arXiv research

The paper constructs exotic knotted surfaces and curves in 4-manifolds.

Study of symplectic manifolds degenerating into singular spaces.

In this article, we construct the first example of a simply connected minimal symplectic 4-manifold homeomorphic but not diffeomorphic to 3CP^2#7CP^2b. We also construct the first exotic symplectic structure on CP^2#5CP^2b.

The study constructs symplectic 4-manifolds with exotic structures.

We give a short proof of the (known) result that there are no Kaehler structures on exotic tori. This yields a negative solution to a problem posed by Benson and Gordon. W discuss the symplectic version of the problem and analyze results which yield an evidence for the conjecture that there are no symplectic structures…

The study creates symplectic Lefschetz fibrations and rational blowdowns for new 4-manifolds.

Given a symplectomorphism f of a symplectic manifold X, one can form the `symplectic mapping cylinder' $X_f = (X \times R \times S^1)/Z$ where the Z action is generated by $(x,s,t)\mapsto (f(x),s+1,t)$. In this paper we compute the Gromov invariants of the manifolds $X_f$ and of fiber sums of the $X_f$ with other sympl…

Let $M$ be $\CP#2\CPb$, $3\CP#4\CPb$ or $(2n-1)\CP#2n\CPb$ for any integer $n\geq 3$. We construct an irreducible symplectic 4-manifold homeomorphic to $M$ and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to $M$. We also construct such exotic smooth structure…

Study of unchaining surgery operation in symplectic 4-manifolds.

An explicit example of an exotic symplectic $\mathbf{R}^6$ is given. Together with an earlier known example on $\mathbf{R}^4$, this yields an explicit exotic symplectic form on $\mathbf{R}^{2n}$ for all $n\geq2$.

We prove that the rational blowdown, a surgery on smooth 4-manifolds introduced by Fintushel and Stern, can be performed in the symplectic category. As a consequence, interesting families of smooth 4-manifolds, including the exotic $K3$ surfaces of Gompf and Mrowka, admit symplectic structures.

We define a new 4-dimensional symplectic cut and paste operation which is analogous to Fintushel and Stern's rational blow-down. We use this operation to produce multiple constructions of symplectic smoothly exotic complex projective space blown-up eight times, seven times, and six times. We also show how this operatio…

New exotic 4-manifolds with zero signature found.

Constructs Lefschetz fibrations on exotic 4-manifolds, providing bounds and equalities for singular fibers.

We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic 4-manifolds as multisections, such as Seiberg-Witten basic classes and except…

A study on the relation between the smooth structure of a symplectic homotopy K3 surface and its symplectic symmetries is initiated. A measurement of exoticness of a symplectic homotopy K3 surface is introduced, and the influence of an effective action of a K3 group via symplectic symmetries is investigated. It is show…

We introduce the $2$-nodal spherical deformation of certain singular fibers of genus $2$ fibrations, and use such deformations to construct various examples of simply connected minimal symplectic $4$-manifolds with small topology. More specifically, we construct new exotic minimal symplectic $4$-manifolds homeomorphic …

New tools classify symplectic fillings of contact 3-manifolds.

Paper defines end Khovanov homology to detect exotic planes.

Study obstructs symplectic structures on Mazur manifolds.

We introduce new symplectic cut-and-paste operations that generalize the rational blowdown. In particular, we will define $k$-replaceable plumbings to be those that, heuristically, can be symplectically replaced by Euler characteristic $k$ 4-manifolds. We will then classify 2-replaceable linear plumbings, construct 2-r…

We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex si…

The study proves exotic smooth structures and equivalent genus functions in 4-manifolds.

Constructs genus-3 Lefschetz fibrations and exotic 4-manifolds with specific $b_2^+$ values.

In this article we construct a minimal symplectic 4-manifold R that has small Euler characteristic (e(R)=8) and two essential Lagrangian tori with nice properties. These properties make R particularly suitable for constructing interesting examples of symplectic manifolds with small Euler characteristic. In particular, …

In [2], the first author constructed the first known examples of exotic minimal symplectic $\CP#5\CPb$ and minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to $3\CP#7\CPb$. The construction in [2] uses Y. Matsumoto's genus two Lefschetz fibrations on $M = \mathbb{T}^{2}\times \mathbb{S}^{2} #4\C…

Let M be either CP^2#3CP^2bar or 3CP^2#5CP^2bar. We construct the first example of a simply-connected symplectic 4-manifold that is homeomorphic but not diffeomorphic to M.

Embeds 3-manifolds in symplectic 4-manifolds with constraints.

