Many open problems and important theorems in low-dimensional topology have been formulated as statements about certain 2--complexes called gropes. This paper describes a precise correspondence between embedded gropes in 4--manifolds and the failure of the Whitney move in terms of iterated `towers' of Whitney disks. The…
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48 results for “geometric Whitney problem”
Notes on Whitney towers in 4-manifolds, focusing on local surface manipulations and invariants.
problem Classifying and understanding Whitney towers in 4-manifolds.
method Local manipulations of surfaces, definitions of Whitney towers and trees, geometric Jacobi identities, classification of twisted Whitney towers.
result Classification of order n twisted Whitney towers in the 4-ball and related invariants.
We give a geometric proof of existence of Whitney stratifications of definable sets in o-minimal structures.
Study curves on a Whitney umbrella using geometric invariants.
problem Analyzing curves passing through a specific geometric shape.
method Using Darboux frame and Frenet-Serre formulas, define invariants and investigate their properties.
result Identified degrees of divergence and top-terms of invariants.
Constructs Riemannian manifolds to approximate metric spaces.
problem Approximating a metric space with a Riemannian manifold.
method Constructs a Riemannian manifold to approximate a metric space .
result Characterizes metric spaces that can be approximated by Riemannian manifolds.
Geometric trick simplifies link homotopy and concordance.
problem Homotopy and concordance of links in homology spheres.
method Relative Whitney trick to remove double points.
result Links in homology spheres can be simplified to topologically slice links.
This paper computes Whitney tower filtrations of classical links. Whitney towers consist of iterated stages of Whitney disks and allow a tree-valued intersection theory, showing that the associated graded quotients of the filtration are finitely generated abelian groups. Twisted Whitney towers are studied and a new qua…
Study shows Whitney sphere collapses to a point in finite time.
problem Understanding the evolution of Whitney sphere under mean curvature flow.
method Investigated equivariant Lagrangian spheres in \(\mathbb{C}^n\) using mean curvature flow.
result Equivariant Lagrangian spheres collapse to a point in finite time and converge to a plane with multiplicity two.
Classifies links in 3-sphere using Whitney towers in rational homology 4-ball.
problem Classifying links in 3-sphere using geometric methods.
method Complete classifications of links using Whitney towers in rational homology 4-ball.
result Geometric characterization of Milnor invariants and higher order Arf invariants.
Study on unknotted gropes and Whitney towers in 4-sphere.
problem Understanding unknottedness of gropes and Whitney towers.
method Introduced unknottedness of gropes and Whitney towers, proved preservation under modifications, constructed handlebody structures.
result Bi-filtrations of knots do not stabilize, approximating double sliceness.
Study geometric properties of S1 singularities and their deformations.
problem Understanding differential geometric properties of S1 singularities and deformations.
method Representing deformation using diffeomorphisms and isometries, studying geometric properties.
result Differential geometric properties of S1 singularities and Whitney umbrellas in deformations.
Study the geometry of a surface formed by extending a Whitney umbrella.
problem Investigate the geometric properties of a specific surface formed by extending a Whitney umbrella.
method Analyze the intersection with the normal plane, geodesic and normal curvatures, Gaussian and mean curvatures.
result Determine the zeros of curvature functions and deduce geometric relationships.
The Whitney-Graustein theorem states that regular closed curves in the 2-plane are classified, up to regular homotopy, by their rotation number. Here we give a simple proof based on contact geometry.
New proof of Kondo-Tanaka theorem using geometric measure theory.
problem Existence of special systems of Whitney flat 1-forms on homology manifolds.
method Geometric measure theory and tools from non-smooth analysis.
result Simple new proof of Kondo-Tanaka theorem and its converse.
A geometric characterization of the Arf invariant of a knot in the 3-sphere is given in terms of two kinds of 4-dimensional bordisms, half-gropes and Whitney towers. These types of bordisms have associated complexities class and order which filter the condition of bordism by an embedded annulus, i.e. knot concordance, …
The paper develops obstructions for embedding 2D complexes into 4D space.
problem Embedding 2D complexes into 4D space and understanding obstructions.
method Uses Goodwillie-Weiss calculus and intersections of Whitney disks.
result Two approaches to obstructions lead to the same result.
A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety is given by means of the conormal cycle of an embedding of in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. W…
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…
Overview of manifolds of mappings for continuum mechanics.
problem Understanding smooth mappings between manifolds.
method Presentation of manifolds of mappings and their properties.
result Smooth convenient manifold of mappings between manifolds.
This paper describes grope and Whitney tower filtrations on the set of concordance classes of classical links in terms of class and order respectively. Using the tree-valued intersection theory of Whitney towers, the associated graded quotients are shown to be finitely generated abelian groups under a (surprisingly) we…
Computes the group of link homotopy classes of 2-spheres in 4-space.
problem Computing the group of link homotopy classes of link maps of 2-spheres into 4-space.
method Geometric constructions and algebraic duals of immersed Whitney disks.
result The group is free abelian, generated by specific constructions and detected by invariants.
New bounds and examples for sphere unknotting numbers.
problem Comparing unknotting numbers for 2-spheres in 4-space.
method Algebraic and geometric techniques.
result Stabilization number is bounded above by one more than Casson-Whitney number.
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…
This paper describes the relationship between the first non-vanishing Milnor invariants of a classical link and the intersection invariant of a twisted Whitney tower. This is a certain 2-complex in the 4-ball, built from immersed disks bounded by the given link in the 3-sphere together with finitely many `layers' of Wh…
Rough geometric integrationmath.DG
Integrates rough geometric forms on manifolds.
problem Integrating rough forms on complex manifolds.
method Combines Whitney's geometric integration and sewing approaches.
result Introduced distributional k-forms for integration.
