We study the intersection theory of punctured pseudoholomorphic curves in
–dimensional
symplectic cobordisms. Using the asymptotic results of the author [Comm. Pure
Appl. Math. 61(2008) 1631–84], 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 [JEMS 4(2002)
313–61].
We then turn our attention to curves in the symplectization
of a
–manifold
admitting a
stable Hamiltonian structure. We investigate controls on intersections of the projections of curves
to the –manifold
and we present conditions that will guarantee the projection of a curve to the
–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, as
introduced by Hofer [Geom. Func. Anal. (2000) 674–704] .
Keywords
pseudoholomorphic curves, symplectic field theory, Floer
homology, intersection theory