In this paper we present a way of computing a lower bound for genus of any smooth representative of a homology class of positive self-intersection in a smooth four-manifold with second positive Betti number . We study the solutions of the Seiberg-Witten equations on the cylindrical end manifold which is…
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48 results for “positive second intersection form”
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.
The paper calculates 4-manifold properties from a trisection diagram.
problem Computing the homology and intersection form of a 4-manifold from a trisection diagram.
method Expressing the homology and intersection form in terms of curves and intersection pairing on the diagram surface.
result Explicit formulas for the second and third homology groups of and an algorithm to compute the intersection form.
Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. We do it simplicially in the setting of a filtered version of face sets, also cal…
Paper calculates homology and intersection form of trisected 4-manifolds with boundary.
problem Calculating homology and intersection form of trisected 4-manifolds with boundary.
method Uses relative trisection diagrams to calculate homology and intersection form.
result Describes a representative of the second Stiefel-Whitney class using relative trisection diagrams.
Proves intersection properties of minimal hypersurfaces in various spaces.
problem Intersection properties of minimal hypersurfaces in different geometric settings.
method Two approaches: classifications of stable minimal hypersurfaces and conformal change with comparison geometry.
result Intersection properties for minimal hypersurfaces in specific geometric settings, including free boundary minimal hypersurfaces.
Classifies lattices from knot surgeries, defining a concordance invariant.
problem Classifying lattices from knot surgeries.
method Classifies lattices as the intersection form of a four manifold with boundary.
result Defines a concordance invariant and generalizes a theorem on lens space surgeries.
The first part of this paper exposits a simple geometric description of the Kirby-Siebenmann invariant of a 4--manifold in terms of a quadratic refinement of its intersection form. This is the first in a sequence of higher-order intersection invariants of Whitney towers studied by the authors, particularly for the 4--b…
Forms and Complex Manifoldsmath.AG
Study the intersection form on Kähler manifolds of dimension 4 and above.
problem Understand the intersection form on higher-dimensional Kähler manifolds.
method Investigate fundamental properties and applications to birational geometry.
result Present open problems in the relationship between birational invariants and topological invariants.
We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and give its applications to intersection forms with local coefficients of 4-manifolds. The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients. The second is a local …
Sharp curvature condition implies spherical space form structure.
problem Characterizing manifolds with specific curvature properties.
method Proving diffeomorphism and homeomorphism to spherical space forms using curvature conditions.
result Closed manifolds with $4 rac{1}{2}$-positive curvature operator of the second kind are spherical space forms.
Study shows finiteness of certain definite 4-manifolds bounded by rational homology 3-spheres.
problem Understanding the intersection forms of definite 4-manifolds bounded by rational homology 3-spheres.
method Utilized constraints from Donaldson's diagonalization theorem, Frøyshov's correction term invariants, and Ozsváth and Szabó's invariants.
result There are finitely many negative definite lattices up to stable-equivalence that can be realized as intersection forms of 4-manifolds bounded by a rational homology 3-sphere.
Let (M,g) be a compact oriented Einstein 4-manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M,g) is CP2, equipped with its standard Fubini-Study metric.
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.
Formula for dihedral cover signatures of four-manifolds.
problem Signature of dihedral covers of four-manifolds.
method Formula derivation and necessary conditions for intersection forms.
result Sharp necessary condition for two-bridge slice singularities.
Positivity of intersections in 4-manifolds leads to taming symplectic structures.
problem Taming symplectic structures in almost complex 4-manifolds.
method Proof of positivity of intersections of pseudoholomorphic curves.
result Positivity of intersections is stable and leads to taming symplectic structures.
Study the intersection of positive closed currents using tangent currents and King's residue formula.
problem Investigate the intersection of positive closed currents in complex manifolds.
method Employ tangent currents and King's residue formula to establish a natural condition for intersection.
result Derive an integral representation of the intersection of positive closed currents.
