Related papers: On the rank of $\pi_1(\text{Ham})$

A conjecture of Thomassen from 1982 states that for every k there is an f(k) so that every strongly f(k)-connected tournament contains k edge-disjoint Hamilton cycles. A classical theorem of Camion, that every strongly connected tournament…

We present a new proof of a result due to Taubes: if X is a closed symplectic four-manifold with b_+(X) > 1+b_1(X) and with some positive multiple of the symplectic form a rational class, then the Poincare dual of the canonical class of X…

We prove that for a certain class of closed monotone symplectic manifolds any Hamiltonian diffeomorphism with a hyperbolic fixed point must necessarily have infinitely many periodic orbits. Among the manifolds in this class are complex…

We address a conjecture that $\pi_1$-surjective maps between closed aspherical 3-manifolds having the same rank on $\pi_1$ must be of non-zero degree. The conjecture is proved for Seifert manifolds, which is used in constructing the first…

We prove that if two closed, connected, regular cosymplectic manifolds have isomorphic groups of cosymplectomorphisms (as topological groups), then the underlying manifolds are diffeomorphic. The proof proceeds by characterizing the Reeb…

Let $\text{Ham(M,L)}$ denote the group of Hamiltonian diffeomorphisms on a symplectic manifold $M$, leaving a Lagrangian submanifold $L\subset M$ invariant. In this paper, we show that $\text{Ham(M,L)}$ has the fragmentation property, using…

We prove that, under a simple condition on the cohomology ring, every closed 4-manifold has mod 2 Seiberg-Witten simple type. This result shows that there exists a large class of topological 4-manifolds such that all smooth structures have…

Consider a Hamiltonian action of a compact Lie group H on a compact symplectic manifold (M,w) and let G be a subgroup of the diffeomorphism group Diff(M). We develop techniques to decide when the maps on rational homotopy and rational…

We show that every closed, locally homogeneous Riemannian manifold with positive simplicial volume must be homeomorphic to a locally symmetric space of non-compact type.

Let (M, {\pi} ) be a Poisson manifold. A Poisson submanifold $P \in M$ gives rise to an algebroid $AP \rightarrow P$, to which we associate certain chomology groups which control formal deformations of {\pi} around P . Assuming that these…

For Hamiltonian circle actions on compact, connected, four-dimensional manifolds, we give a generators and relations description for the even part of the equivariant cohomology, as an algebra over the equivariant cohomology of a point. This…

For r at least 3, p at least 2, we classify all actions of the groups Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the circle. This is the same as describing all nontrivial group homomorphisms between groups of…

We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of K\"ahler surfaces. This leads to a new proof of the deformation-invariance of…

Let (M,\omega) be a four dimensional compact connected symplectic manifold. We prove that (M,\omega) admits only finitely many inequivalent Hamiltonian effective 2-torus actions. Consequently, if M is simply connected, the number of…

We classify the possible elementary amenable fundamental groups of compact aspherical 4-manifolds with boundary and conclude that they are either polycyclic or solvable Baumslag- Solitar. Since these groups are good and satisfy the…

We show that two orientable, four-dimensional folded symplectic toric manifolds are isomorphic provided that their orbit spaces have trivial degree-two integral cohomology and there exists a diffeomorphism of the orbit spaces (as manifolds…

In this paper, we gave some explicit relations between absolute and relative Gromov-Witten invariants. We proved that a symplectic manifold is symplectic rationally connected if it contains a positive divisor symplectomorphic to $P^n$.

We prove that a closed regular $(4k+1)$-Sasakian manifold $(M,h_0)$ admits a family of non-isometric metrics $h_\rho, \rho\geq 0,$ such that $\pi_1({\rm Isom}(M, h_\rho))$, the fundamental group of the isometry group, is infinite for…

The present paper proves that finite symplectic groups of automorphisms of hyperk\"ahler fourfolds deformation equivalent to the Hilbert scheme of two points on a $K3$ surface are contained in the simple group $Co_1$. Then we give an…

We show that for all integers $m\geq 2$, and all integers $k\geq 2$, the orthogonal groups $\Orth^{\pm}(2m,\Fk)$ act on abstract regular polytopes of rank $2m$, and the symplectic groups $\Sp(2m,\Fk)$ act on abstract regular polytopes of…