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.
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.
New method detects non-cobordant surface-links undetectable by other invariants.
problem Detecting non-cobordant surface-links using various invariants.
method A novel approach to distinguish non-cobordant surface-links.
result A new pair of non-cobordant surface-links cannot be distinguished by existing invariants.
Lagrangian cobordisms are three-dimensional compact oriented cobordisms between once-punctured surfaces, subject to some homological conditions. We extend the Le-Murakami-Ohtsuki invariant of homology three-spheres to a functor from the category of Lagrangian cobordisms to a certain category of Jacobi diagrams. We prov…
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.
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.
GRID invariants block certain Lagrangian cobordisms in 3D.
problem Obstructing decomposable Lagrangian cobordisms in 3D.
method Filtered GRID invariants and link Floer homology.
result GRID invariants obstruct decomposable Lagrangian cobordisms.
The study connects knot Floer homology and Lagrangian cobordisms.
problem Obstructing decomposable Lagrangian cobordisms in Weinstein cobordisms.
method Using knot Floer homology and transverse invariants, the study constructs maps between knot Floer homology groups.
result The invariant obstructs decomposable Lagrangian cobordisms of arbitrary genus.
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.
Categorifies quantum invariants using cobordism categories and operads.
problem Categorify quantum invariants using cobordism categories and operads.
method Constructs a cobordism category with a colored operad action, categorifies quantum sln invariants. result Consistency of the cobordism category and explicit functor to matrix factorizations conjectured.
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.
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. New invariant stops certain types of geometric transformations.
problem Obstructing decomposable Lagrangian cobordisms.
method Defined an invariant for Legendrian links.
result Obstructs decomposable Lagrangian cobordisms.
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. 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.
Diagrammatic method calculates knot invariant from tangle decompositions.
problem Computing Rasmussen's invariant for knots.
method Diagrammatic approach using tangles and cobordisms.
result Computed s-invariants of all 3-strand pretzel knots. Lower bounds for a knot invariant are derived using computations and cobordism inequality.
problem Calculating the concordance invariant s# for knots. method Computation for torus knots, cobordism inequality of s#, and arguments for slice-torus invariants. result Lower bounds for s# are derived for knots. New cobordism invariants derived from BPS q-series.
problem Understanding BPS partition functions with different structures.
method Analyzing limits of BPS q-series and summing over Spinc structures. result Derived invariants of homology cobordisms from BPS q-series.
We introduce a relation of cobordism for knots in thickened surfaces and study cobordism invariants of such knots.
Trisections help compute Heegaard Floer homology invariants.
problem Computing Heegaard Floer homology invariants for four-manifolds.
method Using trisection maps to induce cobordism maps on Heegaard Floer homology.
result Trisections provide a method to compute cobordism invariants.
Study eigenvalues of curvature operators to annihilate cobordism invariants.
problem Annihilating rational cobordism invariants on spin manifolds.
method Linear inequalities on curvature operator eigenvalues.
result Curvature conditions stabilize to annihilate invariants.
A function Γ_Y relates Chern-Simons functional to homology cobordism invariants.
problem Understanding the relationship between Chern-Simons functional and homology cobordism.
method Constructing a function Γ_Y on homology spheres and relating it to Chern-Simons functional and invariants.
result Γ_Y recovers the Frøyshov invariant and is related to Fintushel-Stern's R-invariant.
We introduce an equivalence relation, called cobordism, for words and study cobordism invariants of words inspired by methods of low-dimensional topology.
This is a survey paper of author's results on cobordism groups and semigroups of fold maps and simple fold maps. The results include: establishing a relation between fold maps and immersions through geometrical invariants of cobordism classes of fold maps and simple fold maps in terms of immersions with prescribed norm…
Proves super-version of index theorem from algebraic cobordism invariants.
problem Cobordism invariants in supersymmetric quantum mechanics.
method Trace methods for deformation quantization.
result Recovery of cobordism invariant using trace methods.
Geometric invariants of fold maps and their signatures.
problem Characterizing cobordism groups of fold maps.
method Combining immersion data and stable partial framings.
result A Poincare-Hopf type formula for the signature of manifolds.
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.
Efficient cobordisms show minimal signature values on certain links.
problem Finding minimal signature values for specific link types.
method Constructing topological cobordisms between torus links and connected sums of trefoil knots.
result The signature invariant σω at ω=ζ6 takes minimal values on torus links. Functoriality proved for colored link invariants.
problem Link and tangle invariants functoriality proof.
method Functoriality proved for colored Khovanov-Rozansky invariants.
result Functoriality of colored link homologies proved.
