Study defines PFH cobordism maps from Lefschetz fibrations.
problem Indirectly defined PFH cobordism maps using Seiberg-Witten theory.
method Defined PFH cobordism maps using holomorphic curves.
result Computed PFH cobordism maps for specific cases.
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48 results for “elementary cobordism”
Researchers develop methods to construct Lagrangian cobordisms between Legendrian knots.
problem Understanding the relationship between Legendrian knots through Lagrangian cobordisms.
method Combinatorial and geometric methods, including Heegaard Floer Homology and contact surgery.
result Construction of nondecomposable Lagrangian cobordisms between Legendrian knots.
We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot…
Graph TQFT for Heegaard Floer homology with applications.
problem Computing Heegaard Floer homology for graph-cobordisms.
method Using Juhász's sutured Floer TQFT, we construct maps for graph-cobordisms.
result Computed the action of fundamental group on hat Heegaard Floer homology.
Proves exact triangle for Pin(2)-monopole Floer homology.
problem Existence of exact triangle in Pin(2)-monopole Floer homology.
method Using specific Dehn surgeries and elementary cobordism.
result Describes invariants of homology spheres.
Doodles link to commutator identities in a 2-sphere.
problem Understanding commutator identities in free groups via doodles.
method Analyzing doodles with proper noose systems and establishing bijections.
result A bijection between doodles and commutator identities.
Researchers construct gluing maps and prove duality in sutured Floer homology.
problem Computing and understanding sutured Floer homology.
method Explicit construction of gluing maps using contact handles and sutured manifold cobordisms.
result Agreement of cobordism maps in link Floer homology.
New methods link Legendrian satellites to Lagrangian cobordisms.
problem Understanding relations between Legendrian and Lagrangian knots.
method Constructing Lagrangian concordances through satellite operations.
result Maximum Thurston-Bennequin number restricts Legendrian satellite Lagrangian sliceness.
We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair consisting of an exact symplectic manifold and an exact Lagrangian cobordism which agrees with cylinders over Legendrian links $…
We develop a calculus of surgery data, called bridged links, which involves besides links also pairs of balls that describe one-handle attachements. As opposed to the usual link calculi of Kirby and others this description uses only elementary, local moves(namely modifications and isolated cancellations), and it is val…
We investigate the Dolbeault operator on a pair of pants, i.e., an elementary cobordism between a circle and the disjoint union of two circles. This operator induces a canonical selfadjoint Dirac operator on each regular level set of a fixed Morse function defining this cobordism. We show that as we approac…
We present a new, more elementary proof of the Freedman-Teichner result that the geometric classification techniques (surgery, s-cobordism, and pseudoisotopy) hold for topological 4-manifolds with groups of subexponential growth. In an appendix Freedman and Teichner give a correction to their original proof, and reform…
Study on the cobordism distance between knots and their reverses.
problem Understanding the relationship between a knot and its reverse in terms of cobordism distance.
method Examined the cobordism distance d(K,K^r) for various knots and their properties.
result For knots with g_4(K) = g_3(K), d(K,K^r) is strictly less than twice g_4(K).
Ascent concordancemath.GT
New links found that challenge traditional complexity assumptions.
problem Traditional complexity assumptions in concordance of links are challenged.
method Study of cobordisms between links in thickened surfaces, use of augmented Khovanov homology.
result Existence of concordant links that require thickenings of higher genus surfaces.
Smooth 4-manifolds can be represented by loops in the pants complex.
problem Representing smooth 4-manifolds using a specific geometric structure.
method Using loops in the pants complex to represent and analyze 4-manifolds.
result Smooth 4-manifolds are smoothly cobordant to .
New knot Floer homology gives bounds on 4-ball genus.
problem Calculating the smooth 4-ball genus of knots.
method Elementary methods based on grid diagrams and surface cobordisms.
result Shows unoriented knot Floer homology gives a lower bound for 4-ball genus.
The paper presents a new method to create exotic 4-manifolds and surfaces that remain exotic after stabilization.
problem Stabilization of exotic 4-dimensional phenomena and knotted surfaces.
method Elementary approach to constructing exotic 4-manifolds and surfaces, including examples in closed, simply connected 4-manifolds.
result The construction yields exotic surfaces in the 4-ball that remain exotic after stabilization, detected by Khovanov homology.