Let be the group of automorphisms on the first homology group with coefficient of a closed non-orientable surface preserving the mod intersection form. In this paper, we obtain a finite presentation for $\operatorname{Aut}(H_1(N_g;\mathbb Z_2),\c…
Paper characterizes CP² using Ricci flow and conformal invariants.
problem Characterize CP² using geometric invariants and Ricci flow.
method Introduce conformal invariant β, use Ricci flow to relate β bounds to curvature.
result Proves CP² is the only 4-manifold with positive second positive intersection form and specific β bound.
This paper considers aspects of 4-manifold topology from the point of view of the null cone of a neutral metric, a point of view we call neutral causal topology. In particular, we construct and investigate neutral 4-manifolds with null boundaries that arise from canonical 3- and 4-dimensional settings. A null hypersurf…
Study calculates intersection forms of almost-flat 4-manifolds.
problem Understanding the intersection forms of almost-flat 4-manifolds.
method Calculation of intersection forms for all 4-dimensional almost-flat manifolds.
result Intersection forms of all 4-dimensional almost-flat manifolds have been calculated.
This paper classifies minimal fillings of lens spaces.
problem Determining the smallest smooth negative-definite fillings of lens spaces.
method Classification of minimal fillings based on forbidden subgraphs in plumbing graphs.
result Classification of lens spaces with minimal negative-definite canonical plumbing.
We prove a generalisation of Elkies' theorem to nonunimodular definite forms (and lattices). Combined with inequalities of Froyshov and of Ozsvath and Szabo, this gives a simple test of whether a rational homology 3-sphere may bound a definite four-manifold. As an example we show that small positive surgeries on torus …
Study on second homology group of genus 3 hyperelliptic Torelli group.
problem Understanding the structure of second homology group of genus 3 hyperelliptic Torelli group.
method Analyzing abelian cycles associated with disjoint separating curves and their algebraic properties.
result Simple abelian cycles are linearly independent in the second homology group.
The object of study of this article is compact surfaces in the three-dimensional hyperbolic space with a positive-definite second fundamental form. It is shown that several conditions on the Gaussian curvature of the second fundamental form can be satisfied only by extrinsic spheres.
The paper shows that certain 4D manifolds with specific harmonic forms are diffeomorphic to CP2.
problem Characterizing 4D Riemannian manifolds with harmonic 2-forms of constant length.
method Analyzing properties of manifolds with positive sectional curvature and harmonic 2-forms.
result Compact 4D manifolds with harmonic 2-forms of constant length are diffeomorphic to CP2 under certain conditions.
Proves Riemannian positive mass theorem with singularities.
problem Proves Riemannian positive mass theorem for specific types of singular manifolds.
method Uses initial data sets with a second fundamental form to transfer convexity between different singularity components.
result Proves the theorem for manifolds with some mean-concave components and others mean-convex.
A topologically minimal surface may be isotoped into a normal form with respect to a fixed triangulation. If the intersection with each tetrahedron is simply connected, then the pieces of this normal form are triangles, quadrilaterals, and helicoids. Helical pieces can have any number of positive or negative twists. We…
Smoothing curves carefullymath.GT
Paper proves curves can be smoothed to reduce self-intersection by exactly 1.
problem Prove that the shortest closed geodesic self-intersects exactly k times.
method Carefully smoothing intersection points reduces self-intersection by exactly 1.
result The shortest closed geodesic self-intersects exactly k times for hyperbolic and Riemannian metrics.
Study curvature operator on Riemannian manifolds, proving new classification results.
problem Classifying Riemannian manifolds based on the curvature operator of the second kind.
method Analyzing the curvature operator and proving classification theorems.
result Closed manifolds with specific curvature properties are classified.
Classifies definite forms from surgeries on knots with small slice genus.
problem Classifying definite forms from surgeries on knots with specific properties.
method Uses Yang--Mills instanton gauge theory and Heegaard Floer correction terms.
result Classifies positive definite intersection forms for surgeries on knots with slice genus at most 2.
We define non-pluripolar products of closed positive currents on a compact Kaehler manifold. We show that a positive non-pluripolar measure can be written in a unique way as the top degree self-intersection (in the non-pluripolar sense) of a closed positive current in given big cohomology class. The solution is shown t…
The study classifies immersed surfaces with knot group Z in simply-connected 4-manifolds.
problem Classifying immersed surfaces with specific knot groups in simply-connected 4-manifolds.
method Classification via equivariant intersection form and secondary invariant.
result Criteria for isotopy and enumeration of Z-disks in D^4.