The classical Godbillon-Vey invariant is an odd degree cohomology class that is a cobordism invariant of a single foliation. Here we investigate cohomology classes of even degree that are cobordism invariants of (germs of) 1-parameter families of foliations.
The paper uses Heegaard Floer homology to study homology cobordism groups.
problem Understanding the structure of homology cobordism groups.
method Defined an invariant using Heegaard Floer homology, analogous to Stoffregen's connected Seiberg-Witten Floer homology.
result Computed involutive correction terms for certain three-manifolds.
The Upsilon invariant bounds cobordisms between knots and their braid index.
problem Bounding cobordisms between knots and their braid index.
method Using Ozsváth, Stipsicz, and Szabó's Upsilon-invariant.
result Established inductive formulas for the Upsilon invariant of torus knots.
New homology cobordism invariants from instanton Floer theory.
problem Understanding the structure of homology cobordism groups.
method Introduced homology cobordism invariants rs(Y) using instanton Floer theory. result Found new homology cobordism group properties and computed specific values.
New invariant for classifying 4-manifolds up to cobordism.
problem Classifying closed, oriented topological 4-manifolds up to s-cobordism. method Introducing a stable range invariant after stabilization by a fixed number of S2imesS2. result A new invariant for classifying 4-manifolds up to s-cobordism. Squeezed knots are slices of minimal cobordisms; obstructions come from quantum knot invariants.
problem Characterizing and obstructing squeezed knots.
method Analysis of cobordisms, quantum knot invariants, and stable cohomology operations.
result Effective obstructions to squeezedness come from quantum knot invariants, notably Rasmussen invariant refinements.
This paper continues our exploration of homology cobordism of 3-manifolds using our recent results on Cheeger-Gromov rho-invariants associated to amenable representations. We introduce a new type of torsion in 3-manifold groups we call hidden torsion, and an algebraic approximation we call local hidden torsion. We cons…
We explain the notion of a grope cobordism between two knots in a 3-manifold. Each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the Goussarov-Habiro approach to finite type invariants of knots is closely related to our notion of …
Study cobordisms of algebraic knots using Heegaard Floer homology.
problem Understanding cobordisms of algebraic knots.
method Use Heegaard Floer homology and the concordance invariant ν+.
result Explicit formula for ν+ in L-space knots and subadditivity in general.
Researchers prove index invariance under cobordism for Callias-type operators.
problem Index calculation for elliptic operators on compact manifolds.
method Introduce cobordism of Callias-type operators and prove index invariance.
result Index is preserved by cobordism of Callias-type operators.
Defines a new Upsilon torsion function for knot Floer homology.
problem Obtaining constraints on knot cobordisms.
method Defines a one-parameter family of Heegaard Floer torsion invariants.
result Provides new obstructions related to the Gordian distance between knots.
Lipshitz, Ozsváth and Thurston defined a bordered Heegaard Floer invariant CFDA for 3-manifolds with two boundary components, including mapping cylinders for surface diffeomorphisms. We define a related invariant for certain 4-dimensional cobordisms with corners, by associating a morphism F from CFDA(f) to CFDA(g) to e…
New invariants from annular links help classify braids and cobordisms.
problem Classifying braids and cobordisms using annular invariants.
method Defined annular Rasmussen invariants from bifiltered Khovanov-Lee complexes.
result Necessary and sufficient conditions for braid properties derived from invariants.
New invariants lift Milnor invariants for 3-component links.
problem Classifying 3-component links using Milnor invariants.
method Defined and proved invariants γk(L), introduced h(L), and showed their equivalence. result Invariants γk(L) lift certain Milnor invariants and are invariant under weak cobordism. Classifies knots that bound equivariant surfaces with free symmetries.
problem Classifying knots that bound equivariant surfaces with free symmetries.
method Homology cobordism classification of lens spaces using d-invariants.
result Numerical condition determining free periods for torus knots.
We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable …
The contact invariant in monopole Floer homology behaves naturally under strong symplectic cobordisms.
problem Understanding the behavior of the contact invariant under symplectic cobordisms.
method Extending the gluing argument to handle cylindrical ends, showing naturality.
result The contact invariant is natural under strong symplectic cobordisms.
The homology cobordism group of homology cylinders is a generalization of the mapping class group and the string link concordance group. We study this group and its filtrations by subgroups by developing new homomorphisms. First, we define extended Milnor invariants by combining the ideas of Milnor's link invariants an…