Let be a field and let be a multiplicative subgroup. We consider the category of -dimensional cobordisms equipped with a representation of their fundamental group in , and the category of -linear maps defin…
The study extends cobordism theory to complex sections, defining and calculating cobordism groups.
problem Understanding when almost complex manifolds can have complex sections.
method Defined complex section cobordism, determined groups, and introduced an obstruction.
result The obstruction vanishes for certain multiplicative generators in the complex cobordism ring.
The paper introduces a new invariant for cobordism classes of manifolds and extends cobordism groups.
problem Developing a new invariant for cobordism classes of manifolds.
method Introducing an invariant for null-cobordant -manifolds and constructing cobordism groups .
result The invariant is a complete invariant of -cobordism classes of null-cobordant -manifolds.
Study cobordisms of nested manifolds and their invariants.
problem Understanding cobordisms of nested manifolds and their invariants.
method Identify a nested analog of the Pontryagin-Thom construction and find spaces homotopy equivalent to nested Pontryagin-Thom spaces.
result Discover nested cobordism invariants and provide an alternative proof of Wall's splitting result.
Study maps on link Floer homology for unoriented link cobordisms.
problem Defining maps on link Floer homology induced by unoriented link cobordisms.
method Introduced disoriented link cobordism to track critical points, constructed a map on unoriented link Floer homology.
result Comparison with existing constructions for unoriented band moves.
Defines cobordism maps connecting Khovanov and instanton homologies.
problem No direct correspondence between Khovanov and instanton homology cobordism maps.
method Defines a cobordism map on the instanton cube complex as a filtered chain map.
result Proves the cobordism map recovers both Khovanov and instanton homology cobordism maps.
The study explores conditions for -cobordisms between smooth 4-manifolds.
problem Determining when smoothly -cobordant 4-manifolds are also -cobordant.
method Providing new conditions for positive answers to the -cobordism theorem.
result Conditions under which standard methods to construct -cobordisms fail.
In this paper we develop a Morse-like theory in order to decompose birational maps and morphisms of smooth projective varieties defined over a field of characteristic zero into more elementary steps which are locally étale isomorphic to equivariant flips, blow-ups and blow-downs of toric varieties. A crucial role in th…
Study shows Khovanov homology's relation to decomposable Lagrangian cobordisms.
problem Understanding the relationship between Khovanov homology and decomposable Lagrangian cobordisms.
method Utilized previously defined filtered invariants to give obstructions.
result Partial answer to Ekholm, Honda, and Kálmán's question about Khovanov homology and decomposable Lagrangian cobordisms.
New proof of Genauer fibration sequence in cobordism categories.
problem Relating cobordism categories of closed and boundary-manifolds.
method Inspired from algebraic K-theory, using cobordism categories and delooping.
result Generalization of Waldhausen's Additivity theorem to cobordism categories.
The article examines how many stabilizations are needed to transform 5D s-cobordisms into product cobordisms.
problem Understanding the number of stabilizations required to convert 5D s-cobordisms into product cobordisms.
method Utilizes Gabai's 4D light bulb theorem and Freedman Quinn invariant.
result Determines the number of stabilizations needed for 5D s-cobordisms.
Study compares -cobordism categories to standard spaces.
problem Comparing -cobordism categories to standard spaces.
method Topological category of -cobordisms between manifolds.
result Homotopy type comparison of -cobordism categories.
Topology change and selection rules for high-dimensional $\Spin(1, n)_0$-Lorentzian cobordismsmath.GT
The study examines conditions for high-dimensional Lorentzian cobordisms with specific structure groups.
problem Conditions for the existence of Lorentzian and weak Lorentzian cobordisms with $\Spin(1, n)_0$ structure group.
method Analysis of necessary and sufficient conditions, computation of cobordism groups, restrictions on gravitational kinks.
result Restrictions on the gravitational kink numbers of $\Spin(1, n)_0$-weak Lorentzian cobordisms.
Extends cobordism groups of immersions to projections with new results.
problem Understanding cobordism groups of projected immersions.
method Classifying space construction by Szűcs and Terpai, Salomonsen's exact sequence.
result Obtains results on cobordism groups in small and large codimensional cases.
