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. Geometric proof shows topological invariance of handle homology.
problem Topological invariance of handle homology in manifolds.
method Entirely geometric proof using Cerf theory.
result Proof of ∂2=0 in chain complex defined by handle decomposition. Proves complex homology three-spheres can be bounded by many handles.
problem Complex homology three-spheres and their bounding four-manifolds.
method Uses homology cobordism invariant Γ from instanton Floer homology.
result Any bounding four-manifold must be built out of many handles.
Kirby color defined in Khovanov homology for 4D handlebodies.
problem Invariance of 4D handlebodies under Kirby moves.
method Functoriality and cabling properties of Khovanov homology, handle slide isomorphism.
result Kirby-colored Khovanov homology is invariant under handle slide moves.
Study of 3-handle attachments in 4D manifolds using Kirby calculus.
problem Inclusion of 3-handles in Kirby diagrams for 4D manifolds.
method Developed moves to extend Kirby calculus, established homological criterion.
result Identified geometric basis of 3-handle attachments, proved uniqueness theorem.
Study 4D homology cobordisms without 3-handles, deriving obstructions.
problem Understanding 4D homology cobordisms without 3-handles.
method Utilizing Thurston geometries, character varieties, and homologies, derive obstructions.
result Generalize knot Floer homology obstructions for ribbon concordances.
New method uses Khovanov homology to distinguish exotic 4-manifolds.
problem Distinguishing exotic compact, orientable 4-manifolds.
method Introduces a new, simple way of using Khovanov homology.
result First analysis-free proof of existence of exotic Mazur manifolds.
The study distinguishes exotic R^4's using knot Floer homology and Casson handles.
problem Distinguishing exotic R^4's with different knot Floer homologies.
method Using Gadgil's end Floer homology and Casson handles.
result A countably infinite family of pairwise nondiffeomorphic chiral exotic R^4's.
Given a rational homology sphere which bounds rational homology balls, we investigate the complexity of these balls as measured by the number of 1-handles in a handle decomposition. We use Casson-Gordon invariants to obtain lower bounds which also lead to lower bounds on the fusion number of ribbon knots. We use Levine…
We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, …
Study uses twisted Alexander polynomials to link fibered classes in 3-manifolds.
problem Linking fibered classes in 3-manifolds via Alexander polynomials.
method Applies twisted Alexander polynomials to fibered classes in ribbon homology cobordisms.
result Fibered classes of Y+ map to those of Y−. Real Milnor fibres become contractible after attaching handles, matching classical results.
problem Topology of real isolated hypersurface singularities.
method Proving real Milnor fibres contractible after attaching handles, defining real vanishing cycles.
result Integer homology groups of real Milnor fibres are isomorphic to those of a bouquet of spheres under certain conditions.
Constructs exotic proper 2-knots from open 2-handles.
problem Proving topological equivalence of exotic proper knotted surfaces.
method Flexible construction of exotic open 2-handles, distinguishing through genus functions and end Floer homology.
result Constructs infinite families of irreducible exotic proper knotted surfaces.
This is the last of five papers that construct an isomorphism between the Seiberg-Witten Floer homology and the Heegaard Floer homology of a given compact, oriented 3-manifold.
Skein lasagna module calculates 4-manifold invariants using handle decompositions.
problem Calculating invariants of 4-manifolds and links.
method Express skein lasagna module in terms of handle decompositions.
result Skein lasagna module can be locally infinite dimensional.
Two codimension-one submanifolds are cobordant if they have the same homology class.
problem Understanding cobordism equivalence of codimension-one submanifolds.
method Using handle decompositions and surgeries, we relate cobordisms to homology classes and prove equivalence for Seifert surfaces.
result Seifert surfaces are related by tube attachments and removals, providing a conceptual proof.
