Study on mod 2 Betti numbers of complex hyperplane arrangements and Milnor fiber homology.
problem Determining mod 2 Betti numbers of complex hyperplane arrangement complements.
method Combining the mod 2 Aomoto complex and the transfer long exact sequence.
result First homology of Milnor fiber has non-trivial 2-torsion for the icosidodecahedral arrangement.
Consider the moduli space of framed flat U(2) connections with fixed odd determinant over a surface. Newstead combined some fundamental facts about this moduli space with the Mayer-Vietoris sequence to compute its betti numbers over any field not of characteristic two. We adapt his method in characteristic two to pro…
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(AXBXT) and determine their indices via perfect fillings of tables associated with the multiplicities of the eigenvalues of A and $B…
These short lecture notes provide a brief introduction to the field of homology growth. They are composed out of two lectures, which I have given at the Borel seminar 2017 in Les Diablerets. We give a proof of Lück's approximation theorem, discuss generalizations and mention some related open problems. Then we discuss …
Study on cohomology of level 4 braid group, proving representation stability.
problem Investigating cohomology of level 4 braid group.
method Exact formula for first Betti number, representation stability, representation theory development.
result Representation stability in level 4 braid group.
Let p and l be two distinct prime numbers and let G be a group. We study the asymptotic behaviour of the mod-l Betti numbers in p-adic analytic towers of finite index subgroups. If X is a finite l-group of automorphisms of G, our main theorem allows to lift lower bounds for the mod-l cohomology growth in the fixed poin…
We provide upper bounds on the size of the homology of a closed aspherical Riemannian manifold that only depend on the systole and the volume of balls. Further, we show that linear growth of mod p Betti numbers or exponential growth of torsion homology imply that a closed aspherical manifold is "large".
This paper provides a topological interpretation for number theoretic properties of quantum invariants of 3-manifolds. In particular, it is shown that the p-adic valuation of the quantum SO(3)-invariant of a 3-manifold M, for odd primes p, is bounded below by a linear function of the mod p first betti number of M. Shar…
Consider a Riemann surface X of genus g≥2 equipped with an antiholomorphic involution τ. This induces a natural involution on the moduli space M(r,d) of semistable Higgs bundles of rank r and degree d. If D is a divisor such that τ(D)=D, this restricts to an involution on the moduli space $M(r,D)…
Upper bounds on Betti numbers via curvature constraints.
problem Bounding Betti numbers of Riemannian manifolds.
method Integral bounds on curvature eigenvalues, Bochner technique.
result New curvature condition for vanishing Betti numbers.
Study shows Betti numbers of curves and orbifolds relate to volume and genus.
problem Understanding Betti numbers of Shimura curves and 3-orbifolds.
method Analyzes asymptotic behavior of Betti numbers in relation to volume and genus.
result Gauss-Bonnet equality for Shimura curves and vanishing Betti numbers for 3-orbifolds.
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.
problem Verify a conjecture about 2-complex towers using L^2-Betti numbers.
method Systematically study L^2-Betti numbers in zero and prime characteristic.
result Apply L^2-Betti numbers to verify a conjecture about 2-complex towers.
Research confirms a conjecture about complex manifolds with total Betti number three.
problem Understanding the minimal total Betti number of closed almost complex manifolds.
method Analyzing properties of almost complex manifolds and using topological results.
result The only simply connected closed complex manifold with total Betti number three is the complex projective plane.
Lower Ricci curvature bound prevents first Betti number from dropping more than dimension in collapsing manifolds.
problem Understanding how the first Betti number behaves under manifold collapse with Ricci curvature bounds.
method Analyzing sequences of Riemannian manifolds with lower Ricci curvature bounds.
result The first Betti number cannot drop more than the dimension in collapsing manifolds.
New relation found between embolic volume and Betti numbers.
problem Relating embolic volume to Betti numbers of compact manifolds.
method Proof based on Gromov's method in systolic geometry.
result Improved result compared to previous work by Durumeric.
Flat open manifolds with full first Betti number have zero curvature.
problem Maximal first Betti number rigidity for open manifolds with nonnegative Ricci curvature.
method Proving rigidity for open manifolds with specific curvature conditions and Betti numbers.
result Open manifolds with maximal first Betti number are flat.
Estimates Betti numbers of loop spaces of compact manifolds.
problem Estimating Betti numbers of loop spaces of compact manifolds.
method Using finite Grauert tubes to provide an effective estimate.
result Implication of polynomial estimate in the limit of tube radius.
Study on lens spaces bounding 4-manifolds with specific Betti numbers.
problem Which lens spaces can bound 4-manifolds with second Betti number one?
method Construction of specific 4-manifolds and analysis of lens space boundaries.
result Infinite families of lens spaces can bound 4-manifolds with second Betti number one, but not all.
Study on normalized Betti numbers in non-positively curved manifolds.
problem Convergence of normalized Betti numbers in non-positively curved manifolds.
method Benjamini-Schramm convergence, volume-normalization, irreducible symmetric spaces of noncompact type.
result Normalized Betti numbers converge for non-compact manifolds.
Twists agrarian and ℓ2-Betti numbers for locally indicable groups.
problem Understanding ℓ2-Betti numbers of locally indicable groups. method Using generalised agrarian invariants and twisted Alexander-Thurston norms.
result Twisted ℓ2-Betti numbers are equal to usual ℓ2-Betti numbers rescaled by the dimension of the twisting representation. Study computability of real numbers from group properties.
problem Computability of real numbers from group properties.
method Analyzing L2-Betti numbers and L2-torsion of groups. result Real numbers as L2-Betti numbers or L2-torsion are computable. The paper calculates Betti numbers for special geometric manifolds with curvature constraints.
problem Estimating Betti numbers for nearly G2 and nearly Kähler manifolds with curvature bounds. method Using Weitzenböck formulas and bounds on sectional curvature to estimate Betti numbers.
result Sufficient conditions for vanishing certain Betti numbers based on sectional curvature bounds.
