We construct explicit maximal symplectic packings of minimal rational and ruled symplectic $4$-manifolds by few balls in a very simple way.

Given a compact symplectic manifold $(M,\,\kappa)$, $H^{2}(M,\,{\Bbb{R}})$\, represents, in a natural sense, the tangent space of the moduli space of germs of deformations of the symplectic structure. In the case $(M,\,\kappa,\,J)$ is a compact Kähler manifold, the author provides a complete description of the subset of $H^{2}(M,\,{\Bbb{R}})$ corresponding to Kähler deformations, including the non-generic case, where (at least locally) some hyperkähler manifold factors out from $M$. Several examples are also discussed.

In this work we use localization formulas in equivariant cohomology to show that some symplectic actions on $6$-dimensional manifolds with a finite fixed point set must be Hamiltonian. Moreover, we show that their fixed point data (number of fixed points and their isotropy weights) is the same as in $S^2\times S^2 \times S^2$ equipped with a diagonal circle action, and we compute their cohomology rings.

This is the second part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, \(I_{\rm F}\). (See [Y-J. Lee, Reidemeister torsion in Floer--Novikov theory and counting pseudo-holomorphic tori, I, J. Symplectic Geom. >3 (2005), no. 2, 221--311.] for Part I). Having constructed \(I_{\rm F}\) and outlined a proof of its invariance based on bifurcation analysis in Part I, in this part we prove a series of gluing theorems to confirm the bifurcation behavior predicted in Part I. These gluing theorems are different from (and much harder than) the more conventional versions in that they deal with broken trajectories or broken orbits connected at degenerate rest points which are not Morse--Bott. The issues of orientation and signs are also settled in the last section. This part is strongly >dependent on Part I, and is meant only for readers familiar with the previous part of this article.