Using Floer-theoretic methods, we prove that the non-existence of an exotic symplectomorphism on the standard symplectic ball, $\mathbb{B}^{2n},$ implies a rather strict topological condition on the free contact circle actions on the standard contact sphere, $\mathbb{S}^{2n-1}.$ We also prove an analogue for a Liouville domain and contact circle actions on its boundary. Applications include results concerning the symplectic mapping class group and the fundamental group of the group of contactomorphisms.
We construct a symplectic structure on a disc that admits a compactly supported symplectomorphism which is not smoothly isotopic to the identity. The symplectic structure has an overtwisted concave end; the construction of the symplectomorphism is based on a unitary version of the Milnor-Munkres pairing. En route, we introduce a symplectic analogue of the Gromoll filtration.
We show that, for certain families $ϕ_{\mathbf{s}}$ of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of $T^*S^n$ with the family of pullbacks $ϕ^*_{\mathbf{s}}$ gives a noncontractible family of compactly-supported symplectomorphisms. In particular, we find examples: where the Dehn twist along a parametrised Lagrangian sphere depends up to Hamiltonian isotopy on its parametrisation; where the symplectomorphism group is not simply-connected, and where the symplectomorphism group does not have the homotopy-type of a finite CW-complex. We sho…
Consider any symplectic ruled surface $(M^g_λ,ω_λ)$ given by $(Σ_g \times S^2, λσ_{Σ_g} \oplus σ_{S^2})$. We compute all natural equivariant Gromov-Witten invariants $EGW_{g,0}(M^g_λ;H_k, A-kF)$ for all hamiltonian circle actions $H_k$ on $M^g_λ$, where $A=[Σ_g \times pt]$ and $F= [pt \times S^2]$. We use these invariants to show the nontriviality of certain higher order Whitehead products that live in the homotopy groups of the symplectomorphism groups $G_λ^g$, $g \geq 0$. Our results are sharper when $g=0,1$ and enable us to answer a question posed by D.McDuff in the case $g=1$ and provide a…
We consider the 3-point blow-up of the manifold $ (S^2 \times S^2, σ\oplus σ)$ where $σ$ is the standard symplectic form which gives area 1 to the sphere $S^2$, and study its group of symplectomorphisms $\rm{Symp} ( S^2 \times S^2 \#\, 3\overline{\mathbb C\mathbb P}\,\!^2, ω)$. So far, the monotone case was studied by J. Evans and he proved that this group is contractible. Moreover, J. Li, T. J. Li and W. Wu showed that the group Symp$_{h}(S^2 \times S^2 \#\, 3\overline{ \mathbb C\mathbb P}\,\!^2,ω) $ of symplectomorphisms that act trivially on homology is always connected and recently they al…
We study generators of the fundamental group of the group of symplectomorphisms $\mathrm{Symp}({\mathbb C\mathbb P}^2\#\,5\overline{\mathbb C\mathbb P}\,\!^2, ω)$ for some particular symplectic forms. It was observed by J. Kȩdra that there are many symplectic 4-manifolds $(M, ω)$, where $M$ is neither rational nor ruled, that admit no circle action and $π_1 (\mathrm{Ham} (M,ω))$ is nontrivial. On the other hand, it follows from previous results that the fundamental group of the group $\mathrm{Symp}_h({\mathbb C\mathbb P}^2\#\,k\,\overline{\mathbb C\mathbb P}\,\!^2, ω)$, of symplectomorphisms t…
We prove that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. In higher dimensions such actions by symplectomorphisms do exist, and we give explicit examples based on a construction of Fernandez, Gray and Morgan. We also prove that the only compact complex surfaces admitting free circle actions with contractible orbits are primary Hopf surfaces.