Extends functions on symmetric spaces to analytic functions.
problem Extending functions on symmetric spaces to analytic functions.
method Harmonic analysis on symmetric spaces and representation theory of groups.
result Proves Whitney type extension theorems for symmetric spaces.
Paper shows certain generalized Whitney topologies are Baire.
problem Understanding the Baire property in generalized Whitney topologies.
method Analyzing intersections of open and dense sets.
result Generalized Whitney topologies are Baire.
Stiefel-Whitney classes of moment-angle manifolds are trivial.
problem Analyzing the topological properties of moment-angle manifolds.
method Proving triviality of Stiefel-Whitney classes for moment-angle manifolds, including partial quotients.
result Stiefel-Whitney classes of moment-angle manifolds are trivial.
A criterion for Whitney disks connects intersections in 3-manifold homology.
problem Existence of Whitney disks in Heegaard Floer homology.
method Use Nielsen theory to establish a criterion.
result Simple criterion for the existence of Whitney disks.
Study on curves around a Whitney umbrella focusing on geodesic and normal curvatures.
problem Analyzing geometric properties of curves around a specific surface.
method Examined geodesic and normal curvatures, ruled surfaces, and normal developable surfaces.
result Obtained functions representing geometry on a Whitney umbrella.
Extends Whitney's theorem for functions on rough boundaries.
problem Global extension of manifold-valued functions on domains with rough boundaries.
method Using locally convex spaces of compactly-supported sections of vector bundles, proving the existence of an extension operator.
result The restriction map from everywhere-defined functions is a submersion, allowing local linear splittings.
We construct many examples of non-slice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all…
Study connects knot invariants to Whitney tower data.
problem Understanding abelian invariants of knots.
method Relates invariants to intersection data of Whitney towers.
result New algorithm for computing knot invariants.
New forms generalize Whitney forms with rational coefficients for numerical analysis.
problem Numerical problems with singularities near simplex faces.
method Introduce shadow forms and degrees of freedom for integration over faces of blow-up simplices.
result Obtain isomorphism between shadow forms cohomology and cellular cohomology of blow-up simplices.
Criteria for sharksfin and deltoid singularities from plane to plane, with applications.
problem Identifying and understanding singularities in plane-to-plane mappings.
method Providing criteria and geometric meanings for singularities.
result Geometric meanings and criteria for sharksfin and deltoid singularities.
Lagrangian spheres develop singularities under flow, matching Whitney spheres.
problem Understanding singularities in Lagrangian mean curvature flow.
method Analyzing equivariant Lagrangian spheres with Ricci curvature conditions.
result Whitney spheres develop type-II singularities rescaling to a grim reaper and flat subspace.
Link concordance and Whitney towers linked to Milnor invariants.
problem Link concordance and Whitney towers classification.
method Clasper surgeries, Whitney towers, and Milnor invariants.
result Link concordance and Whitney towers classified in terms of Milnor invariants.
We introduce a notion of symmetric Whitney tower cobordism between bordered 3-manifolds, aiming at the study of homology cobordism and link concordance. It is motivated by the symmetric Whitney tower approach to slicing knots and links initiated by Cochran, Orr, and Teichner. We give amenable Cheeger-Gromov rho-invaria…
The Dold-Whitney theorem helps classify bundles and gives a mod 4 slice obstruction.
problem Classifying -bundles and determining sliceability of links.
method Using the Dold-Whitney theorem to classify bundles and relate to the Sato-Levine invariant.
result The Dold-Whitney theorem's mod 4 obstruction coincides with the Sato-Levine invariant.
New method computes link invariants using special Whitney towers.
problem Computing link invariants for 2-component links.
method Cochran towers, special types of twisted Whitney towers.
result Simultaneous computation of invariants for all from a single tower of order .
Real analytic maps can be unstable even if infinitesimal changes are stable.
problem Unstability of real analytic maps despite infinitesimal stability.
method Used a relative version of Whitney's Analytic Approximation Theorem and H. Cartan's Theorems A and B.
result Infinitesimal stability does not imply stability.
Let be a real-analytic manifold and a proper triangulable subanalytic map. Given a subanalytic -form on whose pull-back to every non singular fiber of is exact, we show tha has a relative primitive: there is a subanalytic -form such that . The p…
The Whitney sphere has a unique energy gap for a specific equation.
problem Energy gap phenomenon for the Whitney sphere.
method Solving the equation on Lagrangian surfaces.
result Proves a gap theorem for the Whitney sphere.
We show that the Artin representation on concordance classes of string links induces a well-defined epimorphism modulo order n twisted Whitney tower concordance, and that the kernel of this map is generated by band sums of iterated Bing-doubles of any string knot with nonzero Arf invariant. We also continue J. Levine's…
The paper characterizes Whitney and contact Whitney spheres in complex and Sasakian space forms.
problem Characterizing spheres in complex and Sasakian space forms.
method Establishing optimal integral inequalities involving Ricci curvature and second fundamental form norms.
result New characterizations of Whitney and contact Whitney spheres in complex and Sasakian space forms.
Characterizes Whitney forms on simplices and proves their uniqueness.
problem Characterizing Whitney forms on simplices.
method Proves the uniqueness of differential forms with affine coefficients.
result Whitney forms are the unique differential forms with affine coefficients.
A 4-manifold is parallelizable if its Stiefel-Whitney and Pontryagin classes vanish.
problem Characterizing parallelizable 4-manifolds.
method Classification of -bundles over the 4-sphere using Euler and first Pontryagin classes.
result A closed orientable 4-manifold is parallelizable if and only if its second Stiefel-Whitney class, first Pontryagin class, and Euler characteristic vanish.
We show how to measure the failure of the Whitney trick in dimension 4 by constructing higher- order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with previous…