arXiv research

Study shows Khovanov homology's relation to decomposable Lagrangian cobordisms.

Regular sliceness implies once-stably decomposable sliceness in symplectizations.

We investigate the ramifications of the Legendrian satellite construction on the relation of Lagrangian cobordism between Legendrian knots. Under a simple hypothesis, we construct a Lagrangian concordance between two Legendrian satellites by stacking up a sequence of elementary cobordisms. This construction narrows the…

The study connects knot Floer homology and Lagrangian cobordisms.

New invariant stops certain types of geometric transformations.

We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb{R}^3$. Our results give new, computable, and effective obstructions to the exist…

GRID invariants block certain Lagrangian cobordisms in 3D.

The purpose of this note is to show that classical cobordism arguments, which go back to the pioneering works of Mandelbaum and Moishezon, provide quick and unified proofs of any knot surgered compact simply-connected 4-manifold X_K becoming diffeomorphic to X after a single stabilization by connected summing with S^2 …

Researchers develop methods to construct Lagrangian cobordisms between Legendrian knots.

We study some properties of decomposable exact Lagrangian cobordisms between Legendrian links in $\mathbb{R}^3$ with the standard contact structure. In particular, for any decomposable exact Lagrangian filling $L$ of a Legendrian link $K$, we may obtain a normal ruling of $K$ associated with $L$. We prove that the asso…

The study extends cobordism theory to complex sections, defining and calculating cobordism groups.

Study cobordisms of nested manifolds and their invariants.

The paper introduces a new invariant for cobordism classes of manifolds and extends cobordism groups.

Defines cobordism maps connecting Khovanov and instanton homologies.

The study explores conditions for $h$-cobordisms between smooth 4-manifolds.

The article examines how many stabilizations are needed to transform 5D s-cobordisms into product cobordisms.

Extends cobordism groups of immersions to projections with new results.

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.

New results on algebraic knots with Brieskorn polynomials.

Study $h$-cobordisms of complexity 2 in 5D, finding obstructions and examples.

Study finds conditions for existence of specific pseudo-Riemannian cobordisms.

We study the problem of defining maps on link Floer homology induced by unoriented link cobordisms. We provide a natural notion of link cobordism, disoriented link cobordism, which tracks the motion of index zero and index three critical points. Then we construct a map on unoriented link Floer homology associated to a …

We study necessary and sufficient conditions for the existence of Lorentzian and weak Lorentzian cobordisms between closed smooth manifolds of arbitrary dimension such that the structure group of the frame bundle of the cobordism is $\Spin(1, n)_0$. This extends a result of Gibbons-Hawking on $\Sl(2, \C)$-Lorentzian co…

Using methods inspired from algebraic $K$-theory, we give a new proof of the Genauer fibration sequence, relating the cobordism categories of closed manifolds with cobordism categories of manifolds with boundaries, and of the Bökstedt-Madsen delooping of the cobordism category. Unlike the existing proofs, this approach…

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…

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…

Periodic Floer homology (PFH) is a Gromov-Floer type invariant for fibered three-manifolds with Hamiltonian structures. The cobordism maps on periodic Floer homology induced by symplectic cobordisms are currently only defined indirectly by using Seiberg-Witten theory. In this paper, we investigate the cobordism maps in…

New examples of Legendrian links with infinitely many fillings.

Paper extends cobordism maps in Khovanov and instanton homologies.

We introduce a relation of cobordism for knots in thickened surfaces and study cobordism invariants of such knots.

New cobordisms found between certain quasipositive knots.

Paper presents new cobordism sequences for singular maps.

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.

3-manifolds with specific homology are cobordant if homeomorphic.

Foam cobordism groups linked to interval exchange automorphisms.

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…

Study on knot concordance and homology cobordism using Heegaard Floer homology.

Study shows mapping class groups differ for $h$-cobordant manifolds.

In this article, we reformulate the cobordism map of embedded contact homology, which is induced by exact symplectic cobordism and defined as direct limit of homomorphisms called filtered ECH cobordism map. The filtered ECH cobordism map is defined by counting embedded holomorphic curves with zero ECH index and we prov…

Paper shows string cobordism at 24 dims can be determined by elliptic genus.

Study non-orientable link cobordisms using Floer homologies to prove inequalities.

Study shows surprising cobordism distances between certain torus knots.

Cobordism of virtual string links on $n$ strands is a combinatorial generalization of link cobordism. There exists a bijection between virtual string links up to cobordisms and elements of the group $\mathbb{Z}^{n(n-1)}$. This paper also shows that virtual string links up to unwelded equivalence are classified by those…

Szűcs introduced cobordism of singular maps to compute groups of immersions and embeddings.

New discrete cobordism category for nested manifolds and relations to algebraic structures.

Study eigenvalues of curvature operators to annihilate cobordism invariants.