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.
Study shows Gromov's Betti number bound fails for certain intermediate Ricci curvatures.
problem Gromov's Betti number bound for sectional curvature bounded below does not hold for intermediate Ricci curvatures.
method Established a surgery result for Riemannian metrics with Rick>0 and showed failure of Gromov's bound for specific ranges of k. result Gromov's Betti number bound fails for Rick>0 when ⌊n/2floor+2≤k≤n−1. 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 …
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.
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.
Paper defines embolic volume and relates it to Betti number using the covering trick.
problem Relating embolic volume to topological invariants.
method Covering trick from systolic geometry applied to Berger's inequality.
result Relates embolic volume to the first Betti number.
The paper proves a gap theorem for almost non-negatively curved manifolds.
problem Proving a gap theorem for almost non-negatively curved manifolds.
method Two novel technical tools: controlling the spreading of minimal geodesics and Ricci flow smoothing.
result Closed manifolds with bounded geometry are diffeomorphic to torus bundles.
Bounds on homology of hyperbolic orbifolds using simplicial models.
problem Bounding homology of hyperbolic orbifolds.
method Efficient simplicial model of the thick part of hyperbolic orbifolds.
result Linear bounds on Betti numbers and torsion homology in terms of volume.
We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the Lück Approximation Theorem \cite{luck}, which is much stronger than the linear upper bounds on Betti numbers given by Gr…
Study shows local topologies of certain geometric spaces.
problem Local topological properties of geometric spaces.
method Analysis of Gromov-Hausdorff limits of manifolds with bounded Ricci curvature.
result Local b1 vanishes for regular loci in limits of non-collapsed manifolds. The Arnold conjecture is proven for integers using Floer theory.
problem Proving the Arnold conjecture for symplectic manifolds over integers.
method Constructing a Hamiltonian Floer theory over the Novikov ring with integer coefficients.
result The number of 1-periodic orbits is bounded by the total Betti number over Z of the ambient space.
Upper bounds on revised first Betti number and torus stability for RCD spaces.
problem Bounding the revised first Betti number and stability of RCD spaces.
method Proving an upper bound on the rank of the abelianised revised fundamental group and establishing torus stability.
result Spaces with saturated upper bound on revised first Betti number are mGH-close to flat tori.
We prove that Riemannian metrics with a uniform weak norm can be smoothed to having arbitrarily high regularity. This generalizes all previous smoothing results. As a consequence we obtain a generalization of Gromov's almost flat manifold theorem. A uniform Betti number estimate is also obtained.
Studied L2−invariants on stratified spaces, proving their stability.
problem Stability of L2−invariants on stratified spaces. method Defined and analyzed L2-Betti numbers and Novikov-Shubin invariants for compact smoothly stratified pseudo-manifolds with a wedge metric, extending results to these pseudo-manifolds. result Invariance of L2-Betti numbers and Novikov-Shubin invariants under smoothly stratified, strongly stratum preserving homotopy equivalence. Proves almost profinite rigidity for certain free-by-cyclic groups.
problem Profinite rigidity of free-by-cyclic groups.
method Analyzes ranks of fibres, characteristic polynomials, and stretch factors of monodromies.
result Generic free-by-cyclic groups are almost profinitely rigid.
New topological restrictions found for spaces with nonnegative Ricci curvature.
problem Understanding topological properties of spaces with nonnegative Ricci curvature.
method Analyzing complete Riemannian manifolds and RCD(0,n) spaces, applying rigidity and vanishing theorems.
result Proved a Betti number rigidity theorem and a vanishing theorem for simplicial volume.
In the present paper, we consider the family of all compact Alexandrov spaces with curvature bound below having a definite upper diameter bound of a fixed dimension. We introduce the notion of essential coverings by contractible metric balls, and provide a uniform bound on the numbers of contractible metric balls formi…
Study shows closed manifolds close to flat tori under Kato Ricci curvature bounds.
problem Stability of closed Riemannian manifolds with small Kato Ricci curvature.
method Geometric and diffeomorphic stability results for manifolds with small Kato Ricci curvature.
result Closed manifolds with small Kato Ricci curvature are close to flat tori and diffeomorphic to tori.
The aim of this text is to present the concept of systole of a compact riemannian manifold and to give an overview of systolic geometry. I will also present the "regularization technique", which leads to major results in systolic geometry. I will detail how this technique allows to link the systolic volume of some clos…
Gromov showed that there is an upper bound on the Betti numbers of all closed Riemannian n-manifolds of nonnegative sectional curvature. Grove asked whether such manifolds (if simply connected) fall into only finitely many rational homotopy types. We give a negative answer, in fact in dimension 6, which is the smallest…
We shall prove a new non-vanishing theorem for the stable cohomotopy Seiberg-Witten invariant of connected sums of 4-manifolds with positive first Betti number. The non-vanishing theorem enables us to find many new examples of 4-manifolds with non-trivial stable cohomotopy Seiberg-Witten invariants and it also gives a …
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.
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.
We give new counterexamples to a question of Karsten Grove, whether there are only finitely many rational homotopy types among simply connected manifolds satisfying the assumptions of Gromov's Betti number theorem. Our counterexamples are homogeneous Riemannian manifolds, in contrast to previous ones. They consist of t…
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.
It is a well known result of Gromov that all manifolds of a given dimension with positive sectional curvature are subject to a universal bound on the sum of their Betti numbers. On the other hand, there is no such bound for manifolds with positive Ricci curvature: indeed, Perelman constructed positive Ricci metrics on …
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.
In this article, we produce infinite families of 4-manifolds with positive first betti numbers and meeting certain conditions on their homotopy and smooth types so as to conclude the non-vanishing of the stable cohomotopy Seiberg-Witten invariants of their connected sums. Elementary building blocks used in the earlier …
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.
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.
Study fundamental groups of RCD spaces without smoothness or curvature bounds.
problem Understanding fundamental groups of RCD spaces without additional conditions.
method Combining tools from RCD spaces, Gromov-Hausdorff convergence, and splitting theorems.
result Fundamental groups of RCD spaces are controlled by a finite number of generators and have specific properties under convergence.
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.
The paper proves curvature-related dimension bounds for manifolds.
problem Proving dimension bounds for manifolds with positive scalar curvature.
method Using asymptotic cones and linear growth harmonic functions.
result Upper bounds on essential and Hausdorff dimensions of manifolds.
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.