An oriented compact 4-manifold with boundary is called a positon (resp. negaton) if its intersection form is positive definite (resp. negative definite) and it is simply connected. In this paper, we prove that there exist infinitely many knots which cannot bound null-homologous disks either in positons or in …
The paper resolves a conjecture about curvature conditions on manifolds.
problem Investigating curvature conditions on manifolds to settle a conjecture.
method Analyzing curvature of the second kind and using Brendle's PIC1 condition.
result Manifolds with positive curvature of the second kind are diffeomorphic to a sphere.
In this paper we consider three-manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if the initial manifold has positive Ricci curvature and the boundary is convex (nonnegative Second Fundamental form), its metric can be deformed via th…
Complexity in Bolza surfacemath.DG
Study on complexity of systolic geodesics on Bolza surface.
problem Complexity of systolic geodesics on the Bolza surface.
method Analysis of intersections and triangulation of geodesics.
result There are 12 second systolic geodesics forming a triangulation of the surface.
A method to compute intersection numbers with matrices of positive corank.
problem Computing the intersection number of maps with matrices of positive corank.
method Extending intersection number definition to stratified sets, using homotopy invariants and Stiefel manifolds.
result An effective method to compute the intersection number in polynomial cases.
Wall's result extended to 4-manifolds with definite intersection forms.
problem Realizing automorphisms of definite intersection forms.
method Using a specific 4-manifold construction and Wall's original result.
result Automorphisms of definite intersection forms are realized by diffeomorphisms of the constructed 4-manifold.
The paper introduces generalized Lelong numbers for currents and their applications in intersection theory.
problem Defining and studying generalized Lelong numbers for currents in intersection theory.
method Formulating generalized Lelong numbers for closed smooth (j,j)-forms, defining horizontal dimension, and establishing properties and formulas.
result Effective sufficient conditions for defining and continuity of intersections of positive closed currents.
We prove that any compact selfdual Einstein 4-orbifold of positive scalar curvature whose isometry group contains a 2-torus is, up to an orbifold covering, a quaternion Kaehler quotient of (k-1)-dimensional quaternionic projective space by a (k-2)-torus for some . We also obtain a topological classification in…
Basket links are shown to be isotopic to .
problem Understanding isotopy of basket links to torus links.
method Using symmetrized Seifert form congruence to matrix.
result Basket links are isotopic to .
New polynomials defined for virtual knots, calculated up to crossing 4.
problem Defining and calculating invariants for virtual knots.
method Intersection number of curves on a closed surface.
result Intersection polynomials calculated up to crossing 4.
Paper proves positivity of Chern-Weil forms for certain vector bundles.
problem Proving positivity of Chern-Weil forms for Griffiths semipositive vector bundles.
method Analyzing characteristic differential forms and Schur polynomials.
result Positivity of established.
The first part of this paper completes the classification of Whitney towers in the 4-ball that was started in three related papers. We provide an algebraic framework allowing the computations of the graded groups associated to geometric filtrations of classical link concordance by order n (twisted) Whitney towers in th…
The paper proves a category of dg manifolds with finite positive amplitude.
problem Understanding the structure of dg manifolds with finite positive amplitude.
method Using path spaces and homotopy transfer theorem for curved -algebras.
result Proves that dg manifolds of finite positive amplitude form a category of fibrant objects.
Special Lagrangian webbingmath.SG
Constructs geodesics near intersection points of Lagrangian submanifolds.
problem Geodesics of positive Lagrangian submanifolds near intersection points.
method Cylindrical transform and regularity proof.
result First examples of geodesics in arbitrary dimensions.
Minimum algebraic intersection found in hyperbolic surfaces, growing with genus.
problem Finding the minimum algebraic intersection form in hyperbolic surfaces.
method Analyzing algebraic intersection form in moduli space of hyperbolic surfaces.
result Minimum grows in the order of with genus.