Unified framework for T-duality in both trivial and non-trivial topologies.
problem Unified description of T-duality for metrics and B-fields in non-trivial topology.
method Developed a new unifying framework for T-duality.
result Unified description of T-duality for metrics and B-fields in non-trivial topology.
New families of embeddings in 4-manifolds, topologically trivial but smoothly non-trivial.
problem Constructing non-trivial smooth embeddings of 3-manifolds in 4-manifolds.
method Parameterized families of embeddings, using high-dimensional spheres.
result Embeddings of homology spheres and any 3-manifold in blown-up K3 surfaces.
Paper proves families of singularities can be topologically trivialized.
problem Understanding behavior of singularities under small perturbations.
method Establishes sufficient conditions for embedded topological trivialization.
result New instances of topological stability, including μ-constant deformations. The paper explores connections between topological dynamics and groupoids.
problem Understanding connections between topological dynamics and groupoids.
method Investigates connections between Topological Dynamics, G-Principal Bundles, and Locally Trivial Groupoids.
result Connections between topological dynamics and groupoids are explored.
The paper deals with topologically trivial Legendrian knots in tight and overtwisted contact 3-manifolds. The first part contains a thorough exposition of the proof of the classification of topologically trivial Legendrian knots (i.e. Legendrian knots bounding embedded 2-disks) in tight contact 3-manifolds. This part w…
Surveying methods to create spaces with non-trivial self covers.
problem Creating spaces with non-trivial self covers.
method Surveying techniques for constructing spaces, including low and high dimension continua.
result Discussion of open problems related to self covering phenomena.
Study of minimal surfaces with non-trivial topology in Heisenberg group using loop group method.
problem Constructing and classifying minimal surfaces with non-trivial topology in the Heisenberg group.
method Generalized Weierstrass type representation and loop group method.
result Classification of equivariant minimal surfaces given by one-parameter subgroups.
A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of monodromy groupoid of a topological groupoid generalises those of fundamental groupoi…
Clarifies global structure of Stokes-Dirac structures on manifolds.
problem Global structure of Stokes-Dirac structures on manifolds with non-trivial topology.
method Clarification through consistent boundary conditions.
result Clearer understanding of Stokes-Dirac structures on manifolds.
Classifies topological quandles on simple manifolds.
problem Classifying topological quandles on specific manifolds.
method Analyzing Alexander quandles on real line and unit circle; conjecturing and providing evidence for trivial quandle on closed unit interval.
result Conjecture that only one topological quandle structure exists on closed unit interval.
Study shows knots can be reversed without becoming trivially concordant.
problem Understanding when knots can be reversed without changing their concordance class.
method Constructed an infinite family of knots, analyzed their concordance properties.
result Found an infinitely generated free subgroup in the concordance group of topologically slice knots.
Classifies and identifies Legendrian Θ-graphs and their embeddings.
problem Classifying and identifying Legendrian Θ-graphs and their embeddings.
method Introducing vertex stabilization and twist moves, classifying topologically trivial graphs, and identifying nondestabilizeable embeddings.
result An infinite family of nondestabilizeable Legendrian realizations in the topological class of Θ-graphs.
Study string topology on symmetric spaces, showing non-triviality and nilpotency results.
problem Understanding the structure of string topology on symmetric spaces.
method Used cycles from Bott-Samelson and Ziller to study coproduct and product.
result Showed non-triviality and nilpotency of Chas-Sullivan product and coproduct for higher rank symmetric spaces.
New proof for some knots being topologically slice.
problem Understanding which knots are topologically slice.
method Equivariant topological slice disks for strongly negative amphichiral knots.
result Strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice.
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.
Critiques incorrect fixed point assertions in digital topology.
problem Incorrect or incorrectly proven fixed point assertions in digital topology.
method Critical review of existing assertions.
result Identifies and critiques incorrect fixed point assertions.
We prove the Weinstein conjecture for non-trivial contact connected sums under either of two topological conditions: non-trivial fundamental group or torsion-free homology.
The paper shows homeomorphic encoders are impossible for non-trivial manifolds.
problem Designing a homeomorphic encoder for non-trivial manifolds.
method Topological arguments and constraints derived for manifold-specific encoders.
result Homeomorphic encoders are impractical for non-trivial manifolds.
A generalization of the topological fundamental group is developed in order to exhibit a topologically complete braid group containing Artin's braid group on infinitely many strands with respect to the following notion of convergence: A sequence of braids b(n) converges to the trivial braid iff for each M>0 eventually …
Smoothly isotopic surfaces in a 4-manifold are stable isotopic if trivial in the 2-homology.
problem Investigating whether topologically isotopic surfaces are smoothly isotopic in stabilised 4-manifolds.
method Analyzing the triviality of surfaces in the 2-homology of the ambient 4-manifold and constructing stabilisations.
result The result holds for surfaces trivial in the 2-homology of the 4-manifold and for certain fundamental groups.
Explicit formulas for pretzel knots' Alexander polynomials.
problem Alexander polynomial of pretzel knots
method Provided explicit formulas
result Characterization of pretzel knots with trivial Alexander polynomial
Kotschick and Morita recently discovered factorisations of characteristic classes of transversally symplectic foliations that yield new characteristic classes in foliated cohomology. We describe an alternative construction of such factorisations and construct examples of topologically trivial foliated vector bundles fo…
Homology handles with trivial Alexander polynomial bound a 3D sphere.
problem Understanding when homology handles bound 3D spheres.
method Using Freedman and Quinn's result for Z-homology 3-spheres. result A distinguished homology handle with trivial Alexander polynomial bounds a homology S1imesD3. It is a major unsolved problem as to whether unknot recognition - that is, testing whether a given closed loop in R^3 can be untangled to form a plain circle - has a polynomial time algorithm. In practice, trivial knots (which can be untangled) are typically easy to identify using fast simplification techniques, wherea…
We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. This proof uses only classic results in topology and combinatorial group theory.
