arXiv research

Defines cobordism maps connecting Khovanov and instanton homologies.

Study defines PFH cobordism maps from Lefschetz fibrations.

Study maps on link Floer homology for unoriented link cobordisms.

Alternative definition of cobordism map in ECH theory.

Paper extends cobordism maps in Khovanov and instanton homologies.

Paper shows maps from m-sphere to n-sphere are trivial cobordant.

Paper constructs maps for sutured monopole Floer homology.

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

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…

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…

Link cobordism maps are graded and determine surface genus.

Link Floer homology maps are constructed for link cobordisms.

Paper presents new cobordism sequences for singular maps.

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

Study defines and proves cobordism maps on periodic Floer homology using Seiberg-Witten and holomorphic curve methods.

A generic smooth map of a closed $2k$-manifold into $(3k-1)$-space has a finite number of cusps ($Σ^{1,1}$-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities ($Σ^{1,0}$-singularities). Two fold maps are fold bordant if the…

Python tool calculates cobordism maps in Khovanov homology.

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 …

Proves naturalness of Symplectic Instanton homology and defines cobordism maps.

We compute the oriented cobordism group of fold maps of 4-manifolds into the space with all the possible restrictions (and also with no restriction) to the singular fibers. We also give geometric invariants which describe completely the cobordism group of fold maps.

Two constructions of classifying spaces for singular maps are connected.

Let S be a compact oriented surface. A homology cobordism of S is a cobordism C between two copies of S, such that both the "top" inclusion and the "bottom" inclusion of S in C induce isomorphisms in homology. Homology cobordisms of S form a monoid, into which the mapping class group of S embeds by the mapping cylinder…

Author defines products of elements in cobordism-like modules induced from generic maps.

Researchers construct gluing maps and prove duality in sutured Floer homology.

New techniques in Khovanov homology help distinguish exotic surfaces in 4-ball.

Geometric invariants of fold maps and their signatures.

Trisections help compute Heegaard Floer homology invariants.

The paper studies how Lagrangian cobordisms affect DGAs of Legendrian ends.

Heegaard Floer homology study confirms composition maps match up to homotopy.

Proves conditions for generating families on Lagrangian cobordisms.

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…

For each integer q>0 there is a cohomology theory such that the zero cohomology group of a manifold N of dimension n is a certain group of cobordism classes of proper fold maps of manifolds of dimension n+q into N. We prove a splitting theorem for the spectrum representing the cohomology theory of fold maps. For even q…

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair $(X,L)$ consisting of an exact symplectic manifold $X$ and an exact Lagrangian cobordism $L\subset X$ which agrees with cylinders over Legendrian links $…

The study connects knot Floer homology and Lagrangian cobordisms.

We correct the proof in the unoriented case of Theorem 1.2 in the paper "Cobordisms of maps with singularities of given class"

We construct maps on hat Heegaard Floer homology for cobordisms decorated with graphs. The graph TQFT allows for cobordisms with disconnected ends. Our construction uses Juhász's sutured Floer TQFT. We compute the maps for several elementary graph cobordisms. As an application, we compute the action of the fundamental …

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 prove a conjecture due to M. Kazarian, connecting two classifying spaces in singularity theory. These spaces are: - Kazarian's space (generalizing Vassiliev's algebraic complex and) showing which cohomology classes are represented by singularity strata. - Author's space $X_τ$ giving homotopy representation of cobord…

Generalizes Pontryagin's construction for proper maps in stable dimensions.

A new spectrum recovers cobordism cut and paste groups of manifolds with boundary.

In a previous paper, Sarkar and the third author gave a combinatorial description of the hat version of Heegaard Floer homology for three-manifolds. Given a cobordism between two connected three-manifolds, there is an induced map between their Heegaard Floer homologies. Assume that the first homology group of each boun…

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 …

We define a moment map associated to a smooth torus action on a smooth manifold, without a two-form. We define cobordisms of such structures, allowing non compact manifolds as long as the moment maps are proper. We prove that a compact manifold with a torus action and a moment map is cobordant to the disjoint union of …

New cobordism maps for instanton knot homology help compute surgeries on knots.

A geometric interpretation of a circle transfer map in cobordism categories.

Let $N$ and $P$ be smooth closed manifolds of dimensions $n$ and $p$ respectively. Given a Thom-Boardman symbol $I$, a smooth map $f:N\to P$ is called an $Ω^{I}$-regular map if and only if the Thom-Boardman symbol of each singular point of $f$ is not greater than $I$ in the lexicographic order. We will represent the gr…

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…