Related papers: Hamiltonian circle actions with minimal isolated f…

We formulate the Chern-Simons action for any compact Lie group using Deligne cohomology. This action is defined as a certain function on the space of smooth maps from the underlying 3-manifold to the classifying space for principal bundles.…

The main result of this note is that every closed Hamiltonian S^1 manifold is uniruled, i.e. it has a nonzero Gromov--Witten invariant one of whose constraints is a point. The proof uses the Seidel representation of \pi_1 of the Hamiltonian…

Let S be a connected orientable surface with finitely many punctures, finitely many boundary components, and genus at least 6. Then any C^1 action of the mapping class group of S on the circle is trivial. The techniques used in the proof of…

Cohomogeneity one actions on irreducible Riemannian symmetric spaces of compact type are classified into three cases: Hermann actions, actions induced by the linear isotropy representation of a Riemannian symmetric space of rank 2, and…

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…

In the present paper, we consider an action of the circle group on a compact oriented 4-manifold. We derive the Atiyah-Hirzebruch formula for the manifold, and associate a graph in terms of data on the fixed point set. We show in the case…

We study isometric actions of Steinberg groups on Hadamard manifolds. We prove some rigidity properties related to these actions. In Particular we show that every isometric action of $St_n(F_p\langle t_1,\ldots ,t_k \rangle)$ on Hadamard…

We propose actions for non-linear sigma models on cosets $G/H$ in 2+1 dimensions that include the most general non-linear realizations of Chern-Simons terms. When $G$ is simply connected and $H$ contains $r$ commuting U(1) factors, there…

Let $S^1$ act on a symplectic manifold in a Hamiltonian fashion with momentum map $\Psi$. Fix a value $a$ of $\Psi$. There is a question of whether the symplectic quotient at $a$ is diffeomorphic to the orbit space of some proper Lie group…

We prove the following to results: (1) A subgroup G of the isometry group of a Riemannian manifold M acts properly on M if and only if G is closed in the isometry group of M. (2) The orbits of an isometric action are closed if and only if…

The aim of this article is to present unifying proofs for results in geometric quantisation with real polarisations by exploring the existence of symplectic circle actions. It provides an extension of Rawnsley's results on the Kostant…

On an open, connected symplectic manifold $(M,\omega)$, the group of Hamiltonian diffeomorphisms forms an infinite-dimensional Fr\'echet Lie group with Lie algebra $C^{\infty}_c(M)$ and adjoint action given by pullbacks. We prove that this…

We develop the basic theory of Maurer-Cartan simplicial sets associated to (shifted complete) $L_\infty$ algebras equipped with the action of a finite group. Our main result asserts that the inclusion of the fixed points of this equivariant…

Let P be a closed triangulated manifold, dim P=n. We consider the group of simplicial 1-chains C_1(P) and the homology group H_1(P). We also use some nonnegative weighting function L on C_1(P). For any homological class from H_1(P) method…

Classical mechanical systems are modeled by a symplectic manifold $(M,\omega)$, and their symmetries, encoded in the action of a Lie group $G$ on $M$ by diffeomorphisms that preserves $\omega$. These actions, which are called "symplectic",…

We study loops of symplectic diffeomorphisms of closed symplectic manifolds. Our main result, which is valid for a large class of symplectic manifolds, shows that the flux of a symplectic loop vanishes whenever its orbits are contractible.…

We study smooth locally free actions of ${\mathbb R}^n$ on manifolds $M$ of dimension $n+1$. We are interested in compact orbits and in compact actions: actions with all orbits compact. Given a compact orbit in a neighborhood of compact…

To begin the paper we revisit a cohomological localization result of Jones-Rawnsley which was subsequently improved by Farber, further generalizing the result. We then proceed to improve a previous result of the author on complete…

The unimodality conjecture posed by Tolman in the conference `Moment maps in Various Geometry" in 2005 states that if (M,w) is a 2n-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated…

We calculate geometric and homotopical (or stable) bordism rings associated to semi-free $S^1$ actions on complex manifolds, giving explicit generators for the geometric theory. The classification of semi-free actions with isolated fixed…