New stabilization methods for exotic surfaces in 4-manifolds.
problem Understanding and stabilizing exotic surfaces in 4-manifolds.
method Relating internal and external stabilization, proving results using this relation.
result Many exotic surfaces become smoothly isotopic after external stabilization.
Given a simply-connected closed 4-manifold X and a smoothly embedded oriented surface Σ, various constructions based on Fintushel-Stern knot surgery have produced new surfaces in X that are pairwise homeomorphic to Σ, but not diffeomorphic. We prove that for all known examples of surface knots constructed from …
Horizontal surgery on pseudo-Anosov flows yields almost equivalent flows.
problem Understanding and manipulating pseudo-Anosov flows.
method Performing horizontal surgery on pseudo-Anosov flows by cutting along specific annuli and regluing with a Dehn twist.
result Horizontal Goodman surgery on transitive pseudo-Anosov flows yields an almost equivalent flow.
The study shows how stabilizing manifolds with projective spaces affects their homotopy structure.
problem Understanding the homotopy of manifolds stabilized by projective spaces.
method Trace the effect of surgery on product manifolds, showing a loop homotopy decomposition after localization.
result A loop homotopy decomposition of a manifold after stabilization by a projective space is provided.
We provide a proof of the controlled surgery sequence, including stability, in the special case that the local fundamental groups are trivial. Stability is a key ingredient in the construction of exotic homology manifolds by Bryant, Ferry, Mio and Weinberger, but no proof has been available. The development given here …
In this article we use the mean curvature flow with surgery to derive regularity estimates for the level set flow going past Brakke regularity in certain special conditions allowing for 2-convex regions of high density. We also show a stability result for the plane under the level set flow.
Study investigates metrics on manifolds with stable curvature conditions.
problem Investigating the space of Riemannian metrics with surgery stable curvature conditions.
method Utilized surgery stability condition and Gromov-Lawson construction.
result Homotopy type of the space of metrics is invariant under surgeries.
We use the Ozsváth-Szabó contact invariants to distinguish between tight contact structures obtained by Legendrian surgeries on stabilized Legendrian links in tight contact 3-manifolds. We also discuss the implication of our result on the tight contact structures on the Brieskon homology spheres −Σ(2,3,6n−1).
Scl in groups acting on trees is rational and converges to limits.
problem Understanding stable commutator length in group actions on trees.
method Analyzing groups acting on trees with cyclic stabilizers, focusing on stable commutator length and its limits.
result Stable commutator length is rational and converges to limits in surgery families.
We study embedded spheres in 4-manifolds (2-knots) via doubly pointed trisection diagrams, showing that such descriptions are unique up to stabilization and handleslides, and we describe how to obtain trisection diagrams for certain cut-and-paste operations along 2-knots directly from doubly pointed trisection diagrams…
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.
Rotationally invariant Ricci flows are constructed and shown to converge to spacetimes.
problem Constructing and understanding Ricci flows through surgery on rotationally invariant manifolds.
method Rotationally invariant Ricci flow through surgery, convergence to spacetimes, blowup rate analysis.
result Rotationally invariant Ricci flows converge to spacetimes with controlled curvature blowup.
The paper extends stabilization methods to Poincaré Duality complexes.
problem Stabilization of Poincaré Duality complexes and homotopy gyrations.
method Develops new methods for stabilization of Poincaré Duality complexes, including a homotopy theoretic generalization of a gyration.
result Shows there are only finitely many possible homotopy types of gyrations for a fixed Poincaré Duality complex.
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…
We define integral measures of complexity for Heegaard splittings based on the graph dual to the curve complex and on the pants complex defined by Hatcher and Thurston. As the Heegaard splitting is stabilized, the sequence of complexities turns out to converge to a non-trivial limit depending only on the manifold. We t…
We study codimension one foliations with singularities defined locally by Bott-Morse functions on closed oriented manifolds. We carry to this setting the classical concepts of holonomy of invariant sets and stability, and prove a stability theorem in the spirit of the local stability theorem of Reeb. This yields, among…
New mapping classes of knotted surfaces are computed via surgery.
problem Computing extendable mapping classes of knotted surfaces after surgery.
method Using ordinary untwisted rim surgery, compute the exact extendable mapping-class subgroup.
result The extendable mapping-class subgroup is computed precisely.