Found the smallest 4-manifold with a specific Betti number.
problem Finding a 4-manifold with a specific Betti number.
method Provided an explicit example of a cork for a 4-manifold.
result First explicit example of a cork with second Betti number 9.
Study Euler and Betti numbers of homology groups for a specific type of superalgebra.
problem Calculating Euler and Betti numbers for homology groups of pre Lie superalgebras.
method Introduced double weighted chain spaces to analyze pre Lie superalgebras of multi-vector fields with polynomial coefficients. Calculated Euler and Betti numbers for these homology groups.
result Derived formulas for Euler and Betti numbers of homology groups of pre Lie superalgebras.
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.
problem Bounding Betti numbers of negatively curved orbifolds.
method Quantitative bound on the homology of spherical quotients.
result Linear growth of Betti numbers with volume, arbitrary field coefficients.
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.
problem Minimal hypersurfaces with large Betti numbers in manifolds with positive Ricci curvature.
method Constructing sequences of manifolds with embedded minimal hypersurfaces.
result Minimal hypersurfaces have unbounded first Betti numbers.
Study Bochner formula on metric measure spaces for vanishing Betti numbers.
problem Vanishing Betti numbers on metric measure spaces.
method Introduce weighted curvature conditions.
result Vanishing of all Betti numbers.
Study estimates index of minimal hypersurfaces using Betti numbers.
problem Estimating the index of unstable minimal hypersurfaces.
method Extends previous method using first Betti number.
result Morse index is bounded by first Betti number.
The paper estimates Betti numbers for graphs with specific curvatures, proving bounds and characterizing rigidity.
problem Estimating Betti numbers for graphs with non-negative curvatures.
method Establishing Betti number estimates for graphs with non-negative Ollivier and Bakry-Émery curvatures.
result Upper bounds on the first Betti number for graphs with non-negative curvatures, with characterizations of 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 L2 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.
problem Understanding the structure of torus fibrations under first Betti number restrictions.
method Established rigidity results and necessary/sufficient conditions for topological splitting.
result Classification of torus bundles under specific 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 L2-Betti numbers of Dehn fillings for special groups.
problem Investigate L2-Betti numbers of Dehn fillings. method Prove L2-Betti numbers equality for virtually special groups. result Verify Singer Conjecture for certain Einstein manifolds.
Study extends Elkalla's work on subnormal subgroups to PD3-groups, but L2-Betti numbers need verification.
problem Verifying L2-Betti numbers for PD3-groups and group pairs. method Algebraic arguments extending Elkalla's work, but reliant on unproven L2-Betti number hypothesis. result Need further research on L2-Betti numbers for general PD3-groups. New bounds on manifold Betti numbers derived from semigroup norms.
problem Estimating the first Betti number of compact Riemannian manifolds.
method Birman-Schwinger principle and Schatten norm estimates for semigroup differences, without ultracontractivity assumptions.
result Explicit bounds on Betti numbers depend on Ricci tensor norms.
Arithmetic spaces' thin parts are negligible, impacting Betti numbers.
problem Understanding the structure of arithmetic locally symmetric spaces.
method Analyzing thin parts and deducing asymptotic results on Betti numbers.
result Arithmetic spaces' thin parts are negligible, impacting Betti numbers.
Study connects lens spaces' fundamental group to their symplectic fillings' second Betti numbers.
problem Relationship between lens spaces' fundamental group and symplectic fillings' second Betti numbers.
method Exploration of minimal symplectic fillings of lens spaces.
result Unified and generalized results on lens spaces' fundamental group and symplectic fillings' second Betti numbers.
Study of twisted L2-Betti numbers in manifolds.
problem Vanishing of twisted L2-Betti numbers in lower dimensions. method Analysis of infinite cyclic covers and cohomology classes.
result Proves a conjecture connecting twisted L2-Euler characteristic to Thurston norm for certain manifolds. 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 …
Constructs non-Kähler Calabi-Yau 3-folds with large Betti numbers.
problem Finding non-Kähler Calabi-Yau 3-folds with high Betti numbers.
method Smoothing normal crossing varieties with trivial dualizing sheaves.
result Constructs non-Kähler Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers.
Gromov-Thurston covers have Betti numbers as expected.
problem Understanding Betti numbers of branched covers of hyperbolic manifolds.
method Analyzing Gromov-Thurston branched covers and their Betti numbers.
result Betti numbers match expectations for non-divisible degree covers.
We prove that every finitely presented group with positive first ℓ2-Betti number that virtually surjects onto Z is acylindrically hyperbolic. In particular, this implies acylindrical hyperbolicity of finitely presented residually finite groups with positive first ℓ2-Betti number as well as groups …
Study f-Laplace bounds on gradient Ricci shrinkers, applying to Betti numbers.
problem Bounding eigenvalues of f-Laplacian on gradient Ricci shrinkers. method Upper and lower bounds established using volume growth rate; extends to vector bundles.
result Explicit upper bounds for Betti numbers derived.
The study bounds growth of Hodge numbers and computes L2-Betti numbers for irregular varieties.
problem Bounding growth of normalized Hodge numbers and computing L2-Betti numbers for irregular varieties. method Analysis of abelian covers, weak generic Nakano vanishing theorem, and convergence of plurigenera.
result Optimal bounds on the growth of normalized Hodge numbers and computation of L2-Betti numbers.