Intersection theory of punctured pseudoholomorphic curves

TL;DR

This paper develops a comprehensive intersection theory for punctured pseudoholomorphic curves in 4-dimensional symplectic cobordisms, including local intersection properties, topological controls, and embedding criteria, with applications to open book decompositions.

Contribution

It introduces new topological and intersection control tools for punctured pseudoholomorphic curves, extending previous results and providing criteria for embeddings and applications to holomorphic open books.

Findings

Established local intersection properties at punctures.

Developed an adjunction formula for embeddedness criteria.

Presented conditions for projections to be embeddings.

Abstract

We study the intersection theory of punctured pseudoholomorphic curves in $4$-dimensional symplectic cobordisms. We first study the local intersection properties of such curves at the punctures. We then use this to develop topological controls on the intersection number of two curves. We also prove an adjunction formula which gives a topological condition that will guarantee a curve in a given homotopy class is embedded, extending previous work of Hutchings. We then turn our attention to curves in the symplectization $\mathbb{R}\times M$ of a $3$-manifold $M$ admitting a stable Hamiltonian structure. We investigate controls on intersections of the projections of curves to the $3$-manifold, and we present conditions that will guarantee the projection of a curve to the $3$-manifold is an embedding. Finally we consider an application concerning pseudoholomorphic curves in manifolds admitting a certain class of holomorphic open book decomposition, and an application concerning the existence of generalized pseudoholomorphic curves.