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 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.
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 prove that the second Betti number of a compact Riemannian manifold vanishes under certain Ricci curved restriction.
New curvature conditions imply vanishing of Betti numbers for certain manifolds.
problem Understanding Betti numbers and curvature conditions for Riemannian manifolds.
method Analyzing curvature operators of the second kind and their implications on Betti numbers.
result Curvature conditions lead to vanishing of Betti numbers for specific manifolds.
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.
Let M be an irreducible compact hyperkähler manifold of complex dimension six. Under an assumption on the Looijenga-Lunts-Verbitsky decomposition of the cohomology of M, we prove that the second Betti number of M is at most 23.
New method to decompose 4-manifolds with positive scalar curvature.
problem Understanding and decomposing 4-manifolds with positive scalar curvature.
method 0 and 1-surgeries on topologically PSC 4-orbifolds.
result Every closed, oriented, topologically PSC 4-manifold can be obtained from a specific type of 4-orbifold.
We classify simply connected rationally elliptic manifolds of dimension five and those of dimension six with small Betti numbers from the point of view of their rational cohomology structure. We also prove that a geometrically formal rationally elliptic six dimensional manifold, whose second Betti number is two, is rat…
Let M be a compact irreducible hyperkahler manifold, from Bogomolov inequality [V1] we obtain forbidden values of the second Betti number b2 in arbitrary dimension. UPD: Unfortunately, decomposition of dual to BBF-form is not right in the main theorem. Instead of this work, take a look on recent preprints of Sawon…
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.
Study on 4-manifolds with exotic smooth structures and Z_2 fundamental group.
problem Exploring smooth structures on 4-manifolds with specific fundamental groups.
method Construction of irreducible, smooth, oriented, closed, definite 4-manifolds with Z_2 fundamental group and specific Betti numbers.
result Proves existence of infinitely many smooth structures on definite 4-manifolds with positive second Betti number and Z_2 fundamental group.
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. The study proves curvature bounds for hyperkähler manifolds.
problem Proving curvature invariants of hyperkähler manifolds.
method Analytical proof in complex dimension four, experimental proof in higher dimensions, verification for known manifolds.
result The conjectured curvature invariants are proven to be positive/negative for all known hyperkähler manifolds up to dimension eight.
We prove that any simply connected compact 3-Sasakian manifold, of dimension seven, is formal if and only if its second Betti number is b2<2. In the opposite, we show an example of a 7-dimensional Sasaki-Einstein manifold, with second Betti number b2≥2, which is formal. Therefore, such an example does not adm…
Study of Betti numbers in prodsimplicial complexes for directed graphs, focusing on DNA recombination.
problem Analyzing Betti numbers in directed graphs for DNA recombination.
method Custom prodsimplicial complexes for acyclic directed graphs, investigating Betti numbers.
result Investigated Betti numbers and cycles in prodsimplicial complexes for DNA recombination.
This paper proves that on any tamed closed almost complex four-manifold (M,J) whose dimension of J-anti-invariant cohomology is equal to the self-dual second Betti number minus one, there exists a new symplectic form compatible with the given almost complex structure J. In particular, if the self-dual second Bett…
Given a simply connected, closed four manifold, we associate to it a simply connected, closed, spin five manifold. This leads to several consequences : the stable and unstable homotopy groups of such a four manifold is determined by its second Betti number, and the ranks of the homotopy groups can be explicitly calcula…
Study pinched submanifolds, proving homology vanishing results.
problem Understanding the geometry and topology of pinched submanifolds.
method Investigates submanifolds with a pinching condition on extrinsic invariants.
result Homology vanishing theorems for pinched submanifolds.
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 show that the lower bounds for Betti numbers given in math.GT/9909161 are equalities for a class of racks that includes dihedral and Alexander racks. We confirm a conjecture from the same paper by defining a splitting for the short exact sequence of quandle chain complexes. We define isomorphisms between Alexander r…
We give a complete characterization of all possible pairs (v,e), where v is the number of vertices and e is the number of edges, of any simplicial triangulation of an S^k-bundle over S^1. The main point is that Kuhnel's triangulations of S^{2k+1} x S^1 and the nonorientable S^{2k}-bundle over S^1 are unique among all t…
Authors find small triangulations for specific 4-manifolds.
problem Finding optimal triangulations for 4-manifolds.
method Triangulated connected sums of CP^2 and S^2×S^2, conjectured minimal pentachora.
result Triangulations have the smallest number of pentachora for their types.