Non-isotopic Heegaard splittings of non-minimal genus were known previously only for very special 3-manifolds. We show in this paper that they are in fact a wide spread phenomenon in 3-manifold theory: We exhibit a large class of knots and manifolds obtained by Dehn surgery on these knots which admit such splittings. M…
The paper finds small exotic 4-manifolds with free abelian groups.
problem Finding small exotic 4-manifolds with specific fundamental groups.
method Pairwise non-diffeomorphic closed irreducible 4-manifolds with free abelian fundamental group of rank less than three.
result Explicit mechanism to compute equivariant intersection form and stabilization result for exotic structures.
The study examines how twisting a knot affects its homology and stability properties.
problem Investigating the impact of twisting a knot on its homology and stability.
method Use bordered Floer homology and immersed curve invariants.
result Total dimension, τ(K_m), and thickness of K_m are linear functions of m for large m.
Study of cable links of uniformly thick knots, revealing new isotopy phenomena.
problem Understanding Legendrian isotopy in cable links of uniformly thick knots.
method Introduced new technique of Legendrian surgeries to classify Legendrian knots in negative cables of twist knots.
result Found new phenomena of stabilized Legendrian links that are smoothly isotopic but not Legendrian isotopic.
A virtual n-string α is a collection of n oriented smooth generic loops on a surface M. A stabilization of α is a surgery that results in attaching a handle to M along disks avoiding α, and the inverse operation is a destabilization of α. We consider virtual n-strings up to virtual homotopy, i.e., seq…
An elementary stabilization of a Legendrian link L in the spherical cotangent bundle ST∗M of a surface M is a surgery that results in attaching a handle to M along two discs away from the image in M of the projection of the link L. A virtual Legendrian isotopy is a composition of stabilizations, destabiliz…
Formula connects surgeries to Seiberg-Witten invariants.
problem Understanding how surgeries affect Seiberg-Witten invariants.
method Proves surgery formulas for Seiberg-Witten invariants and families.
result Expresses new invariants in terms of original ones.
Let K be a knot in the 3--sphere. An r-surgery on K is left-orderable if the resulting 3--manifold K(r) of the surgery has left-orderable fundamental group, and an r-surgery on K is called an L-space surgery if K(r) is an L-space. A conjecture of Boyer, Gordon and Watson says that non-reducing surgeries on K can be cla…
Study confirms contact cosmetic surgery for most knots, with exceptions.
problem Determining contact cosmetic surgeries for Legendrian knots.
method Analyzing Legendrian knots, including unknots, and using contact surgery theory.
result Some Legendrian unknots have unique contact surgeries without a cosmetic pair.
Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.
problem Cosmetic surgeries on knots in homology spheres and their constraints.
method Rational surgery formula of the Casson-Walker invariant for 2-component links.
result Constraints on knots and surgery slopes for cosmetic surgeries.
Cosmetic surgeries on pretzel knots are unique.
problem Understanding unique three-manifolds from surgeries on knots.
method Analyzing pretzel knots through Dehn surgeries.
result All pretzel knots satisfy the cosmetic surgery conjecture.
Study pochette surgery on 4-manifolds, focusing on 4-spheres.
problem Understanding pochette surgery on 4-spheres and its effects.
method Using linking number of pochette embeddings, compute homology and analyze surgeries.
result Pochette surgery on any homology 4-sphere can be computed via homology, and trivial cord surgeries do not change diffeomorphism type.
New method proves cosmetic surgery conjecture for certain knots.
problem Proving the cosmetic surgery conjecture for specific knots.
method Using filtered instanton homology and Chern-Simons filtration.
result Reduced the conjecture to surgeries of ±2 on genus 2 knots. New Heegaard Floer homology findings block chirally cosmetic surgeries.
problem Chirally cosmetic surgeries on knots and manifolds.
method Heegaard Floer homology, immersed curve formulations, and surgery formula.
result New obstructions to chirally cosmetic surgeries identified.