We identify a family of torus representations such that the corresponding singular symplectic quotients at the $0$-level of the moment map are graded regularly symplectomorphic to symplectic quotients associated to representations of the circle. For a subfamily of these torus representations, we give an explicit description of each symplectic quotient as a Poisson differential space with global chart as well as a complete classification of the graded regular diffeomorphism and symplectomorphism classes. Finally, we give explicit examples to indicate that symplectic quotients in this class may …
In a recent preprint Yael Karshon showed that there exist non-conjugate tori in a group of symplectomorphisms of a Hirzebruch surface. She counted them in terms of the cohomology class of the symplectic structure. We show that a similar phenomenon exists in the contactomorphism groups of pre-quantum circle bundles over Hirzebruch surfaces. Note that the contact structures in question are fillable. This may be contrasted with an earlier paper where we showed that there are infinitely many non-conjugate tori in the contactomorphism groups of certain overtwisted lens spaces (Contact cuts, Israel …
We consider a fixed contact 3-manifold that admits infinitely many compact Stein fillings which are all homeomorphic but pairwise non-diffeomorphic. Each of these fillings gives rise to a closed contact 5-manifold described as a contact open book whose page is the filling at hand and whose monodromy is the identity symplectomorphism. We show that the resulting infinitely many contact 5-manifolds are all diffeomorphic but pairwise non-contactomorphic. Moreover, we explicitly determine these contact 5-manifolds.
We consider the action of symplectic monodromy on chain-level enhancements of quantum cohomology. First, we construct a family version of $A_\infty$-structure on quantum cohomology (this should morally correspond to Hochschild cohomology of a "family of Fukaya categories over the circle"). Following Kaledin, we look at the obstruction class of this structure, and argue that it can be related to a quantum version of Massey products on the one hand (which, in nice cases, can be related to actual counts of rational curves) and to the classical Andreadakis-Johnson theory of the Torelli group on th…
This paper studies Hamiltonian circle actions, i.e. circle subgroups of the group Ham(M,\om) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M,\om). Our main tool is the Seidel representation of π_1(\Ham(M,\om)) in the units of the quantum homology ring. We show that if the weights of the action at the points at which the moment map is a maximum are sufficiently small then the circle represents a nonzero element of π_1(\Ham(M,\om)). Further, if the isotropy has order at most two and the circle contracts in \Ham(M,\om) then the homology of M is invariant under an involution. …
Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if $(M, \om)$ is a coadjoint orbit of a compact Lie group $G$ then every element of $π_1(G)$ may be represented by a semifree $S^1$-action. A theorem of McDuff--Slimowitz then implies that $π_1(G)$ injects into $π_1(\Ham(M, \om))$, which answers a question raised by Weinstein. We also show that a circle action on a manifold $M$ which is semifree near a fixed point $x$ cannot contract in a compact Lie subgroup $G$ of the diffeomorphism group unless the action is reversed by an …
In this note, we prove that two constructions of exotic monotone Lagrangian tori, namely the one by Chekanov and Schlenk and the one obtained by the circle bundle construction of Biran are Hamiltonian isotopic in $\mathbb{C}P^2$ and $S^2 \times S^2$.
Let $M$ be a compact orientable surface equipped with a volume form $ω$, $P$ be either $\mathbb{R}$ or $S^1$, $f:M\to P$ be a $C^{\infty}$ Morse map, and $H$ be the Hamiltonian vector field of $f$ with respect to $ω$. Let also $\mathcal{Z}_ω(f) \subset C^{\infty}(M,\mathbb{R})$ be set of all functions taking constant values along orbits of $H$, and $\mathcal{S}_{\mathrm{id}}(f,ω)$ be the identity path component of the group of diffeomorphisms of $M$ mutually preserving $ω$ and $f$. We construct a canonical map $φ: \mathcal{Z}_ω(f) \to \mathcal{S}_{\mathrm{id}}(f,ω)$ being a homeomorphism whene…
We prove that there exist diffeomorphisms of tori, supported in a disc, which are not isotopic to symplectomorphisms with respect to any symplectic structure. This yields a partial negative answer to a question of Benson and Gordon about the existence of symplectic structures on tori with exotic differential structure.
Let $K$ be a compact Lie group of positive dimension. We show that for most unitary $K$-modules the corresponding symplectic quotient is not regularly symplectomorphic to a linear symplectic orbifold (the quotient of a unitary module of a finite group). When $K$ is connected, we show that even a symplectomorphism to a linear symplectic orbifold does not exist. Our results yield conditions that preclude the symplectic quotient of a Hamiltonian $K$-manifold from being locally isomorphic to an orbifold. As an application, we determine which unitary $\operatorname{SU}_2$-modules yield symplectic q…