Corrects incorrect assertions about fixed points in digital topology.
problem Incorrect or incorrectly proven assertions about fixed points in digital metric spaces.
method Analysis of existing assertions and proofs.
result Identifies and corrects errors in published assertions.
Study on spherical CR manifolds with non-trivial Chern classes.
problem Understanding Chern classes in spherical CR manifolds.
method Construction and proof of constraints on Chern classes.
result Topological obstruction to spherical CR structures on contact manifolds.
Study shows differences between smooth and topological almost concordance of knots.
problem Disparity between smooth and topological almost concordance of knots.
method Defined and analyzed almost concordance in 3-manifolds, proving results through infinite families and specific examples.
result Infinite families of knots exist that are topologically concordant but not smoothly almost concordant.
Study on constraints for topological and smooth realizations of line arrangements and configurations.
problem Investigating constraints on topological and smooth realizations of combinatorial line arrangements and (nk)-configurations. method Exploring constraints via locally-flatly or smoothly embedded 2-spheres, using Furuta's 10/8-Theorem, and G-signature theorem.
result Established a new lower bound for (nk)-configurations, showing n≥k2−5 for topological realizations. The space of closed subgroups of a locally compact topological group is endowed with a natural topology, called the Chabauty topology. We completely describe the space of closed sugroups of the group RxZ, which is not trivial : for example, its fundamental group is uncountable.
The paper improves bounds on topological complexity for certain manifolds.
problem Determining bounds on topological complexity for specific manifolds.
method Analyzing cohomology classes and their pullbacks, using Gromov norm.
result Improved bounds on topological complexity for connected sums.
The paper constructs surfaces with trivial mapping class groups in 4-manifolds.
problem Constructing surfaces with specific topological and smooth properties in simply connected 4-manifolds.
method Constructing infinitely many pairwise topologically inequivalent surfaces with trivial extendable mapping class groups.
result The surfaces have trivial extendable mapping class groups and vanishing first Alexander modules.
We use the 2-loop term of the Kontsevich integral to show that there are (many) knots with trivial Alexander polynomial which don't have a Seifert surface whose genus equals the rank of the Seifert form. This is one of the first applications of the Kontsevich integral to intrinsically 3-dimensional questions in topolog…
Incorrect fixed point results in digital topology are corrected.
problem Incorrect fixed point results in digital topology.
method Analysis of recent papers in digital topology.
result Corrected incorrect conclusions in fixed point results.
We relate the Andrews-Curtis conjecture to the triviality problem for balanced presentations of groups using algorithms from 3-manifold topology. Implementing this algorithm could lead to counterexamples to the Andrews-Curtis conjecture.
Incorrect fixed point results in digital topology are debunked.
problem Incorrect fixed point results in digital topology.
method Mimicking classical topology tools.
result Many recent fixed point results are incorrect or trivial.
The paper corrects and improves previous assertions in digital topology.
problem Incorrect or poorly proven assertions in digital topology.
method Review and correction of existing assertions.
result Improved and corrected assertions in digital topology.
Non-trivial obstructions found for topological solitons in Yang-Mills-Chern-Simons theories.
problem Existence of topological solitons in Yang-Mills-Chern-Simons theories on compact manifolds.
method Cohomological formulations of the calculus of variations, focusing on Yang-Mills-Chern-Simons theories on compact manifolds in odd dimensions.
result Non-trivial obstructions leading to a strong non-existence theorem for topological solitons.
The operation of (untwisted) Whitehead doubling trivializes the Alexander module of a knot (and consequently, all known abelian invariants), and converts knots to topologically slice ones. In this note we show that Whitehead doubling does not trivialize the rational function that equals to the 2-loop part of the Kontse…
We study rank 1 flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank 1 flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.
We announce a solution to several enumeration problems in topology of surfaces. This includes an enumeration of homotopy classes of sections of locally trivial fiber bundles over surfaces and a computation of non-abelian 1-cohomology of surfaces.
We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities Sn−2⊂Sn, n≥5, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement…
The paper corrects and improves previous assertions in digital topology.
problem Incorrect or insufficiently proven assertions in digital topology.
method Review and correction of existing assertions.
result Improved and corrected assertions in digital topology.
In this paper we exhibit infinite families of embedded tori in 4-manifolds that are topologically isotopic but smoothly distinct. The interesting thing about these tori is that they are topologically trivial in the sense that each bounds a topologically embedded solid handlebody. This implies that there are stably ribb…
We show that every non-trivial tame knot or link in R^3 has a quadrisecant, i.e. four collinear points. The quadrisecant must be topologically non-trivial in a precise sense. As an application, we show that a nonsingular, algebraic surface in R^3 which is a knotted torus must have degree at least eight.
Gerbes encode spectral gaps in topological insulators.
problem Capturing spectral gaps in topological insulators.
method Geometric encoding through 'Real' gerbes.
result Gerbes precisely capture spectral gaps.
Algorithms find second and third shortest non-trivial closed walks on surfaces.
problem Finding non-trivial closed walks on surfaces efficiently.
method Algorithms based on careful analysis of shortest curves and configurations.
result Second shortest walk found in O(n2logn) time, third in O(n3) time. Infinitely many links are top concordant to Hopf but not smoothly.
problem Finding links topologically concordant to Hopf but not smoothly.
method Provided infinitely many 2-component links.
result Links have nontrivial Alexander polynomial.