We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…
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48 results for “rational homology disk”
New examples show non-locally-flat PL-disk bounds in rational homology balls but not in integer homology balls.
problem Characterizing knots that bound PL-disks in integer homology balls.
method Involutive Heegaard Floer homology formal properties.
result Found infinitely many manifold-knot pairs (Y, J) where J does not bound a PL-disk in an integer homology ball but does in a rational homology ball.
In this paper we show that if the minimal good resolution graph of a normal surface singularity contains at least two nodes (i.e. vertex with valency at least 3) then the singularity does not admit a smoothing with Milnor fiber having rational homology equal to the rational homology of the 4-disk (called a ration…
Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
problem Characterizing manifolds that are both rational homology spheres and double disk bundles.
method Analyzing the structure of manifolds as unions of disk bundles and using properties of rational homology and cohomology.
result Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
Generalizes Lefschetz fibrations with rational homology disk smoothings.
problem Understanding rational homology disk smoothings of surface singularities.
method Introduces a genus to generic fibers of Lefschetz fibrations.
result Families of relations in mapping class groups represent smoothings.
In this paper we give a necessary combinatorial condition for a negative--definite plumbing tree to be suitable for rational blow--down, or to be the graph of a complex surface singularity which admits a rational homology disk smoothing. New examples of surface singularities with rational homology disk smoothings are a…
This paper completes the classification of certain surface singularities with rational homology disk smoothings.
problem Classifying surface singularities with rational homology disk smoothings.
method Study of configurations of rational curves on projective rational surfaces.
result There is a unique rational homology disk smoothing component except in the cases of an obvious symmetry of the resolution dual graph.
Classifies degenerations of complex projective plane with rational singularities.
problem Classifying singularities of complex projective plane.
method Assuming Wahl's conjecture, classifies degenerations using rational homology disk smoothing.
result Classifies surfaces with rational singularities, including new degenerations with non-log canonical singularities.
Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.
problem Understanding which Seifert fibered spaces can be boundaries of symplectic rational homology balls.
method Analyzes convex boundaries and Lagrangian disk fillings of Legendrian knots.
result Strong restrictions on which Seifert fibered spaces can bound symplectic rational homology balls.
New findings on knots that are both topologically and rationally slice.
problem Understanding knots that are both topologically and rationally slice.
method Analyzing the concordance group of knots in .
result There are infinitely many topologically slice knots that are strongly rationally slice but not slice.
Fintushel and Stern showed that the Brieskorn sphere bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to p…
New 4-manifold accounts for rationally slice knots.
problem Characterize rationally slice knots.
method Show independence of construction of rational homology ball .
result Single 4-manifold accounts for all known rationally slice knots.
New knots show linear independence in slice concordance.
problem Understanding the structure of rationally slice knots.
method Provided an infinite family of knots that are linearly independent.
result Found knots that are linearly independent and infinite order.
New Stein fillings found for non-weighted homogeneous singularities.
problem Finding Stein fillings for certain non-weighted homogeneous singularities.
method Using spinal open books and nearly Lefschetz fibrations.
result Stein rational homology disk fillings for non-weighted homogeneous singularities.
We construct a combinatorial invariant of Legendrian knots in standard contact three-space. This invariant, which encodes rational relative Symplectic Field Theory and extends contact homology, counts holomorphic disks with an arbitrary number of positive punctures. The construction uses ideas from string topology.
On symplectic capsmath.SG
An important class of contact 3--manifolds are those that arise as links of rational surface singularities with reduced fundamental cycle. We explicitly describe symplectic caps (concave fillings) of such contact 3--manifolds. As an application, we present a new obstruction for such singularities to admit rational homo…
The paper resolves fundamental groups for three exceptional surface singularity families.
problem Determining the fundamental groups for three exceptional families of surface singularities.
method New explicit constructions and the Pinkham method for some families.
result Fundamental groups for three exceptional families are determined.
Study on disk configurations in strips shows stability patterns.
problem Understanding stability patterns in disk configurations in strips.
method Finite presentation of rational homology groups, representation stability.
result Disk configuration space exhibits first-order representation stability.
We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface , where is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of . For …
We prove that if M is a closed, connected, oriented, rationally inessential manifold, then the Hofer-Zehnder capacity of the unit disk bundle of the cotangent bundle of M is finite.
Let be a regular neighborhood of a negative chain of -spheres (i.e. exceptional divisor of a cyclic quotient singularity), and let be a rational homology ball which is smoothly embedded in . Assume that the embedding is simple, i.e. the corresponding rational blow-up can be obtained by just a sequen…
Khovanov homology fails to differentiate certain slice disks.
problem Differentiating roll-spun slice disks from trivial ones.
method Using Khovanov homology and Morse theory.
result Khovanov homology cannot distinguish roll-spun slice disks from trivial ones.
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.
Thanks to the recent work of Bhupal, Stipsicz, Szabo, and the author, one has a complete list of resolution graphs of weighted homogeneous complex surface singularities admitting a rational homology disk ("QHD") smoothing, i.e., one with Milnor number 0. They fall into several classes, the most interesting of which are…
The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.
problem Constructing rational homology 3-spheres that bound rational homology 4-balls.
method Exploring plumbed 3-manifolds and using rational homology circles.
result Infinite families of rational homology 3-spheres that bound rational homology 4-balls.
The study classifies manifolds that can be split into two disk bundles.
problem Understanding manifolds that can be decomposed into two disk bundles.
method Established through topological restrictions and rational ellipticity.
result Classification of manifolds up to diffeomorphism in dimensions five and six.