We give several criteria on a closed, oriented 3-manifold that will imply that it is the boundary of a (simply connected) 4-manifold that admits infinitely many distinct smooth structures. We also show that any weakly fillable contact 3-manifold, or contact 3-manifolds with non-vanishing Heegaard Floer invariant, is th…

A survey of finite group actions on symplectic 4-manifolds is given with a special emphasis on results and questions concerning smooth or symplectic classification of group actions, group actions and exotic smooth structures, and homological rigidity and boundedness of group actions. We also take this opportunity to in…

In this paper we construct a family of simply connected, spin, non-complex, symplectic 4-manifolds which cover all but finitely many allowed lattice points $(χ, c)$ lying in $0 \leq c \leq 8.76χ$. Furthermore, as a corollary, we prove that there exist infinitely many exotic smooth structures on $(2n+1)(S^2 \times S^2)$…

Paper proves a symplectic inequality using trisections and contact geometry.

The paper proves symplectic neighbourhood theorems for stratified subspaces.

In this article we construct a new family of simply connected symplectic 4-manifolds with $b_2^+ =1$ and $c_1^2 =2$ which are not diffeomorphic to rational surfaces by using rational blow-down technique. As a corollary, we conclude that a rational surface ${\mathbf CP}^2 \sharp 7{\bar{\mathbf CP}^2}$ admits an exotic s…

By applying the lantern relation substitutions to the positive relation of the genus two Lefschetz fibration over $\mathbb{S}^{2}$. We show that $K3\#2 \overline{\mathbb{CP}}{}^{2}$ can be rationally blown down along seven disjoint copies of the configuration $C_2$. We compute the Seiberg-Witten invariant of the result…

We explicitly construct genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic to complex rational surfaces CP^2 # p (-CP^2) for p=7, 8, 9, and to 3 CP^2 #q (-CP^2) for q =12,...,19. Complementarily, we prove that there are no minimal genus-2 Lefschetz fibrations whose total spa…

New symplectic 4-manifolds with non-negative signatures are constructed using complex surfaces and quotients.

We prove a coisotropic intersection result and deduce the following: 1. Lower bounds on the displacement energy of a subset of a symplectic manifold, in particular a sharp stable energy-Gromov-width inequality. 2. A stable non-squeezing result for neighborhoods of products of unit spheres. 3. Existence of a "badly sque…

In an article from 2008, A. Akhmedov and B. D. Park constructed irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to the manifolds CP^2#3CP^2bar and 3CP^2#5CP^2bar. These manifolds are constructed by using generalized fibre sums. In this note we describe an explicit splitting of the second (co-)homo…

In \cite{AP3, AHP}, the first author and his collaborators constructed the irreducible symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to $(2n-1){\mathbb{CP}}^{2}\#(2n-1)\overline{\mathbb{CP}}^{2}$ for each integer $n \geq 25$, and the families of simply connected irreducible nonspin symplectic $4$…

This paper explores twisted Lagrangian tori in C^2 and their Hamiltonian stationarity.

It is known that every compact Stein 4-manifolds can be embedded into a simply connected, minimal, closed, symplectic 4-manifold. By using this property, we discuss a new method of constructing corks. This method generates a large class of new corks including all the previously known ones. We prove that every one of th…

We show that, for any two orientable smooth open 4-manifolds $X_0,X_1$ which are homeomorphic, their cotangent bundles $T^*X_0,T^*X_1$ are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold $R$ homeomorphic to $\mathbb{R}^4$, the standard Stein structure on $T^*R$ is Stei…

In dimensions congruent to 1 modulo 4, we prove that the cotangent bundle of an exotic sphere which does not bound a parallelisable manifold is not symplectomorphic to the cotangent bundle of the standard sphere. More precisely, we prove that such an exotic sphere cannot embed as a Lagrangian in the cotangent bundle of…

The purpose of this paper is to introduce Liouville hypersurfaces in contact manifolds, which generalize ribbons of Legendrian graphs and pages of supporting open books. Liouville hypersurfaces are used to define a gluing operation for contact manifolds called the Liouville connect sum. Performing this operation on a c…

The purpose of this article is twofold. First we outline a general construction scheme for producing simply-connected minimal symplectic 4-manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to $\CP#(2k+1)\CPb$ …

Consider cotangent bundles of exotic spheres, with their canonical symplectic structure. They admit automorphisms which preserve the part at infinity of one fibre, and which are analogous to the square of a Dehn twist. Pursuing that analogy, we show that they have infinite order up to isotopy (inside the group of all a…

Smooth and symplectic symmetries of an infinite family of distinct exotic $K3$ surfaces are studied, and comparison with the corresponding symmetries of the standard $K3$ is made. The action on the $K3$ lattice induced by a smooth finite group action is shown to be strongly restricted, and as a result, nonsmoothability…