The Dold-Whitney theorem helps classify bundles and gives a mod 4 slice obstruction.
problem Classifying SO(3)-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.
A 4-manifold is parallelizable if its Stiefel-Whitney and Pontryagin classes vanish.
problem Characterizing parallelizable 4-manifolds.
method Classification of SO(4)-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.
Non-trivial Clifford bundle from loop space tangent bundle.
problem Triviality obstruction of Clifford bundle on loop space.
method Constructing Clifford algebra bundle from loop space tangent bundle, showing non-triviality through Stiefel-Whitney and Pontrjagin classes.
result Clifford bundle is non-trivial, obstructed by manifold's Stiefel-Whitney and Pontrjagin classes.
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.
Proves Massey's theorems on complex structure obstructions.
problem Finding complex structures on real vector bundles.
method Fractional Stiefel-Whitney classes and combinations of Pontryagin, Chern, and Euler classes.
result Determines the second obstruction for rank six bundles.
Non-orientable 4-manifolds are simple branched coverings of RP^4 and twisted S^3-bundles.
problem Characterizing non-orientable 4-manifolds as branched coverings.
method Showing that every closed connected non-orientable PL 4-manifold is a simple branched covering of $\RP^4$ and a twisted S3-bundle. result Non-orientable 4-manifolds are simple branched coverings of $\RP^4$ and twisted S3-bundles, with specific conditions on the degree and branch set. The moduli space of Higgs bundles is stratified into complex symplectic submanifolds.
problem Constructing a complex Whitney stratification of the moduli space of Higgs bundles.
method Showed that the orbit type decomposition is a complex Whitney stratification with each stratum being a complex symplectic submanifold.
result The moduli space of Higgs bundles is a stratified complex symplectic space.
Study Hamiltonian formalisms of degenerate gravity Lagrangians.
problem Hamiltonian analysis of degenerate gravity Lagrangians.
method Employed Dirac-Bergmann constraint algorithm and Gotay-Nester-Hinds algorithm.
result Total Hamiltonian functions and Hamilton's equations derived.
The paper studies graded manifolds and their functorial relationship.
problem Understanding the functor between two categories of graded manifolds.
method Examines polynomial filtrations and homogeneity structures, applying the Batchelor-Gawedzki theorem and Borel-Whitney theorem.
result The functor is full and surjective on objects between the categories of graded vector bundles and manifolds.
A canonically defined mod 2 linear dependency current is associated to each collection of m sections of a real rank n vector bundle. This current is supported on the linear dependency set of the collection of sections. It is defined whenever the collection satisfies a weak measure theoretic condition called "atomicity"…
Adding a large trivial bundle gives nonnegative curvature on symmetric spaces.
problem Finding metrics with nonnegative curvature on vector bundles.
method Adding a large trivial bundle to vector bundles over compact rank one symmetric spaces.
result Nonnegative sectional curvature achieved after adding a sufficiently large trivial bundle.
Smooth distributions on subcartesian spaces can be globally finitely generated.
problem Understanding smooth distributions on subcartesian spaces.
method Embedding in Euclidean space, Whitney Embedding Theorem, and distribution theory.
result Smooth generalized distributions and subbundles on connected subcartesian spaces are globally finitely generated.
New dHYM connections found on complex vector bundles.
problem Existence of dHYM connections on higher rank vector bundles.
method Constructing explicit non-trivial examples and providing algebraic conditions.
result First explicit non-trivial dHYM connections on higher rank holomorphic vector bundles.
The paper examines Spin-structures on real Bott manifolds and provides conditions for their existence.
problem Determining the existence of Spin-structures on real Bott manifolds.
method Analyzes the structure of real Bott towers and uses homomorphisms and Stiefel-Whitney classes.
result Formulates necessary and sufficient conditions for Spin-structures on real Bott manifolds when k is even.
The paper identifies 2^(k+1) distinct components of hyperbolic representations.
problem Understanding the structure of representations of non-orientable surfaces.
method Analysis of square map and Stiefel-Whitney classes.
result There are 2^(k+1) connected components of representations.
