Title: Cieliebak's Invariance Theorem and contact structures via connected sums,
Abstract: We present a strong version of Abouzaid's No-Escape Lemma, which allows
varying contact forms on the boundary and which can be used instead of the
Maximum Principle. Moreover, we give a clarified proof of Cieliebak's
Invariance Theorem for Symplectic homology under subcritical handle attachment.
Finally, we introduce the notion of asymptotically finitely generated contact
structures, which states essentially that the Symplectic homology in a certain
degree of any filling of such contact manifolds is uniformly generated by only
finitely many Reeb orbits. This property is then used to show that a large
class of manifolds carries infinitely many exactly fillable contact structures.