Fintushel and Stern showed that the Brieskorn sphere Σ(2,3,7) 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…
The paper calculates a new invariant for 4-manifolds using handle decompositions and skein relations.
problem Computing invariants for 4-manifolds built from handles.
method Handle attachment formulas, cabled colimits, lasso relation.
result Explicit calculations and partial vanishing results for specific 4-manifolds.
New insights into knot fusion numbers via cabling.
problem Understanding fusion numbers of ribbon knots and their behavior under cabling.
method Utilizing knot Floer homology and cabling formulas to analyze fusion numbers.
result The fusion number and strong homotopy fusion number of (p,1)-cable knots are preserved.
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.
The study solves a 2008 problem by Kalai about infinitely many homology-spheres.
problem Proving infinitely many homology-spheres with g3=0 in dimensions higher than four. method Using handlebody decompositions and PL manifolds with specific triangulations.
result There are infinitely many homology-spheres with g3=0 in dimensions higher than four. 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 …
The paper bounds the handle number of sutured manifolds using Morse-Novikov numbers and tunnel numbers.
problem Bounding the handle number of sutured manifolds.
method Developed bounds on the Morse-Novikov number of a link in terms of its tunnel number, and used these to bound the handle number of Heegaard splittings.
result The handle number function is bounded, constant on rays from the origin, and locally maximal.
We use contact handle decompositions and a stabilization process to compute the cylindrical contact homology of a subcritical Stein-fillable contact manifold with vanishing first Chern class, and show that it is completely determined by the homology of a subcritical Stein-filling of the contact manifold.
Extends Khovanov homology to links in a specific manifold.
problem Defining Khovanov homology for links in a new manifold.
method Extends Rozansky's construction to $M^r=#^r(S^2 imes S^1)$, defining Kh(L) and exploring its properties. result Recovery of WRT invariants and evaluation in skein module for null-homologous links.
We give an explicit construction of the Honda--Kazez--Matić gluing maps in terms of contact handles. We use this to prove a duality result for turning a sutured manifold cobordism around, and to compute the trace in the sutured Floer TQFT. We also show that the decorated link cobordism maps on the hat version of link F…
Bordered Heegaard Floer homology is an invariant for three-manifolds with boundary. In particular, this invariant associates to a handle decomposition of a surface F a differential graded algebra, and to an arc slide between two handle decompositions, a bimodule over the two algebras. In this paper, we describe these b…
Sutured manifolds defined by Gabai are useful in the geometrical study of knots and 3-dimensional manifolds. On the other hand, homology cylinders are in an important position in the recent theory of homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by focusing on sutured …
Introduces Floer lasagna modules using link Floer homology.
problem No specific problem stated; focuses on new mathematical concept.
method Inspired by skein lasagna module, uses link Floer homology.
result Computes Floer lasagna modules for specific 4-manifolds.
There are a number of homological knot invariants, each satisfying an unoriented skein exact sequence, which can be realized as the limit page of a spectral sequence starting at a version of the Khovanov chain complex. Compositions of elementary 1-handle movie moves induce a morphism of spectral sequences. These morphi…
A new method converts knot Floer homology to immersed curves.
problem Calculating bordered invariants of knot complements.
method Handle attachment to convert twice-punctured disk to once-punctured torus.
result Recovers Lipshitz-Ozsváth-Thurston's bordered invariant result.
We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in S1×S2 or any connected sum #k(S1×S2), viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplec…
Topological Bayesian Optimization finds optimal structures using topological data.
problem Optimizing complex structured data like material or neural network structures.
method Extract topological information from structures using persistent homology, apply Bayesian optimization with kernels for persistence diagrams.
result Topological information improves search efficiency for optimal structures.
