arXiv research

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.

Study topological invariants of complexes for Riemannian manifolds.

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

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…

Zeta functions extended to nonorientable surfaces, order of vanishing computed.

Morse inequalities for noncompact manifolds with group action.

Study bounds index of minimal hypersurfaces in curved spaces.

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

New harmonic functions show nodal sets can be topologically complex despite frequency and regularity constraints.

In this article we study algebraic properties of the specific class of groups $\mathcal{G}$ generated by direct products and wreath products. Such class of groups appears in calculation of fundamental groups of orbits of Morse functions on compact manifolds. We prove that for any group $G\in\mathcal{G}$ the ranks of th…

Smooth maps bound Betti numbers of zero sets.

For hyperbolic 3-manifolds, Ruelle zeta function vanishing order is 4 minus Betti number.

We compute the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we study their…

We describe loss surfaces using topological Betti numbers.

A new method uses vectorized summaries of persistence diagrams for efficient hypothesis testing.

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…

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.

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.

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.

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 uses Betti curves to confirm hyperbolic geometry in brain, climate, and financial networks.

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

Study computability of real numbers from group properties.

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 make a detailed study of various (quadratic and linear) Morse-Bott trace functions on the orthogonal groups $O(n)$. We describe the critical loci of the quadratic trace function Tr$(AXBX^T)$ and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of $A$ and $B…

Arithmetic spaces' thin parts are negligible, impacting 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…

The study of Morse functions on 3-manifolds and their Reeb graphs.

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 …