Quantum groups created from disk configuration space homologies.
problem Creating quantum groups from algebraic structures.
method Reconstructing quantum groups from homologies of configuration spaces of disks.
result New combinatorics and actual submanifolds of configuration spaces.
New Khovanov homology for links with multiple punctures.
problem Defining a new Khovanov homology for links with multiple punctures.
method Defined a variant of Khovanov homology for links in thickened disks with multiple punctures, related to previous work by spectral sequences.
result Spectral sequences recover annular Khovanov homology to Khovanov homology.
The homotopy fiber of the inclusion from the long embedding space to the long immersion space is known to be an iterated based loop space (if the codimension is greater than two). In this paper we deloop the homotopy fiber to obtain the topological Stiefel manifold, combining results of Lashof and of Lees. We also give…
The paper explores slice disks and their properties using satellite operations and knot Floer homology.
problem Tackles the conjecture about slice disks and their positive Whitehead doubles.
method Uses techniques from knot Floer homology, Seiberg-Witten theory, and Khovanov homology.
result Provides evidence for the conjecture and constructs exotic disks.
Classifies torus bundles bounding 4-manifolds with rational homology.
problem Classifying torus bundles over the circle that bound 4-manifolds with rational homology.
method Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
result Completely classified torus bundles over the circle that bound 4-manifolds with rational homology.
We study the classification of slice disks of knots up to isotopy and diffeomorphism using an invariant in knot Floer homology. We compute the invariant of a slice disk obtained by deform-spinning, and show that it can be effectively used to distinguish non-isotopic slice disks with diffeomorphic complements. Given a s…
Spectral sequence connects knot homologies to quotient knots.
problem Distinguishing knots using homology.
method Construct spectral sequence relating Khovanov homology to quotient knots.
result Khovanov homology distinguishes certain slice disks.
Paper proves a conjecture about a Heegaard Floer invariant for certain rational homology spheres.
problem Proving a conjecture about the Heegaard Floer d-invariant for negative-definite plumbed rational homology spheres.
method Using Zemke's isomorphism between lattice and Heegaard Floer homology, the paper proves Némethi's conjecture.
result The conjecture about the Heegaard Floer d-invariant for negative-definite plumbed rational homology spheres is proven.
We give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology bal…
Factorization homology theories of topological manifolds, after Beilinson, Drinfeld and Lurie, are homology-type theories for topological -manifolds whose coefficient systems are -disk algebras or -disk stacks. In this work we prove a precise formulation of this idea, giving an axiomatic characterization of fa…
Instanton Floer homology matches Heegaard Floer for almost-rational plumbings.
problem Matching instanton Floer homology with Heegaard Floer for specific 3-manifolds.
method Utilizes lattice homology and a recent cobordism map decomposition theorem.
result Isomorphism between framed instanton Floer homology and Heegaard Floer for almost-rational plumbings.
New 3-manifolds bound rational 4-balls through specific operations.
problem Finding rational homology 3-spheres that bound rational homology 4-balls.
method Two operations that preserve lattice embedding obstruction to bounding rational homology balls.
result Explicit examples of rational surgeries on torus knots that bound rational homology balls.
Homological stability fails for Cremona groups, rational varieties, and function fields.
problem Homological stability in Cremona groups fails in both possible ways.
method Explained the failure of homological stability for Cremona groups.
result Homological stability fails for Cremona groups in both possible ways.
Classifies surgeries on torus knots and cables that bound rational homology balls.
problem Which surgeries on torus knots and cables bound rational homology balls?
method Classification based on integral surgeries and rational numbers q/p for cables.
result Set of rational numbers q/p for cables of a given knot K is bounded.
Researchers found the second homology of Torelli groups for large g.
problem Calculating the second rational homology of Torelli groups.
method Explicit calculations for .
result The second rational homology group is determined for Torelli groups with g >= 6.
New exotic 4D spaces found using knot slicing techniques.
problem Finding non-diffeomorphic exotic 4D spaces.
method Using RBG links to create slice knots with non-diffeomorphic complements.
result Distinguished new exotic 4D spaces using end Floer homology.
In this paper we demonstrate the existence of Sasakian-Einstein structures on certain 2-connected rational homology 7-spheres. These appear to be the first non-regular examples of Sasakian-Einstein metrics on simply connected rational homology spheres. We also briefly describe the rational homology 7-spheres that admit…
Study shows no smooth embeddings of rational homology balls into complex projective plane.
problem Embedding rational homology balls into complex projective plane.
method Elementary arguments to prove non-existence of almost complex embeddings.
result No smooth embeddings of rational homology balls into complex projective plane.
Pinwheels in symplectic rational and ruled surfaces and non-squeezing of rational homology ballsmath.SG
The study explores pinwheels in symplectic surfaces and non-squeezing of rational homology balls.
problem Understanding when Lagrangian pinwheels embed in symplectic rational and ruled surfaces.
method Almost toric fibrations and symplectic rational blow-up.
result A rational homology ball embeds into a rational homology cylinder if and only if the parameter is greater than or equal to 1.
We consider the question of when a rational homology 3-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology embeds in any other …
Study shows Seifert fibered spaces don't bound rational homology balls.
problem Understanding when Seifert fibered spaces bound rational homology balls.
method Analyzes Seifert fibered spaces with different conditions and orientations.
result Characterizes conditions for Seifert fibered spaces to bound rational homology balls.
Study on rational projective planes with small index singularities.
problem Existence and classification of rational homology projective planes with small index quotient singularities.
method Topological and smooth obstructions analysis, classification of singularities.
result Classification of quotient singularities for rational homology projective planes with indices up to three.