The study examines the stability of Einstein metrics on Sasaki Einstein and nearly parallel G2 manifolds.
problem Linear instability of Einstein metrics on Sasaki Einstein and nearly parallel G2 manifolds.
method Analysis of the second and third Betti numbers for Sasaki Einstein and nearly parallel G2 manifolds.
result Positive second and third Betti numbers lead to linear instability for the respective manifolds.
Singular fibrations over surfaces generalize Lefschetz fibrations and have new construction methods.
problem Understanding and constructing singular fibrations over surfaces.
method Explains how to construct examples of singular fibrations with a single singularity and outlines previous results.
result Closed orientable 4-manifolds with large first Betti number and vanishing second Betti number do not admit singular fibrations.
Study handles in 4-manifolds with cyclic fundamental group.
problem Understanding handle decompositions for specific 4-manifolds.
method Constructed handle decompositions based on fundamental group and Betti numbers.
result Handle decomposition formula for specific 4-manifolds.
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.
The second Betti number of a smooth, closed, connected and simply connected, four-dimensional spin manifold is greater or equal 11/8 times the abolute value of its signature.
The study classifies manifolds that can be split into two disk bundles.
problem Understanding manifolds that can be decomposed into two disk bundles.
method Established through topological restrictions and rational ellipticity.
result Classification of manifolds up to diffeomorphism in dimensions five and six.
Study shows gap between de Rham and symplectic-Bott-Chern harmonic forms for specific almost-Kähler manifolds.
problem Understanding the gap between de Rham and symplectic-Bott-Chern harmonic forms on specific almost-Kähler manifolds.
method Analyzing the space of de Rham harmonic forms and symplectic-Bott-Chern harmonic forms on closed almost-Kähler manifolds.
result The second non-HLC degree measures the gap between de Rham and symplectic-Bott-Chern harmonic forms.
We classify the systems of T-roots of the flag manifolds M of the exceptional compact simple Lie groups with the second Betti number b2(M)≥2.
New computations for third and fourth cohomology of Lie group spaces.
problem Computing cohomology groups of homogeneous spaces of Lie groups.
method Explicit descriptions of third and fourth real de Rham cohomologies in terms of Lie-theoretic data.
result For a large class of homogeneous spaces, the difference between third and fourth Betti numbers equals the difference between the numbers of simple factors of the ambient group and the associated closed subgroup.
The abstract explores Artin presentations and their connection to 4-manifolds using triangle groups.
problem Characterizing and classifying 4-manifolds using Artin presentations and triangle groups.
method Utilizing triangle groups to find Artin presentations that present the trivial group and determining 4-manifolds with specific properties.
result Identified all Artin presentations on two generators that present the trivial group and all smooth, closed, simply-connected 4-manifolds with specific properties.
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.
We show that a strict, nearly Kähler 6-manifold with either second or third Betti number nonzero is linearly unstable with respect to the ν-entropy of Perelman and hence is dynamically unstable for the Ricci flow.
The study connects curvature operators' positivity to manifold topology.
problem Positivity of curvature operators and their geometric implications.
method Analysis of Garding cones and positivity properties of curvature operators.
result Shifted cone conditions on curvature operators constrain manifold topology.
We show that closed, immersed, minimal hypersurfaces in a compact symmetric space satisfy a lower bound on the index plus nullity, which depends linearly on their first Betti number. Moreover, if either the minimal hypersurface satisfies a certain genericity condition, or if the ambient space is a product of two CROSSe…
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 proved that an arbitrary finite group acting locally linearly, homologically trivially, and pseudofreely on a closed, simply connected 4-manifold must in fact be cyclic and act semifreely, provided the second betti number of the manifold is at least three.
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 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.
We give explicit formulas for the ranks of the third and fourth homotopy groups of all oriented closed simply-connected four manifolds in terms of their second Betti numbers. We also show that the rational homotopy type of these manifolds is classified by their rank and signature.
We prove integral curvature bounds in terms of the Betti numbers for compact submanifolds of the Euclidean space with low codimension. As an application, we obtain topological obstructions for δ-pinched immersions. Furthermore, we obtain intrinsic obstructions for minimal submanifolds in spheres with pinched second f…
The paper proves a criterion for virtual Euler class one in hyperbolic 3-manifolds.
problem Determining the virtual Euler class one in hyperbolic 3-manifolds.
method Analyzing Alexander polynomials and constructing taut foliations.
result Constructing examples of hyperbolic 3-manifolds with virtual Euler class one.