>We prove the non-triviality of the Reeb flow for the standard contact spheres $\mathbb{S}^{2n+1}, n \neq 3$, inside the fundamental group of their contactomorphism group. The argument uses the existence of homotopically non-trivial $2$-spheres in the space of contact structures of a $3$-Sasakian manifold.

We show that there exist infinitely many simply connected compact Stein 4-manifolds with $b_2 = 2$ such that they are all homeomorhic but mutually non-diffeomorphic, and they are Stein fillings of the same contact $3$-manifold on their boundaries. We also describe their handlebody pictures.

Recently, extending work by Karshon et al., Borisov and McDuff showed in The topology of toric symplectic manifolds that a given closed symplectic manifold $(M,\omega)$ has a finite number of distinct toric structures. Moreover, in The topology of toric symplectic manifolds McDuff also showed that a product of two projective spaces $\mathbb{C}P^r \times \mathbb{C}P^s$ with any given symplectic form has a unique toric structure provided that $r, s \geq 2$. In contrast, the product $\mathbb{C}P^r \times \mathbb{C}P^1$ can be given infinitely many distinct toric structures, although only a finite number of these are compatible with each given symplectic form $\omega$. In this paper, we extend these results by considering the possible toric structures on a toric symplectic manifold $(M,\omega)$ with $\dim H^2(M) = 2$. In particular, all such manifolds are $\mathbb{C}P^r$ bundles over $\mathbb{C}P^s$ for some $r, s$. We show that there is a unique toric structure if $r \lt s$, and also that if $r, s \geq 2$, $M$ has at most finitely many distinct toric structures that are compatible with any symplectic structure on $M$. Thus, in this case the finiteness result does not depend on fixing the symplectic structure. We will also give other examples where $(M,\omega)$ has a unique toric structure, such as the case where $(M,\omega)$ is monotone.

Given a symplectic manifold $(M,\omega)$ and a Lagrangian submanifold $L$, we construct versions of the symplectic blow-up and blow-down which are defined relative to $L$. We further show that if $M$ admits an anti-symplectic involution $\phi$, i.e., a diffeomorphism such that $\phi^2 = \mathrm{Id}$ and $\phi^* \omega = - \omega$, and we blow-up an appropriately symmetric embedding of symplectic balls, then there exists an antisymplectic involution on the blow-up $\tilde{M}$ as well. We then derive a homological condition for real Lagrangian surfaces $L = \mathrm{Fix} (\phi)$ which determines when the topology of $L$ changes after a blowdown, and we use these constructions to study the relative packing numbers and packing stability for real symplectic four manifolds which are non-Seiberg-Witten simple.

We characterize which closed Reeb orbits of a dynamically convex contact form on the 3-sphere bound disk-like global surfaces of section for the Reeb flow, without any genericity assumptions. We show that these global surfaces of section come in families, organized as open book decompositions. As an application we obtain new global surfaces of section for the Hamiltonian dynamics on strictly convex three-dimensional energy levels.

We prove that a compact log-symplectic manifold has a class in the second cohomology group whose powers, except maybe for the top, are nontrivial. This result gives cohomological obstructions for the existence of log-symplectic structures similar to those in symplectic geometry.

Let $T$ be a compact torus and $X$ a nice compact $T$-space (say a manifold or variety). We introduce a functor assigning to $X$ a GKM-sheaf $\mathcal{F}_X$ over a GKM-hypergraph $\Gamma_X$. Under the condition that $X$ is equivariantly formal, the ring of global sections of $\mathcal{F}_X$ are identified with the equivariant cohomology, $H^*_T (X; \mathbb{C}) \cong H^0(\mathcal{F}_X)$. We show that GKM-sheaves provide a general framework able to incorporate numerous constructions in the GKM-theory literature. In the second half of the paper we apply these ideas to study the equivariant topology of the representation variety $\mathcal{R}K := \mathrm{Hom}(\pi_1 (\Sigma),K)$ under conjugation by $K$, where $\Sigma$ is a nonorientable surface and $K$ is a compact connected Lie group. We prove that $\mathcal{R}_{SU(3)}$ is equivariantly formal for all $\Sigma$ and compute its equivariant cohomology ring. We also produce conjectural betti number formulas for some other Lie groups.