arXiv research

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

We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical theorems of Chern and Lashof to complex projective space.

The paper generalizes curvature bounds for submanifolds with singularities.

For a small cover Q^n and any principal (Z_2)^m-bundle M^n over Q^n, it was shown in a previous work of the author that the total sum of Z_2-Betti numbers of M^n is at least 2^m. In this paper, we prove that when M^n is connected, the total sum of Z_2-Betti numbers of such an M^n exactly equals 2^m if and only if M^n i…

We derive a uniform bound for the total betti number of a closed manifold in terms of a Ricci curvature lower bound, a conjugate radius lower bound and a diameter upper bound. The result is based on an angle version of Toponogov comparison estimate for small triangles in a complete manifold with a Ricci curvature lower…

The Arnold conjecture is proven for integers using Floer theory.

The study examines the asymptotic behavior of cohomology groups of algebraic group subgroups.

Proves a conjecture about graph complexes without specific cycle lengths.

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.

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

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.

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, hyperbol…

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.

Positive braids have a signature bound by their Betti number.

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

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…

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

The basic cohomology of a Riemannian foliation on a complete manifold with all leaves closed is the cohomology of the leaf space. In this paper we introduce various methods to compute the basic cohomology in the presence of both closed and non-closed leaves in the simply-connected case (or more generally for Killing fo…

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.