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 ϰR for null-cobordant n-manifolds and constructing cobordism groups ΩnR. result The invariant ϰR is a complete invariant of R-cobordism classes of null-cobordant n-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 h-cobordisms between smooth 4-manifolds.
problem Determining when smoothly h-cobordant 4-manifolds are also s-cobordant. method Providing new conditions for positive answers to the s-cobordism theorem. result Conditions under which standard methods to construct h-cobordisms fail. 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 h-cobordism categories to standard spaces.
problem Comparing h-cobordism categories to standard spaces. method Topological category of h-cobordisms between manifolds. result Homotopy type comparison of h-cobordism categories. 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.
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.
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.
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.
Study h-cobordisms of complexity 2 in 5D, finding obstructions and examples.
problem Understanding h-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 Zn(n−1). result Virtual string links up to cobordism are classified by elements of Zn(n−1). Study finds conditions for existence of specific pseudo-Riemannian cobordisms.
problem Existence of Spin(n+1)-dimensional cobordisms with specific signature. method Analyzes Spin(n+1)-dimensional cobordisms with signature (2,n−1). result Computes cobordism groups and necessary/sufficient conditions for existence.
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.
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 G-cobordisms, for a torus G, is embedded into the theory of stable complex G-cobordisms of not necessarily compact manifolds equipped with proper abstract moment maps. Thus the introduction of such non-compact cobordisms in the stable complex G-cobordism theory does not…
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.
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.
New cobordisms found between certain quasipositive knots.
problem Existence of complex cobordisms between quasipositive knots.
method Construction of specific knots and cobordisms of specified genus.
result No complex cobordism exists between certain quasipositive knots.
Paper presents new cobordism sequences for singular maps.
problem Understanding cobordism groups of singular maps.
method Develops analogous exact sequences to classical ones.
result Positive answer to Szűcs' question on cobordisms of immersions.
We introduce a relation of cobordism for knots in thickened surfaces and study cobordism invariants of such knots.
The h-cobordism theorem is a noted theorem in differential and PL topology. A generalization of the h-cobordism theorem for possibly non simply connected manifolds is the so called s-cobordism theorem. In this paper, we prove semialgebraic and Nash versions of these theorems. That is, starting with semialgebraic or Nas…
We introduce an equivalence relation, called cobordism, for words and study cobordism invariants of words inspired by methods of low-dimensional topology.
Foam cobordism groups linked to interval exchange automorphisms.
problem Understanding cobordism groups of foams.
method Using the Sah-Arnoux-Fathi invariant and K-theory.
result Identified cobordism groups with homology and K-groups.
It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured F…
3-manifolds with specific homology are cobordant if homeomorphic.
problem Characterizing integer homology cobordism for Sol manifolds.
method Analyzing first homology groups and using cobordism theory.
result Integer homology cobordant Sol manifolds are homeomorphic.
Study rational homology cobordisms of 3-spheres and lens spaces.
problem When rational homology 3-spheres are cobordant to connected sums of lens spaces.
method Knot Floer homology combined with rational homology cobordism theory.
result Every rational homology cobordism class in the subgroup generated by lens spaces is uniquely represented by a connected sum of lens spaces.
Study on knot concordance and homology cobordism using Heegaard Floer homology.
problem Knot concordance and homology cobordism.
method Heegaard Floer homology.
result Recent results in knot concordance and homology cobordism.
Study non-orientable link cobordisms using Floer homologies to prove inequalities.
problem Prove inequalities involving Euler characteristic and local maxima in non-orientable cobordisms.
method Use unoriented instanton and knot Floer homology to introduce unoriented versions of band unknotting number and refined cobordism distance.
result Show that the difference between unoriented refined cobordism distance of a knot from the unknot and non-orientable slice genus can be arbitrarily large.
Paper shows string cobordism at 24 dims can be determined by elliptic genus.
problem Determining string cobordism at dimension 24.
method Using elliptic genus as an invariant.
result Shows string cobordism at 24 dims can be determined by elliptic genus.
Study shows mapping class groups differ for h-cobordant manifolds.
problem Mapping class groups are invariant under h-cobordism. method Introduced moduli spaces of h-block bundles to distinguish manifolds. result Mapping class groups of h-cobordant manifolds can differ. New obstructions found for Lagrangian cobordisms in knot Floer homology.
problem Existence of Lagrangian cobordisms in knot Floer homology.
method Functorial behavior of Legendrian invariants with respect to decomposable Lagrangian cobordisms.
result New computable obstructions to Lagrangian cobordisms.
Study shows surprising cobordism distances between certain torus knots.
problem Determining cobordism distances between thin and thick torus knots.
method Analyzes locally flat cobordisms between torus knots with small and large braid indices.
result Surprising fact about torus knots as cross-sections of almost minimal cobordisms.
New bounds refine how multivariable signature and nullity change under link cobordisms.
problem Understanding how multivariable signature and nullity change under link cobordisms.
method Refined bounds on multivariable signature and nullity using link cobordisms and 0.5-solvable cobordism. result Multivariable signature and nullity are invariant under 0.5-solvable cobordism. 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.
Szűcs introduced cobordism of singular maps to compute groups of immersions and embeddings.
problem Computing cobordism groups of immersions and embeddings in dimensions where classical theory fails.
method Investigation of classifying spaces constructed by Szűcs and Rimányi.
result Collection and organization of results towards computation of cobordism groups of singular maps.
Study Whitehead torsions in inertial h-cobordisms.
problem Understanding Whitehead torsions in specific geometric contexts.
method Analyzing inertial h-cobordisms to identify subsets of the Whitehead group.
result Identified various types of Whitehead torsions representing nested subsets of the Whitehead group.