology, especially the Betti numbers of this module. For …

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 …

We study the convergence of volume-normalized Betti numbers in Benjamini-Schramm convergent sequences of non-positively curved manifolds with finite volume. In particular, we show that if $X$ is an irreducible symmetric space of noncompact type, $X \neq \mathbb H^3$, and $(M_n)$ is any Benjamini-Schramm convergent sequ…

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.

We prove that the second Betti number of a compact Riemannian manifold vanishes under certain Ricci curved restriction.

In [DJL07] it was shown that if A is an affine hyperplane arrangement in C^n, then at most one of the L^2-Betti numbers of its complement is non--zero. We will prove an analogous statement for complements of any algebraic curve in C^2. Furthermore we also recast and extend results of [LM06] in terms of L^2-Betti number…

Constructs non-Kähler Calabi-Yau manifolds with large Betti numbers.

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 …

Noncompact RCD spaces with maximal first Betti number are rigid.

The first Betti number for a lattice in a classifying space for variations of Hodge structures vanishes.

The paper bounds Betti numbers of complex-hyperbolic manifolds.

We give lower bounds on the maximal injectivity radius for a closed orientable hyperbolic 3-manifold M with first Betti number 2, under some additional topological hypotheses. A corollary of the main result is that if M has first Betti number 2 and contains no fibroid surface then its maximal injectivity radius exceeds…

Study shows Gromov's Betti number bound fails for certain intermediate Ricci curvatures.

We give examples of closed hyperbolic 3-manifolds with first Betti number 2 and 3 for which no sequence of finite abelian covering spaces increases the first Betti number. For 3-manifolds $M$ with first Betti number 2 we give a characterization in terms of some generalized self-linking numbers of $M$, for there to exis…

We define for arbitrary modules over a finite von Neumann algebra $\cala$ a dimension taking values in $[0,\infty]$ which extends the classical notion of von Neumann dimension for finitely generated projective $\cala$-modules and inherits all its useful properties such as additivity, cofinality and continuity. This all…