Round surgery diagrams represent 3-manifolds in S3.
problem Representing and manipulating 3-manifolds in S3. method Introducing round surgery diagrams and defining moves to establish Kirby Calculus.
result Any 3-manifold can be obtained by a round surgery on a framed link in S3. The study calculates and analyzes alternating surgeries for various knots.
problem Identifying and understanding alternating surgeries on knots.
method Algorithmic computation and structural analysis of alternating surgery slopes.
result The set of alternating surgery slopes is algorithmically computable and exhibits interesting phenomena.
We find all Heegaard diagrams with the property "alternating" or "weakly alternating" on a genus two orientable closed surface. Using these diagrams we give infinitely many genus two 3--manifolds, each admits an automorphism whose non-wondering set consists of two Williams solenoids, one attractor and one repeller. The…
The paper constructs homotopy 4-spheres using pochette surgery.
problem Creating homotopy 4-spheres from pochette surgeries.
method Pochette surgery generalizes Gluck surgery to construct embeddings of pochettes into the 4-sphere and proves homotopy 4-spheres are diffeomorphic to the 4-sphere.
result Homotopy 4-spheres obtained from pochette surgeries are all diffeomorphic to the 4-sphere.
Contact round surgeries on (S3,ξst) help in constructing and understanding contact 3-manifolds.
problem Constructing contact 3-manifolds using Legendrian surgeries.
method Introducing contact round surgeries of indices 1 and 2, and associating them with surgery diagrams.
result Every closed connected contact 3-manifold can be obtained by a sequence of contact round surgeries on Legendrian knots in (S3,ξst). This paper concerns the truly or purely cosmetic surgery conjecture. We give a survey on exceptional surgeries and cosmetic surgeries. We prove that the slope of an exceptional truly cosmetic surgery on a hyperbolic knot in S3 must be ±1 and the surgery must be toroidal but not Seifert fibred. As consequence we…
Complete exceptional surgeries identified for two-bridge links.
problem Identifying complete exceptional surgeries on two-bridge links.
method Enumerated all hyperbolic two-bridge links with complete exceptional surgeries.
result Listed all candidate slopes for complete exceptional surgeries.
A theorem of Kirby states that two framed links in the 3-sphere produce orientation-preserving homeomorphic results of surgery if they are related by a sequence of stabilization and handle-slide moves. The purpose of the present paper is twofold: First, we give a sufficient condition for a sequence of handle-slides on …
Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.
problem Investigate the homotopy of blow ups in algebraic and symplectic geometry.
method Develops fibrewise surgery theory and a purely homotopy theoretic approach.
result Obtained homotopy decompositions of the based loop space on blow ups.
Study classifies exceptional surgeries on specific links.
problem Classifying exceptional Dehn surgeries on specific links.
method Examined minimally twisted seven-chain links.
result Classified all exceptional Dehn surgeries on the links.
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.
We show that all exceptional surgeries on hyperbolic alternating knots in the 3-sphere are integral surgeries.
A Seifert surgery is an integral surgery on a knot in S^3 producing a Seifert fiber space which may contain an exceptional fiber of index 0. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; its edges correspond to single twistings along "seiferters" or "annular pair…
New proof for a knot type not admitting certain surgeries.
problem Chirally cosmetic surgeries on non-torus 2-bridge knots.
method Proof by contradiction and geometric analysis.
result Proves non-torus 2-bridge knots do not admit chirally cosmetic surgeries.
Study links with specific surgeries in 4-manifolds.
problem Understanding which links have surgeries yielding specific 4-manifolds.
method Examining trisections and link surgeries in 4-manifolds.
result Found families of links with the specified surgeries, but not all.
Study finds chirally cosmetic surgeries on knots and manifolds, contradicting previous conjectures.
problem Disprove conjectures about chirally cosmetic surgeries and purely cosmetic surgeries.
method Construct infinite families of chirally cosmetic surgeries and purely cosmetic surgeries on specific geometric objects.
result Found chirally cosmetic surgeries and purely cosmetic surgeries that contradict previous conjectures.
New surgery operation preserves monotonicity of Lagrangians.
problem Preserving monotonicity of Lagrangians in surgery operations.
method BSP surgery, wall-crossing formula for disk-potentials.
result BSP surgery can preserve monotonicity of Lagrangians.