Related papers: Circle actions on oriented 4-manifolds

Following the idea of Lusztig, Atiyah-Hirzebruch and Kosniowski, we note that the Dolbeault-type operators on compact, almost-complex manifolds are rigid. When the circle action has isolated fixed points, this rigidity result will produce…

Let the circle act in a Hamiltonian fashion on a connected compact symplectic manifold $(M, \omega)$ of dimension $2n$. Then the $S^1$-action has at least $n+1$ fixed points. In a previous paper, we study the case when the fixed point set…

We study isometric actions of compact Lie groups on complete orientable positively curved $n$-manifolds whose orbit spaces have non-empty boundary in the sense of Alexandrov geometry. In particular, we classify quotients of the unit sphere…

Let the circle act holomorphically on a compact K\"ahler manifold $M$ of complex dimension $n$ with moment map $\phi\colon M\to\R$. Assume the critical set of $\phi$ consists of 3 connected components, the extrema being isolated points. We…

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…

For a circle action on a compact almost complex manifold with a fixed point, the lower bound on the number of fixed points is known in dimension up to 12 except 10. In this paper, we show that if the circle group acts on a 10-dimensional…

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…

Let $X_0$ denote a compact, simply-connected smooth $4$-manifold with boundary the Poincar\'e homology $3$-sphere $\Sigma(2,3,5)$ and with even negative definite intersection form $Q_{X_0}=E_8$. We show that free $\mathbb{Z}/p$ actions on…

We study a class of localized indices for the Dirac type operators on a complete Riemannian orbifold, where a discrete group acts properly, co-compactly and isometrically. These localized indices, generalizing the $L^2$-index of Atiyah, are…

Consider a Hamiltonian circle action on a closed $8$-dimensional symplectic manifold $M$ with exactly five fixed points, which is the smallest possible fixed set. In their paper, L. Godinho and S. Sabatini show that if $M$ satisfies an…

We prove that the action of a reductive complex Lie group on a K\"ahler manifold can be linearized in the neighbourhood of a fixed point, provided that the restriction of the action to some compact real form of the group is Hamiltonian with…

This paper contains several results concerning circle action on almost-complex and smooth manifolds. More precisely, we show that, for an almost-complex manifold $M^{2mn}$(resp. a smooth manifold $N^{4mn}$), if there exists a partition…

Let $T$ be a compact fibered $3$--manifold, presented as a mapping torus of a compact, orientable surface $S$ with monodromy $\psi$, and let $M$ be a compact Riemannian manifold. Our main result is that if the induced action $\psi^*$ on…

For an action of a compact torus $T$ on a smooth compact manifold~$X$ with isolated fixed points the number $\frac{1}{2}\dim X-\dim T$ is called the complexity of the action. In this paper we study certain examples of torus actions of…

In this paper I construct, using off the shelf components, a compact symplectic manifold with a non-trivial Hamiltonian circle action that admits no Kaehler structure. The non-triviality of the action is guaranteed by the existence of an…

We discuss how the global geometry and topology of manifolds depend on different group actions of their fundamental groups, and in particular, how properties of a non-trivial compact 4-dimensional cobordism $M$ whose interior has a complete…

The question of what conditions guarantee that a symplectic $S^1$ action is Hamiltonian has been studied for many years. In a 1998 paper, Sue Tolman and Jonathon Weitsman proved that if the action is semifree and has a non-empty set of…

Consider a compact symplectic manifold of dimension $2n$ with a Hamiltionan circle action. Then there are at least $n+1$ fixed points. Motivated by recent works on the case that the fixed point set consists of precisely $n+1$ isolated…

If a (possibly finite) compact Lie group acts effectively, locally linearly, and homologically trivially on a closed, simply-connected four-manifold with second Betti number at least three, then it must be isomorphic to a subgroup of S^1 x…

We will prove the following theorems. The first theorem posits the existence of a fixed point for the actions of nilpotent Lie groups on nonpositively curved compact manifolds. The second theorem states that actions of solvable Lie groups…