Topological surgery occurs in natural phenomena where two points are selected and attracting or repelling forces are applied. The two points are connected via an invisible `thread'. In order to model topologically such phenomena we introduce dynamics in 1-, 2- and 3-dimensional topological surgery, by means of attracti…
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48 results for “topological surgery”
Paper shows Ricci flow with surgery converging to Taub-NUT metric.
problem Existence of Ricci flow with surgery and topology change.
method Ricci flow with surgery and local topology change.
result Flow converges to Taub-NUT metric on \mathbb{R}^4.
Contact surgery distancemath.GT
Defines contact surgery distance and shows it's bounded by topological surgery distance by 5.
problem Comparing contact structures on 3-manifolds.
method Defining contact surgery distance and proving upper bound on its value.
result Contact surgery distance is at most 5 larger than topological surgery distance.
In this paper we try to establish a connection between a three-dimensional Lotka--Volterra dynamical system and two-dimensional topological surgery. There are many physical phenomena exhibiting two-dimensional topological surgery through a `hole drilling' process. By our connection, such phenomena may be modelled mathe…
We introduce a collection of 1/2--null 4-dimensional surgery problems. This is an intermediate notion between the classically studied universal surgery models and the -null kernels which are known to admit a solution in the topological category. Using geometric applications of the group-theoretic 2-Engel rela…
Constructs a mean curvature flow with surgery for compact mean convex hypersurfaces.
problem Mean curvature flow with surgery for compact mean convex hypersurfaces.
method Topological surgeries performed by the flow itself through nondegenerate cylindrical singularities, adjusted at smooth times.
result Extends previous results for 2-convex flows and constructs a flow for compact mean convex hypersurfaces.
Topological surgery is a mathematical technique used for creating new manifolds out of known ones. We observe that it occurs in natural phenomena where a sphere of dimension 0 or 1 is selected, forces are applied and the manifold in which they occur changes type. For example, 1-dimensional surgery happens during chromo…
We give a purely topological definition of the perturbative quantum invariants of links and 3-manifolds associated with Chern-Simons field theory. Our definition is as close as possible to one given by Kontsevich. We will also establish some basic properties of these invariants, in particular that they are universally …
Under certain homological hypotheses on a compact 4-manifold, we prove exactness of the topological surgery sequence at the stably smoothable normal invariants. The main examples are the class of finite connected sums of 4-manifolds with certain product geometries. Most of these compact manifolds have non-vanishing sec…
Study surgeries between lens spaces using Heegaard Floer d-invariant.
problem Classify surgeries between lens spaces of type .
method Developed a -invariant surgery formula for -space knots.
result Classified distance one surgeries between lens spaces of the form .
We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics…
We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics…
An introduction to the applications of algebraic surgery to the structure theory of high-dimensional topological manifolds.
Researchers identify graph components for unicellular collections.
problem Understanding connected components of surgery graph for unicellular collections.
method Group-theoretic approach involving mapping class group action.
result Connected components enumerated by a homological invariant.
Topological 4-dimensional surgery is conjectured to fail, in general, for free fundamental groups. M. Freedman and P. Teichner have shown that surgery problems with an arbitrary fundamental group have a solution, provided they satisfy a certain condition on Dwyer's filtration on second homology. We give a new geometric…
We connect topological changes that can occur in -space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena, including relationships between quantum entanglement and wormhole formation. By considering the initial manifold as the -dimensional spatial sectio…
Surgery HQFTmath.QA
Homotopy Quantum Field Theories (HQFTs) generalize more familiar Topological Quantum Field Theories (TQFTs). In generalization of the surgery construction of 3-dimensional TQFTs from modular categories, we use surgery to derive 3-dimensional HQFTs from G-modular categories.
In this paper, we extend the formal definition of topological surgery by introducing new notions in order to model natural phenomena exhibiting it. On the one hand, the common features of the presented natural processes are captured by our schematic models and, on the other hand, our new definitions provide the theoret…
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.
The surgery unknotting number of a Legendrian link is defined as the minimal number of particular oriented surgeries that are required to convert the link into a Legendrian unknot. Lower bounds for the surgery unknotting number are given in terms of classical invariants of the Legendrian link. The surgery unknotting nu…
We use topological surgery in dimension four to give sufficient conditions for the zero framed surgery manifold of a 3-component link to be homology cobordant to the 3-torus, which arises from zero framed surgery on the Borromean rings, via a topological homology cobordism preserving the homotopy classes of the meridia…
Modeling wormhole creation without singularities in relativity.
problem Creating wormholes without singularities in classical relativity.
method Topological surgery and Morse theory to construct a nonsingular wormhole.
result Wormholes can be created nonsingularly in classical relativity.
We give two applications of the 2-Engel relation, classically studied in finite and Lie groups, to the 4-dimensional topological surgery conjecture. The A-B slice problem, a reformulation of the surgery conjecture for free groups, is shown to admit a homotopy solution. We also exhibit a new collection of universal surg…
Quantum Statistics and Spacetime Surgerycond-mat.str-el
We apply the geometric-topology surgery theory on spacetime manifolds to study the constraints of quantum statistics data in 2+1 and 3+1 spacetime dimensions. First, we introduce the fusion data for worldline and worldsheet operators capable creating anyon excitations of particles and strings, well-defined in gapped st…
Contact surgery numbersmath.GT
The paper defines and studies contact surgery numbers for contact 3-manifolds.
problem Understanding the minimal number of components of a surgery link describing a contact 3-manifold.
method Defined and studied various versions of contact surgery numbers, relating them to other invariants and computing specific cases.
result There exist infinitely many non-isotopic contact structures on certain manifolds that cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere.
Study shows knots with specific properties avoid certain topological surgeries.
problem Understanding knots and their behavior under specific surgeries.
method Analyzes null-homotopic knots in 3-manifolds and their effect on surgeries.
result Zero surgery on null-homotopic knots in certain 3-manifolds avoids summands of .
In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topolog…
Topological surgery in dimension is intrinsically connected with the classification of -manifolds and with patterns of natural phenomena. In this expository paper, we present two different approaches for understanding and visualizing the process of -dimensional surgery. In the first approach, we view the proc…
Browder-Novikov-Sullivan-Wall surgery theory investigates the homotopy types of manifolds, using a combination of algebra and topology. It is the aim of these notes to provide an introduction to the more algebraic aspects of the theory (such as the Wall surgery obstruction groups), without losing sight of the geometric…
In this paper we observe that 2-dimensional 0-surgery occurs in natural processes, such as tornado formation and other phenomena reminiscent of hole drilling. Inspired by such phenomena, we introduce new theoretical concepts which enhance the formal definition of 2-dimensional 0-surgery with the observed dynamics. To d…
We apply results from both contact topology and exceptional surgery theory to study when Legendrian surgery on a knot yields a reducible manifold. As an application, we show that a reducible surgery on a non-cabled positive knot of genus g must have slope 2g-1, leading to a proof of the cabling conjecture for positive …
New findings show infinitely many knots cannot be smoothly round handle slices.
problem Understanding the smoothability of knots in 4-dimensional space.
method Analyzing the properties of knots under surgery conjectures and cobordism conjectures.
result Infinitely many knots fail to be smoothly round handle slices.
In a talk at the Cornell Topology Festival in 2005, W. Thurston discussed a graph which we call "The Big Dehn Surgery Graph", B. Here we explore this graph, particularly the link of S^3, and prove facts about the geometry and topology of B. We also investigate some interesting subgraphs and pose what we believe are imp…
Invariants for colored links found, with topological protection of certain knots.
problem Finding invariants for colored links and understanding their topological stability.
method Equivariant bordism groups of Conner and Floyd, topological surgeries.
result Topological protection of certain tricolored knots, with specific transformations.
Even though the disk embedding theorem is not available in dimension 4 for free fundamental groups, some surgery problems may be shown to have topological solutions. We prove that surgery problems may be solved if one considers closed 4-manifolds and the intersection pairing is extended from the integers, and prove a r…
We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon.
We prove that the canonical 4-dimensional surgery problems can be solved after passing to a double cover. This contrasts the long-standing conjecture about the validity of the topological surgery theorem for arbitrary fundamental groups (without passing to a cover). As a corollary, the surgery conjecture is reformulate…
In this paper we will show that two surfaces of the same genus and homology class in a simply connected 4-manifold are concordant. We will show they are often topologically isotopic when their complements have cyclic fundamental group. Finally, we will show that if they are 0-concordant, then surgery on one is equivale…
The paper explores extensions of local moves on string links and their relation to ribbon surfaces.
problem Finding a unique extension of classical local moves on string links.
method Relating local moves to surgeries on ribbon surfaces in R^4.
result There can be at most one unique welded extension of a classical move that is a ribbon residue.
Higher surgeries preserve Steklov spectra in 3D and above.
problem Effect of topology changes on Steklov eigenvalues in higher dimensions.
method Perform surgeries of codimension 2 or higher on compact manifolds.
result Topology changes do not affect Steklov spectra in dimensions 3 and above.
Determines higher smooth surgery structure sets of complex projective spaces.
problem Understanding the higher smooth surgery structure sets of complex projective spaces.
method Analyzes the free subgroup and torsion in low dimensions.
result Obtains information in all dimensions for the free subgroup.
Modular knots follow Chebotarev law from surgeries on hyperbolic fibered links.
problem Understanding knots that behave like prime numbers.
method Analyzing planetary links and surgeries on hyperbolic fibered links in .
result Modular knots around torus knots follow the Chebotarev law.
Contact round surgery of contact 3-manifolds is introduced in this paper. By using this method, an alternative proof of the existence of a contact structure on any closed orientable 3-manifold is given. It is also proved that any contact structure on any closed orientable 3-manifold is constructed from the standard con…
This paper uses Steenrod homology to simplify surgery on generalized manifolds.
problem Simplifying the surgery process on generalized manifolds.
method Develops Steenrod homology theory to avoid geometric splitting in surgery.
result Constructs elements of Steenrod homology groups for generalized manifolds.
Extends string-net theory to 3D TQFT via surface graphs and surgery.
problem Formulate 3D TQFT using string-net theory.
method Extend string-net construction to 3D TQFT using surface graphs and surgery.
result Alternative description of Turaev-Viro model using string-nets.
Surgery On Foliationsmath.OA
The paper develops surgery theories for foliations and solves a problem posed by Weinberger.
problem Classifying leaves of foliations and verifying Borel conjectures.
method Developed two surgery theories and two kinds of Whitehead torsion for foliations.
result Solved a problem posed by Weinberger and verified conjectures in some cases.
We consider the question: "If the zero-framed surgeries on two oriented knots in the 3-sphere are integral homology cobordant, preserving the homology class of the positive meridians, are the knots themselves concordant?" We show that this question has a negative answer in the smooth category, even for topologically sl…
Cell-Like Maps and Surgerymath.GT
Maps can transform non-homeomorphic manifolds into the same space.
problem Mapping non-homeomorphic manifolds to a common space.
method Cell-like maps for .
result Found non-homeomorphic -manifolds mapping to .