相关论文: The fundamental group of $S^1$-manifolds

This note describes some recent results about the homotopy properties of Hamiltonian loops in various manifolds, including toric manifolds and one point blow ups. We describe conditions under which a circle action does not contract in the…

The main subjects of the paper is studying the fundamental groups of closed symplectically aspherical manifolds. Motivated by some results of Gompf, we introduce two classes of fundamental groups $\pi_1(M)$ of symplectically aspherical…

Let $M$ be a $2n$-dimensional closed symplectic manifold admitting a Hamiltonian circle action with isolated fixed points. We show that if $M$ contains an $S^1$-invariant symplectic hypersurface $D$ such that $M\setminus D$ is a homology…

In this article, we provide an exposition about symplectic toric manifolds, which are symplectic manifolds $(M^{2n}, \omega)$ equipped with an effective Hamiltonian $\mathbb{T}^n\cong (S^1)^n$-action. We summarize the construction of $M$ as…

We give a construction to obtain canonically an ``isotropic average'' of given $C^1$-close isotropic submanifolds of a symplectic manifold. To do so we use an improvement of Weinstein's submanifold averaging theorem (obtained in…

In an earlier paper, the second author resolved a question of McDuff by constructing a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points. In this paper, we…

A simple Hamiltonian manifold is a closed connected symplectic manifold equipped with a Hamiltonian action of a torus T with moment map Phi: M-->t^*, such that the fixed set M^T has exactly two connected components, denoted M_0 and M_1. We…

We study the topology of compact manifolds with a Lie group action for which there are only finitely many non-principal orbits, and describe the possible orbit spaces which can occur. If some non-principal orbit is singular, we show that…

Let $(M, \omega)$ be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian $S^1$ action such that the fixed point set consists of isolated points or surfaces. Assume dim $H^2(M)<3$, in \cite{L}, we…

A symplectic semitoric manifold is a symplectic $4$-manifold endowed with a Hamiltonian $(S^1 \times \mathbb{R})$-action satisfying certain conditions. The goal of this paper is to construct a new symplectic invariant of symplectic…

Symplectic slice theorems elucidate the local structure of symplectic manifolds carrying Hamiltonian actions of compact Lie groups. We generalize these theorems in two natural settings. The first is based on the idea that complex reductive…

We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field.

Let $M$ be a symplectic manifold, equipped with a semifree symplectic circle action with a finite, nonempty fixed point set. We show that the circle action must be Hamiltonian, and $M$ must have the equivariant cohomology and Chern classes…

Let $(M,\omega_M)$ be a six dimensional closed monotone symplectic manifold admitting an effective semifree Hamiltonian $S^1$-action. We show that if the minimal (or maximal) fixed component of the action is an isolated point, then…

Given a symplectic manifold $(M,\omega)$ endowed with a proper Hamiltonian action of a Lie group $G$, we consider the action induced by a Lie subgroup $H$ of $G$. We propose a construction for two compatible Witt-Artin decompositions of the…

In this paper we generalize constructions of non-commutative integrable systems to the context of weakly Hamiltonian actions on Poisson manifolds. In particular we prove that abelian weakly Hamiltonian actions on symplectic manifolds split…

In this paper we introduce invariants of semi-free Hamiltonian actions of $S\sp 1$ on compact symplectic manifolds (which satisfy some technical conditions related to positivity) using the space of solutions to certain gauge theoretical…

Consider a symplectic circle action on a closed symplectic manifold with non-empty isolated fixed points. Associated to each fixed point, there are well-defined non-zero integers, called weights. We prove that the action is Hamiltonian if…

We consider compact connected six dimensional symplectic manifolds with Hamiltonian SU(2) or SO(3) actions with cyclic principal stabilizers. We classify such manifolds up to equivariant symplectomorphisms.

In this paper we compute the homotopy groups of the symplectomorphism groups of the 3-, 4- and 5-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy…