Dmitry Tonkonog

Commuting symplectomorphisms and Dehn twists in divisors

Two commuting symplectomorphisms of a symplectic manifold give rise to actions on Floer cohomologies of each other. We prove the elliptic relation saying that the supertraces of these two actions are equal. In the case when a symplectomorphism

A simple proof of the geometric fractional monodromy theorem

f

Embedding 3-manifolds with boundary into closed 3-manifolds

commutes with a symplectic involution, the elliptic relation provides a lower bound on the dimension of

The closed-open string map for

HF^*(f)

S^1

in terms of the Lefschetz number of

–invariant Lagrangians

f

Nerves Of Good Covers Are Algorithmically Unrecognizable

restricted to the fixed locus of the involution. We apply this bound to prove that Dehn twists around vanishing Lagrangian spheres inside most hypersurfaces in Grassmannians have infinite order in the symplectic mapping class group.

The wall-crossing formula and Lagrangian mutations

A simple proof of the “geometric fractional monodromy theorem” (Broer-Efstathiou-Lukina 2010) is presented. The fractional monodromy of a Liouville integrable Hamiltonian system over a loop γ ⊂ ℝ2 is a generalization of the classic monodromy to the case when the Liouville foliation has singularities over γ. The “geometric fractional monodromy theorem” finds, up to an integral parameter, the fractional monodromy of systems similar to the 1: (−2) resonance system. A handy equivalent definition of fractional monodromy is presented in terms of homology groups for our proof.

From symplectic cohomology to Lagrangian enumerative geometry

Abstract We prove that there is an algorithm which determines whether or not a given 2-polyhedron can be embedded into some integral homology 3-sphere. This is a corollary of the following main result. Let M be a compact connected orientable 3-manifold with boundary. Denote G = Z , G = Z / p Z or G = Q . If H 1 ( M ; G ) ≅ G k and ∂M is a surface of genus g, then the minimal group H 1 ( Q ; G ) for closed 3-manifolds Q containing M is isomorphic to G k − g . Another corollary is that for a graph L the minimal number rk H 1 ( Q ; Z ) for closed orientable 3-manifolds Q containing L × S 1 is twice the orientable genus of the graph.

Low-area Floer theory and non-displaceability

Given a monotone Lagrangian submanifold invariant under a loop of Hamiltonian diffeomorphisms, we compute a piece of the closed-open string map into the Hochschild cohomology of the Lagrangian which captures the homology class of the loop’s orbit on that Lagrangian. Our computation has a range of applications: split-generation and non-formality results for real Lagrangian submanifolds in some toric varieties, like projective spaces or their blow-ups along linear subspaces; and a split-generation result for monotone toric fibres when the Landau-Ginzburg superpotential has a singularity of type A2.