We express any Courant algebroid bracket by means of a metric connection, and construct a Courant algebroid structure on any orthogonal Whitney sum E⊕C where E is a given Courant algebroid and C is a flat, pseudo- Euclidean vector bundle. Then, we establish the general expression of the bracket of a transitive …
Paper introduces stratified vector bundles and their properties.
problem Understanding singular spaces and their vector bundles.
method Characterization via monoid actions and examples from various fields.
result Functorial properties extended to the stratified case.
This paper investigates the projectivization of real vector bundles over small covers. We first give a necessary and sufficient condition for such a projectivization to be a small cover. Then associated with moment-angle manifolds, we further study the structure of such a projectivization as a small cover. As an applic…
Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles, and defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles, as well as a transformation from matroid b…
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.
The purpose of this paper is to compute determinant index bundles of certain families of Real Dirac type operators on Klein surfaces as elements in the corresponding Grothendieck group of Real line bundles in the sense of Atiyah. On a Klein surface these determinant index bundles have a natural holomorphic description …
Computes conditions for real spinor bundles on pseudo-Riemannian manifolds.
problem Obtaining Dirac operators on real spinor bundles of complex type.
method Using Lipschitz structures and Karoubi Stiefel-Whitney classes, reformulating the problem in terms of Spin^o_α structures.
result Explicit construction and characterization of real spinor bundles of irreducible complex type.
Let h be a Real bundle, in the sense of Atiyah, over a space X. This is a complex vector bundle together with an involution which is compatible with complex conjugation. We use the fact that BU is equipped with a structure of conjugation space, as defined by Hausmann, Holm, and Puppe, to construct equivariant Chern cla…
A proper etale Lie groupoid is modelled as a (noncommutative) spectral geometric space. The spectral triple is built on the algebra of smooth functions on the groupoid base which are invariant under the groupoid action. Stiefel-Whitney classes in Lie groupoid cohomology are introduced to measure the orientability of th…
The paper reduces normal curvature and enhances homology recovery via embedded submanifolds.
problem Recovering the homology of submanifolds with narrow cycles.
method Embedding submanifolds into scaled oriented Grassmannian bundles to reduce normal curvature and stabilize Čech persistent homology.
result The Čech persistent homology is stable with respect to the interleaving distance and provides lower bounds on scales for homology recovery.
This work merges 3-anchored bundles into 3-Lie algebroids.
problem Combining two 3-anchored bundles into a unified structure.
method Develops algebraic framework for merging bundles with mutual actions and cocycle terms.
result Unified setting for 3-Lie algebroids and special cases.
In this paper we define a Poincaré-Reidemeister scalar product on the determinant line of the cohomology of any flat vector bundle over a closed orientable odd-dimensional manifold. It is a combinatorial "torsion-type" invariant which refines the PR-metric, introduced earlier by the first author, and contains an additi…
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 real Bott manifold is the total space of iterated RP^1 bundles starting with a point, where each RP^1 bundle is projectivization of a Whitney sum of two real line bundles. We prove that two real Bott manifolds are diffeomorphic if their cohomology rings with Z/2 coefficients are isomorphic. A real Bott manifold is a …
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.
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.
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.
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 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.
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…
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.
This is a slightly expanded version of the talk given by Ch.O. at the conference "Instantons in complex geometry", at the Steklov Institute in Moscow. The purpose of this talk was to explain the algebraic results of our paper "Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces". In this paper w…
Study complex structures with totally real sections, providing integrability equations.
problem Existence and integrability of complex structures with totally real sections.
method Explicit integrability equations derived from fiberwise Taylor expansions.
result Detailed fiberwise Taylor expansions and integrability equations in a geometric case.
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.
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.
We prove a Theorem on homotheties between two given tangent sphere bundles SrM of a Riemannian manifold M,g of dim≥3, assuming different variable radius functions r and weighted Sasaki metrics induced by the conformal class of g. New examples are shown of manifolds with constant positive or with constan…
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.
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…
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…
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.
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.
New method computes link invariants using special Whitney towers.
problem Computing link invariants βi(L) for 2-component links. method Cochran towers, special types of twisted Whitney towers.
result Simultaneous computation of βi invariants for all i≤k from a single tower of order 2k.