We exhibit two symplectic surfaces embedded in the 4-ball which bound the same transverse knot, have the same topology (as abstract surfaces), and are distinguished by the fundamental groups of their complements.

In her PhD thesis, Milin developed a $Z_k$-equivariant version of the contact homology groups constructed in Geometry of contact transformations and domains: orderability vs squeezing, "Geom. Topol." 10 (2006), 1635–1747 and used it to prove a $Z_k$-equivariant contact non-squeezing theorem. In this article, we re-obtain the same result in the setting of generating functions, starting from the homology groups studied in Contact homology, capacity and non-squeezing in $R^2n × S^1$ via generating functions, "Ann. Inst. Fourier (Grenoble)" 61 (2011), 145–185. As Milin showed, this result implies orderability of lens spaces.

Fix a compact 4-dimensional manifold with self-dual second Betti number one and with a given symplectic form. This article proves the following: The Frêchet space of tamed almost complex structures as defined by the given symplectic form has an open and dense subset whose complex structures are compatible with respect to a symplectic form that is cohomologous to the given one. The theorem is proved by constructing the new symplectic form by integrating over a space of currents that are defined by pseudo-holomorphic curves.

In this short article, we find an explicit formula for Maslov index of Whitney $n$-gons joining intersections points of $n$ half-dimensional tori in the symmetric product of a surface. The method also yields a formula for the intersection number of such an $n$-gon with the fat diagonal in the symmetric product.