The introduction motivates the remainder of the book via two specific examples of theorems from the early days of symplectic topology in which intersection theory plays a prominent role. We sketch closely analogous proofs of both theorems, emphasizing the way that intersection theory is used, but point out why the second theorem (on symplectic 4-manifolds that are standard near infinity) requires a nonobvious extension of homological intersection theory to punctured holomorphic curves. We then discuss informally some of the properties this theory will need to have and what kinds of subtle issues may arise.

This appendix is intended as a quick reference on Siefring’s intersection theory for the benefit of researchers who would like to use it and need an easy place to look up the main facts. All results here are stated without proof, with references (mostly to Lectures 3 and 4 in this book) given for further details. Two additional topics are covered that do not appear elsewhere in this book: covering relations for the star-pairing and normal Chern number, and the intersection product between holomorphic buildings. Finally, the appendix concludes with a comparison of notational and terminology conventions between Siefring’s theory and the equivalent notions that often appear in the literature on embedded contact homology: in particular, we clarify the relationship between Siefring’s relative asymptotic contributions and Hutchings’s asymptotic linking number and writhe.

This lecture presents the technical asymptotic results underlying Siefring’s intersection theory for punctured holomorphic curves, including the necessary prerequisites on asymptotic operators and the relations proved by Hofer, Wysocki and Zehnder between winding numbers of asymptotic eigenfunctions and the Conley–Zehnder index. Siefring’s relative asymptotic formulas are stated largely without proof but are motivated in terms of an asymptotic analogue of the similarity principle. The last section then discusses the punctured analogue of the question about holomorphic foliations considered in Lecture 2, which motivates the definition of the normal Chern number for punctured holomorphic curves.

Using the relative asymptotic results of the previous lecture as a black box, this lecture explains the main definitions and results of Siefring’s intersection theory for punctured holomorphic curves. This includes the definition of a homotopy-invariant algebraic intersection number (the so-called “star-pairing”), the notion of hidden intersections at infinity and asymptotic positivity of intersections, and a punctured generalization of the adjunction formula.