Hamiltonian circle actions with minimal isolated fixed points

Let the circle act in a Hamiltonian fashion on a compact symplectic manifold $(M, \omega)$ of dimension $2n$. Then the $S^1$-action has at least $n+1$ fixed points. We study the case when the fixed point set consists of precisely $n+1$ isolated points. We show certain equivalence on the first Chern class of $M$ and some particular weight of the $S^1$-action at some fixed point. We show that the particular weight can completely determine the integral cohomology ring of $M$, the total Chern class of $M$, and the sets of weights of the $S^1$-action at all the fixed points. We will see that all these data are isomorphic to those of known examples, \CP^n, or \Gt_2(\R^{n+2}) with $n\geq 3$ odd, equipped with standard circle actions.Comment: title is slightly changed. Some contents are change

The fundamental groups of contact toric manifolds

Let $M$ be a connected compact contact toric manifold. Most of such manifolds are of Reeb type. We show that if $M$ is of Reeb type, then $\pi_1(M)$ is finite cyclic, and we describe how to obtain the order of $\pi_1(M)$ from the moment map image.Comment: This version is the published on

Twisted Topological Graph Algebras

We define the notion of a twisted topological graph algebra associated to a topological graph and a $1$-cocycle on its edge set. We prove a stronger version of a Vasselli's result. We expand Katsura's results to study twisted topological graph algebras. We prove a version of the Cuntz-Krieger uniqueness theorem, describe the gauge-invariant ideal structure. We find that a twisted topological graph algebra is simple if and only if the corresponding untwisted one is simple.Comment: 27 page