Alternative definition of cobordism map in ECH theory.
problem Defining and proving equivalence of cobordism map in ECH theory.
method Reformulating and proving independence of filtered ECH cobordism map using Seiberg-Witten theory.
result The new definition of cobordism map is equivalent to existing definitions.
Making cobordisms symplecticmath.SG
Establishes existence principle for symplectic cobordisms.
problem Symplectic cobordisms with concave overtwisted boundaries.
method -principle for symplectic cobordisms.
result Existence of symplectic structures on cobordisms.
We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles defined by the author in his earlier works.
Study exact Lagrangian cobordisms in cotangent bundles, proving bounds on sheaf interleaving distance and shadow distance.
problem Understanding Lagrangian cobordisms and their properties in cotangent bundles.
method Use microlocal theory of sheaves, sheaf quantization, and cone decompositions.
result Interleaving distance of sheaves is bounded by the shadow distance of the cobordism.
Proves existence of exact lagrangian cobordisms between legendrians.
problem Existence of exact lagrangian cobordisms between closed legendrians.
method Proves existence through mathematical proof.
result Existence of non-compact exact lagrangian cobordisms.
New results on algebraic knots with Brieskorn polynomials.
problem Understanding cobordisms of algebraic knots defined by Brieskorn polynomials.
method Analyzing Fox--Milnor type relations, decomposing algebraic cobordism classes, and studying cyclic suspensions.
result Spherical algebraic knots associated with Brieskorn polynomials have infinite order in the knot cobordism group.
On -cobordisms of complexity math.GT
Study -cobordisms of complexity 2 in 5D, finding obstructions and examples.
problem Understanding -cobordisms of complexity 2 in 5D.
method Compute monopole Floer homology and action of twisting involution.
result Obtained obstructions and constructed examples of high complexity.
Determine pairs of torus knots with genus one cobordisms, with exceptions.
problem Identify pairs of torus knots with genus one cobordisms.
method Combine obstructions from Heegaard Floer knot complex with explicit constructions.
result Determine pairs of torus knots with genus one cobordisms, with exceptions.
This paper classifies virtual string links up to cobordism using group theory.
problem Classifying virtual string links up to cobordism.
method Using group theory, specifically the group .
result Virtual string links up to cobordism are classified by elements of .
Study finds conditions for existence of specific pseudo-Riemannian cobordisms.
problem Existence of Spin-dimensional cobordisms with specific signature.
method Analyzes Spin-dimensional cobordisms with signature .
result Computes cobordism groups and necessary/sufficient conditions for existence.
The cobordism hypothesismath.AT
Proves a conjecture about factorization homology leading to the cobordism hypothesis.
problem Proving a conjecture about factorization homology.
method Assuming a conjecture about factorization homology with adjoints.
result Proves the cobordism hypothesis.
Cobordism of disk knotsmath.GT
We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots $D^{n-2}\into D^n$. Cobordisms of disk knots that do not fix the boundary sphere knots are easily classified by the cobordism properties of these boundaries, and any two even-dimensional disk knots with isotopic boundary knots are cobordant r…
Paper shows maps from m-sphere to n-sphere are trivial cobordant.
problem Understanding cobordism classes of maps and covers for spheres.
method Analyzes cobordism classes of maps from m-sphere to n-sphere.
result For m>n, cobordism classes of maps from m-sphere to n-sphere are trivial.
We show that the theory of stable complex -cobordisms, for a torus , is embedded into the theory of stable complex -cobordisms of not necessarily compact manifolds equipped with proper abstract moment maps. Thus the introduction of such non-compact cobordisms in the stable complex -cobordism theory does not…
Link cobordism maps are graded and determine surface genus.
problem Understanding graded structures in link Floer homology.
method Grading change formula and cobordism maps.
result Link cobordism maps are determined by surface genus.
Link Floer homology maps are constructed for link cobordisms.
problem Link Floer homology invariance and functoriality.
method Defined cobordism maps on a curved chain homotopy type invariant.
result Invariance of constructed cobordism maps proved.
Paper extends cobordism maps in Khovanov and instanton homologies.
problem Define and prove compatibility of cobordism maps in Khovanov and instanton homologies.
method Extend embedded cobordism map to immersed cobordisms and prove compatibility.
result Induced map on Khovanov homology is injective for certain concordances.