arXiv research

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

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

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

Study rigidifies torus bundles under first Betti number constraints.

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 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 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…

Study estimates index of minimal hypersurfaces using Betti numbers.

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

Noncompact RCD spaces with maximal first Betti number are rigid.

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…

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

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

We give a characterization for the Alexander Polynomials of closed orientable 3-manifolds M with first Betti number 1, as well as some partial results for the characterization problem for M having first Betti number > 1. We first prove an analogue of a theorem of Levine: that the product of an Alexander polynomial of M…

Upper bounds on revised first Betti number and torus stability for RCD spaces.

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…

New classification for Vaisman manifolds with specific properties.

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

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

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

We calculate the Lefschetz number of a Galois automorphism in the cohomology of certain arithmetic congruence groups arising from orders in quaternion algebras over number fields. As an application we give a lower bound for the first Betti number of a class of arithmetically defined hyperbolic 3-manifolds and we deduce…

The paper bounds the first Betti number and discusses properties of Lefschetz fibrations.

Positive braids have a signature bound by their Betti number.

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

Nonorientable spanning surfaces of periodic knots can have arbitrarily high first Betti number.

The study examines conditions for Haken 3-manifolds and their fundamental groups.

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

Let M be an n-dimensional Kähler manifold with numerically effective Ricci class. In this note we prove that, if the first Betti number b_1(M)=2n, then M is biholomorphic to the complex torus T^n_C.

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 …

Geometrically, the first Betti number of orbits is linked to the Kronrod-Reeb graph.

The paper proves a gap theorem for almost non-negatively curved manifolds.

We study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of $f$-minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number. When the fi…

First we recall homology groups of prer Lie superalgebras. Then introducing double weighted chain spaces, we deal with pre Lie superalgebra of multi-vector fields with polynomial coefficients on n-dimensional number space. The bracket is Schouten bracket. We have several results about Euler number and Betti numbers of …

We derive new estimates for the first Betti number of compact Riemannian manifolds. Our approach relies on the Birman-Schwinger principle and Schatten norm estimates for semigroup differences. In contrast to previous works we do not require any a priori ultracontractivity estimates and we provide bounds which explicitl…

Study bounds index of minimal hypersurfaces in curved spaces.

The study proves the existence of many geodesics on complex manifolds.

We consider Lorentzian manifolds with parallel light-like vector field V. Being parallel and light-like, the orthogonal complement of V induces a codimension one foliation. Assuming compactness of the leaves and non-negative Ricci curvature on the leaves it is known that the first Betti number is bounded by the dimensi…

Improved lower bound for geodesics on manifolds.

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…

Proves almost profinite rigidity for certain free-by-cyclic groups.

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…

The paper computes special values of combinatorial zeta functions to reveal topological properties of manifolds.

It is known that alternative links are pseudoalternating. In 1983 Louis Kauffman conjectured that both classes are identical. In this paper we prove that Kauffman Conjecture holds for those links whose first Betti number is at most 2. However, it is not true in general when this value increases, as we also prove by fin…

Study of Betti numbers in prodsimplicial complexes for directed graphs, focusing on DNA recombination.

The study examines rigidity properties of noncompact manifolds with nonnegative Ricci curvature.

We prove the vanishing of the first Betti number on compact manifolds admitting a Weyl structure whose Ricci tensor satisfies certain positivity conditions, thus obtaining a Bochner-type vanishing theorem in Weyl geometry. We also study compact Hermitian-Weyl manifolds with non-negative symmetric part of the Ricci tens…

The Thurston norm is derived from polytopes and applied to group cohomology.

Smooth maps bound Betti numbers of zero sets.