A cut-and-paste approach to contact topology

3-dimensional methods in contact geometry

These are lecture notes on cut-and-paste methods in 3-dimensional contact geometry.

Families of four dimensional manifolds that become mutually diffeomorphic after one stabilization

In this paper, we will introduce a cut and paste move, called a geometrically null log transform, and prove that any two manifolds related by a sequence of these moves become diffeomorphic afte r one stabilization. To motivate the cut and paste move, we will use the symplec tic fiber sum, and a construction of Fintushel and Stern to construct several large families of 4-manifolds. We will then proceed to prove that the members of any one of these families become diffeomorphic after one stabilization. Finally, we will compute the Seiberg-Witten invariants of each member of each of the families.

Bott-Taubes-Vassiliev cohomology classes by cut-and-paste topology

Bott and Taubes used integrals over configuration spaces to produce finite-type a.k.a. Vassiliev knot invariants. Cattaneo, Cotta-Ramusino and Longoni then used these methods together with graph cohomology to construct "Vassiliev classes" in the real cohomology of spaces of knots in higher-dimensional Euclidean spaces, as first promised by Kontsevich. Here we construct integer-valued cohomology classes in spaces of knots and links in Euclidean spaces of dimension greater than three. We construct such a class for any integer-valued graph cocycle, by the method of gluing compactified configurati…

The Milnor triple-linking number of string links by cut-and-paste topology

Bott and Taubes constructed knot invariants by integrating differential forms along the fiber of a bundle over the space of knots, generalizing the Gauss linking integral. Their techniques were later used to construct real cohomology classes in spaces of knots and links in higher-dimensional Euclidean spaces. In previous work, we constructed cohomology classes in knot spaces with arbitrary coefficients by integrating via a Pontrjagin--Thom construction. We carry out a similar construction over the space of string links, but with a refinement in which configuration spaces are glued together acc…

A class of tight contact structures on Sigma_2 x I

We employ cut and paste contact topological techniques to classify some tight contact structures on the closed, oriented genus-2 surface times the interval. A boundary condition is specified so that the Euler class of the of the contact structure vanishes when evaluated on each boundary component. We prove that there exists a unique, non-product tight contact structure in this case.

Multisections of 4-manifolds

We introduce multisections of smooth, closed 4-manifolds, which generalize trisections to decompositions with more than three pieces. This decomposition describes an arbitrary smooth, closed 4-manifold as a sequence of cut systems on a surface. We show how to carry out many smooth cut and paste operations in terms of these cut systems. In particular, we show how to implement a cork twist, whereby we show that an arbitrary exotic pair of smooth 4-manifolds admit 4-sections differing only by one cut system. By carrying out fiber sums and log transforms, we also show that the elliptic fibrations …

Convex plumbings and Lefschetz fibrations

We show that under appropriate hypotheses, a plumbing of symplectic surfaces in a symplectic 4-manifold admits strongly convex neighborhoods. Moreover the neighborhoods are Lefschetz fibered with an easily-described open book on the boundary supporting the induced contact structure. We point out some applications to cut-and-paste constructions of symplectic 4-manifolds.

接触切割图与Weinstein $\mathcal{L}$-不变量

The contact cut graph and a Weinstein $\mathcal{L}$-invariant

本文定义接触切割图并建立其与Lefschetz纤维化及带divides多截面的对应，由此构造Weinstein域的辛不变量$\mathcal{L}$，讨论其与Lefschetz稳定化的关系，并给出$\mathcal{L}=0$的约束条件及$\mathcal{L}$任意大的例子。

Surgery along star shaped plumbings and exotic smooth structures on 4-manifolds

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 operation can be used in conjunction with knot surgery to construct an infinite family of minimal exotic smooth structures on the complex projective space blown-up seven times.