Holomorphic Curves in Symplectic Cobordisms

Seiberg–Witten and Gromov invariants for self-dual harmonic 2–forms

“…This technical lemma was originally proved for closed symplectic 4-manifolds in [41, Lemma 3.1] (see also [77,Theorem 5.1]), and we will prove it at the end of Section 1.3. By intersection positivity of J-holomorphic curves, a J-holomorphic exceptional sphere is the unique J-holomorphic curve in its homology class.…”

“…Proof of Lemma 1.3.12. We follow the proof of [77,Theorem 5.1]. Given a symplectic exceptional sphere in the class e ∈ E ω , we can always construct a (probably non-generic) cobordism-admissible almost complex structure J 0 on (X, ω) which is integrable in a neighborhood of the sphere.…”

Floer theory of disjointly supported Hamiltonians on symplectically aspherical manifolds

“…In this section we study the homology of this subcomplex. We show that for a properly chosen pair .f; J /, the homology of .C U ı .f /; @j U ı / coincides with the homology of U, namely, (24) H .C U ı .f /; @j U ı / Š H .U /:…”