arXiv research

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…

Proves PL cobordism category's homotopy type, analogous to smooth case.

New invariants lift Milnor invariants for 3-component links.

Doodles link to commutator identities in a 2-sphere.

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show…

In this paper, we introduce a bordism category $\mathcal{C}_d^{PL}$ whose objects are bundles of closed $(d-1)$-dimensional piecewise linear manifolds and whose morphisms are bundles of $d$-dimensional piecewise linear cobordisms. In the main theorem of this article, we show that the classifying space $B\mathcal{C}_d^{…

Fix a tangential structure $θ: B \longrightarrow BO(d+1)$ and an integer $k < d/2$. In this paper we determine the homotopy type of a cobordism category $\mathbf{Cob}^{\text{mf}, k}_θ$, where morphisms are given by $θ$-cobordisms $W: P \rightsquigarrow Q$ equipped with a choice of proper Morse function $h_{W}: W \longr…

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…

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.

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

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 …

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…

Researchers develop methods to construct Lagrangian cobordisms between Legendrian knots.

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…

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.

The paper explores connections between braids, links, and cobordisms using algebraic methods.

We consider the topological category of $h$-cobordisms between manifolds with boundary and compare its homotopy type with the standard $h$-cobordism space of a compact smooth manifold.