Working with group homomorphisms, a construction of manifolds is introduced to preserve homology groups. The construction gives as special cases Qullien's plus construction with handles obtained by Hausmann, the existence of one-sided h-cobordism of Guilbault and Tinsley, the existence of homology spheres and higher-…
From any 4-dimensional oriented handlebody X without 3- and 4-handles and with b_2>0, we construct arbitrary many compact Stein 4-manifolds which are mutually homeomorphic but not diffeomorphic to each other, so that their topological invariants (their fundamental groups, homology groups, boundary homology groups, and …
The paper defines and calculates stabilization distances between surfaces in 4-manifolds.
problem Calculating the minimal number of 1-handle stabilizations for surfaces to become isotopic.
method Using homology of cyclic covers and metabelian twisted homology.
result For every nonnegative integer m, there exist pairs of surfaces with a specific stabilization distance.
Study Morse complexity of manifolds and homology classes, proving bounds and implications.
problem Understanding Morse complexity of manifolds and homology classes.
method Used surgery theory and index theory to prove upper and lower bounds.
result Locally symmetric spaces of Lie groups with discrete series representations do not admit open book decompositions.
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.
Constructs dg categories from surfaces using Khovanov homology.
problem Categorify quantum topology using surfaces and Khovanov homology.
method Constructs dg categories from surfaces using structures in Khovanov homology.
result Unified perspective on various categorified quantum topology constructions.
We prove a homological stability theorem for moduli spaces of manifolds of dimension 2n, for attaching handles of index at least n, after these manifolds have been stabilised by countably many copies of Sn×Sn. Combined with previous work of the authors, we obtain an analogue of the Madsen--Weiss theorem …
Link homology theories connect to 4-manifold invariants and TQFTs.
problem Connecting link homology theories to 4-manifold invariants and TQFTs.
method Functorial link homologies, skein modules, handle decompositions.
result Link homology theories become diagram-independent and furnish invariants of 4-manifolds.
Determines conditions for ribbon cobordisms between lens spaces.
problem Conditions for ribbon rational homology cobordisms between lens spaces.
method Analyzes ribbon cobordisms and uses properties of lens spaces and linear lattices.
result If a lens space admits a ribbon rational homology cobordism to a different lens space, it must be homeomorphic to L(n,1), up to orientation-reversal. Study intersection forms of 4-manifolds using trisections and mapping class groups.
problem Understanding intersection forms of 4-manifolds via trisections and mapping class groups.
method Apply mapping class group techniques and trisections to study intersection forms of smooth 4-manifolds.
result Analogous result for trisections of 4-manifolds analogous to Morita's for homology 3-spheres.
The paper calculates homology and intersection pairing of branched covers using disoriented homology.
problem Computing homology and intersection pairing of branched covers of 4-ball.
method Associate disoriented homology groups to projections of links and surfaces, show isomorphism to branched cover homology, define pairing on first disoriented homology of surfaces.
result Disoriented homology is isomorphic to the homology of branched cover and pairing is equal to the intersection pairing.
Develops a method to compute Morse homology for clean but not necessarily transverse intersections.
problem Computing Morse homology for clean but not necessarily transversely intersecting manifolds.
method Constructs minimal semi-global Kuranishi structures for moduli spaces of Morse trajectories, generalizing obstruction bundle gluing.
result Obtains iterated gluing equals simultaneous gluing, maintaining computability.
Study on lens spaces bounding 4-manifolds with specific Betti numbers.
problem Which lens spaces can bound 4-manifolds with second Betti number one?
method Construction of specific 4-manifolds and analysis of lens space boundaries.
result Infinite families of lens spaces can bound 4-manifolds with second Betti number one, but not all.
The paper introduces new invariants for genus one knots and surfaces.
problem Understanding invariants of genus one knots and surfaces.
method Investigating properties of the Alexander form of 3-manifolds to extract invariants of Seifert surfaces.
result Extracted invariants of genus one Seifert surfaces from the Alexander form of their exteriors.
We define invariants ds and ds, which are the maximal and minimal second Betti number divided by 8 among definite spin boundings of a homology sphere. The similar invariants g8 and g8 are defined by the maximal (or minimal) product sum of E8-form of bounding 4-manifold…