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.
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.
Rationally null-homologous links in Seifert fibered spaces may be represented combinatorially via labeled diagrams. We introduce an additional condition on a labeled link diagram and prove that it is equivalent to the existence of a rational Seifert surface for the link. In the case when this condition is satisfied, we…
Proves certain knots are rationally slice and vanishing of certain invariants.
problem Rational sliceness of specific knots.
method Uses Heegaard Floer homology to show vanishing of invariants.
result Certain knots are rationally slice and invariants vanish.
Alexander polynomial detects non-bi-orderability for certain knot groups.
problem Detecting bi-orderability of knot groups.
method Alexander polynomial analysis on fibered knots.
result Alexander polynomial has at least one positive real root for bi-orderable groups.
We verify that the rational blow-down schemes along certain Seifert fibered 3-manifolds found by the second author, Szabo and Wahl are, in fact, symplectic operations.
This paper provides a rational model for fiberwise THH transfer using A-infinity algebras.
problem Rational models for fiberwise THH transfer of fibrations over a base space.
method Explicit description of Hochschild homology transfer in terms of A-infinity algebras.
result Rational models for Becker-Gottlieb transfer and fiberwise THH-simple structures.
New Stein fillings found for rational surface singularities.
problem Exploring Stein fillings of rational surface singularities.
method Using planar open books and Lefschetz fibrations, describe Stein fillings via symplectic disk arrangements.
result Many rational singularities admit Stein fillings not diffeomorphic to Milnor fibers.
Graph manifolds' Thurston norms are sums of linear functionals, and every such norm can be realized.
problem Understanding Thurston norms of graph manifolds and their realizability.
method Analyzing the structure of Thurston norms as sums of linear functionals and showing realizability.
result Every Thurston norm of a graph manifold can be expressed as a sum of absolute values of linear functionals with rational coefficients.
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.
Knots in L-spaces with certain surgeries are rationally fibered and have tight contact structures.
problem Characterizing knots in L-spaces and their surgeries.
method Analyzing Dehn surgeries, Thurston norm minimization, and Floer homology.
result Knots in L-spaces with specific surgeries are rationally fibered and induce tight contact structures.
We write a formula for the LMO invariant of a rational homology sphere presented as a rational surgery on a link in S^3. Our main tool is a careful use of the Aarhus integral and the (now proven) "Wheels" and "Wheeling" conjectures of B-N, Garoufalidis, Rozansky and Thurston. As steps, side benefits and asides we give …
Constructs inequivalent Lefschetz fibrations on various surfaces.
problem Finding inequivalent Lefschetz fibrations on surfaces.
method Explicit construction of fibrations on rational and ruled surfaces.
result Explicit construction of inequivalent fibrations on all rational and ruled surfaces.
We construct an infinite family of knots in rational homology spheres with irreducible, non-fibered complements, for which every non-longitudinal filling is an L-space.
Proves uniqueness of Lefschetz fibrations over discs based on singular fibers.
problem Classifying Lefschetz fibrations over the disc based on their singular fibers.
method Inspired by homological mirror symmetry and Karpov--Nogin's theorem, uses topological equivalence and fiber analysis.
result Lefschetz fibrations over discs are uniquely determined by their singular fibers.
The study creates symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.
problem Creating symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.
method Producing simply connected, minimal, symplectic Lefschetz fibrations and rationally blowing down Lefschetz fibrations with clustered nodal fibers.
result New constructions of small symplectic exotic 4-manifolds.
We prove that the Witten-Reshetikhin-Turaev (WRT) SO(3) invariant of an arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a rational surgery formula for the unified invariant dominating WRT SO(3) invariants of rational homology 3-spheres at roots of unity of order co-prime with the torsion. As an…
It is known that every closed oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3-manifolds …
Study essentiality and simplicial volume of manifolds fibered over spheres.
problem When manifolds fibered over spheres are essential or have positive simplicial volume.
method Analyzing mapping tori and fiber bundles over spheres, using results on macroscopic dimension and characteristic classes.
result Mapping tori of odd-dimensional manifolds with non-zero simplicial volume are essential, while fiber bundles over spheres of dimension d > 1 have zero simplicial volume.
Let E(1)_p denote the rational elliptic surface with a single multiple fiber f_p of multiplicity p. We construct an infinite family of homologous non-isotopic symplectic tori representing the primitive class [f_p] in E(1)_p when p>1. As a consequence, we get infinitely many non-isotopic symplectic tori in the fiber cla…
The study determines fiber homotopy trivial bundles and their impact on curvature.
problem Understanding fiber homotopy trivial bundles and their effect on curvature.
method Classical approach via block bundles and surgery theory.
result Existence of elements of infinite order in homotopy groups of spaces of positive curvature.
The study classifies χ−slice pretzel links and Seifert fiber spaces.
problem Understanding χ−slice pretzel links and their properties. method Analyzing the sliceness of pretzel knots and extending results to pretzel links.
result Complete classifications of positive and negative pretzel links that are χ−slice, and partial classifications of 3-stranded and 4-stranded pretzel links. Study rational homology spheres with two legs, classifying those bounding rational balls.
problem Classifying Seifert rational homology spheres with two complementary legs.
method Complete classification of Seifert manifolds with 3 exceptional fibers and two complementary legs.
result Classification of Seifert manifolds bounding rational homology balls.
Study sharp bounds on knot surgeries in specific manifolds.
problem Cosmetic surgeries on knots in rational homology spheres.
method Character variety theory of PSL2(C). result Sharp bound on number of slopes producing same manifold.
Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers o…
Every negative amphichiral knot is rationally slice.
problem Proving every negative amphichiral knot is rationally slice.
method Systematic analysis of the action induced by the negative amphichiral map on the JSJ decomposition of the link exterior.
result Every negative amphichiral link is rationally slice.
Using the mapping cone of a rational surgery, we give several obstructions for Seifert fibered surgeries, including obstructions on the Alexander polynomial, the knot Floer homology, the surgery coefficient and the Seifert and four-ball genus of the knot.
The paper characterizes symplectic fillings of Seifert 3-manifolds using rational blowdowns.
problem Understanding symplectic fillings of Seifert 3-manifolds.
method Rational blowdown surgery and minimal symplectic fillings.
result A necessary and sufficient condition for minimal symplectic fillings to be obtained by rational blowdowns.
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.
Study the monodromy and center-focus problems for rational maps defined by products of generic lines.
problem Monodromy and center-focus problems for rational maps defined by products of generic lines.
method Analyze the 1-homology group and meromorphic 1-forms to characterize vanishing Abelian integrals.
result Characterize meromorphic 1-forms whose Abelian integrals vanish on cycles around a center singularity.
Study rational homotopy types of embedding spaces of manifolds.
problem Understanding the rational homotopy types of embedding spaces of manifolds.
method Express rational homotopy types through combinatorially defined L-infinity algebras of diagrams.
result Expressed the rational homotopy type of connected components of embedding spaces.
Planar multilinks prove rational singularities in surface geometry.
problem Characterizing surface singularities using planar multilinks.
method Combining topological and combinatorial approaches, including Min--Roy--Wang's work.
result Planar multilinks imply rational singularities and sandwiched singularities.
Study shows infinite order in mapping class groups for certain 3D shapes.
problem Understanding the mapping class groups of certain 3D shapes.
method Analogues of Seiberg-Witten-Floer homology for 3-manifolds.
result Monodromy diffeomorphisms have infinite order in smooth mapping class groups.
We describe rational knots with any of the possible combinations of the properties (a)chirality, (non-)positivity, (non-)fiberedness, and unknotting number one (or higher), and determine exactly their number for a given number of crossings in terms of their generating functions. We show in particular how Fibonacci numb…
Study properties of algebraic threefolds with specific singularities.
problem Characterize and classify algebraic threefolds with certain singularities.
method Use topological invariants, rational homology, Poincaré duality, and Lie algebras.
result Relate topological invariants to Lie algebras and representations.
In this note we complete the discussion of minimality of symplectic fiber sums. We find, that for fiber sums along spheres the minimality of the sum is determined by the cases discussed by M. Usher and one additional case: If the sum is the result of the rational blow-down of a symplectic -4-sphere in X, then it is non…
The study restricts when Seifert fibered spaces can bound definite manifolds.
problem When can Seifert fibered spaces bound definite manifolds?
method Established an inequality for plumbing intersection lattices and applied it to two applications.
result Characterized Seifert fibered spaces that bound rational homology S1imesD3's and answered a question about Donaldson's theorem and Fintushel-Stern's R-invariant. Let M be a fibered 3-manifold with multiple boundary components. We show that the fiber structure of M transforms to closely related transversely oriented taut foliations realizing all rational multislopes in some open neighborhood of the multislope of the fiber. Each such foliation extends to a taut foliation in t…
Identifies Heegaard Floer homology solid tori via Dehn fillings.
problem Characterizing Heegaard Floer homology solid tori.
method Using Dehn fillings to identify solid tori.
result Characterized Seifert fibered Heegaard Floer solid tori.
Study the periods mapping from hyperelliptic curves, revealing fiber topology.
problem Global topology of 2D fibers of the periods mapping.
method Decomposition of moduli space into polyhedra labeled by planar graphs.
result Investigation of low dimensional fibers of the periods mapping.
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 D4 (called a ration…
In this paper, we derive the family switching formula of -n two-sphere fiber bundle embedded in a smooth four-manifold fiber bundle. In the smooth category, it is a partial generalization of Fintushel-Stern's argument for four-manifolds. We also derive an algebraic analogue of the family switching formula, allowing the…
In the note we study Legendrian and transverse knots in rationally null-homologous knot types. In particular we generalize the standard definitions of self-linking number, Thurston-Bennequin invariant and rotation number. We then prove a version of Bennequin's inequality for these knots and classify precisely when the …
The study finds infinitely many special Lagrangian submanifolds in certain complex manifolds.
problem Existence of special Lagrangian submanifolds in log Calabi-Yau manifolds.
method Analyzes the existence of special Lagrangian submanifolds in log Calabi-Yau manifolds using Ricci-flat Kähler metrics.
result Proves existence of infinitely many disjoint special Lagrangian submanifolds in certain log Calabi-Yau manifolds.
A natural generalization of a crossing change is a rational subtangle replacement (RSR). We characterize the fundamental situation of the rational tangles obtained from a given rational tangle via RSR, building on work of Berge and Gabai, and determine the sites where these RSR may occur. In addition we also determine …
New proof shows RFRS groups virtually fibred if their L2-Betti number vanishes.
problem Characterizing residually finite rationally solvable groups.
method Using L2-Betti numbers to prove virtual fibering. result Finitely generated RFRS groups are virtually fibred if their L2-Betti number vanishes. 3D space without definite 4D counterpart found.
problem Finding 3-manifolds that don't bound definite 4-manifolds.
method Analysis of characteristic covectors in bimodular lattices.
result A rational homology 3-sphere that bounds no definite 4-manifold.
We investigate rational homology cobordisms of 3-manifolds with non-zero first Betti number. This is motivated by the natural generalization of the slice-ribbon conjecture to multicomponent links. In particular we consider the problem of which rational homology S1×S2's bound rational homology S1×D3'…