Related papers: Circle actions with two fixed points

We establish a necessary and sufficient condition for pairs of integers to arise as the weights at the fixed points of an effective circle action on a compact almost complex 4-manifold with a discrete fixed point set. As an application, we…

We consider 3-manifolds admitting the action of an involution such that its space of orbits is homeomorphic to $S^3.$ Such involutions are called hyperelliptic as the manifolds admitting such an action. We consider finite groups acting on…

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…

In this paper, we study a circle action on a compact oriented manifold with a discrete fixed point set. The fixed point data consists of the weights of the $S^1$-representations at the fixed points. We prove various results and properties…

Motivated by the theory of Riemann surfaces, we classify all possibilities for finite simple groups acting faithfully on a compact Riemann surface of genus at least 2 in such a way that all non-trivial elements have at most three fixed…

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 show that, for $\varepsilon=\dfrac{1}{4000}$, any action of a finite cyclic group by $(1+\varepsilon)$-bilipschitz homeomorphisms on a closed 3-manifold is conjugated to a smooth action.

Let a torus act on a compact oriented manifold $M$ with isolated fixed points, with an additional mild assumption that its isotropy submanifolds are orientable. We associate a signed labeled multigraph encoding the fixed point data (weights…

We study algebraic properties of groups of PL or smooth homeomorphisms of unit cubes in any dimension, fixed pointwise on the boundary, and more generally PL or smooth groups acting on manifolds and fixing pointwise a submanifold of…

If the circle acts in a Hamiltonian way on a compact symplectic manifold of dimension $2n$, then there are at least $n+1$ fixed points. The case that there are exactly $n+1$ isolated fixed points has its importance due to various reasons.…

We study smooth, closed orientable $S^1$-manifolds $M$ with exactly $3$ fixed points. We show that the dimension of $M$ is of the form $4\cdot 2^a$ or $8\cdot(2^a+2^b)$ with $a,b\geq 0$ and $a\neq b$. Moreover, under the extra assumption…

In this paper, we prove that if a finite group acts smoothly and effectively on an integral homology six-sphere and the fixed point set has an odd Euler characteristic, then the acting group is isomorphic to either the alternating group on…

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…

We use the notion of fixity for representations of finite groups to construct free and smooth actions on products of spheres. In particular we show that a finite p-group (for p>3) will act freely and smoothly on a product of two spheres if…

We consider an effective action of a compact (n-1)-torus on a smooth 2n-manifold with isolated fixed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain…

In this series of two articles, we prove that every action of a finite group $G$ on a finite and contractible $2$-complex has a fixed point. The proof goes by constructing a nontrivial representation of the fundamental group of each of the…

We show that if a holomorphic $n$ dimensional compact torus action on a compact connected complex manifold of complex dimension $n$ has a fixed point then the manifold is equivariantly biholomorphic to a smooth toric variety.

We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is…

Let G be a group acting on the plane by orientation-preserving homeomorphisms. We show that if for some k>0 there is a ball of radius r > k/\sqrt{3} such that each point x in the ball satisfies |gx -hx| < k for all g, h in G, and the action…

The classical multiplicative (Hirzebruch) genera of manifolds have the wonderful property which is called rigidity. Rigidity of a genus h means that if a compact connected Lie group G acts on a manifold X, then the equivariant genus h^G(X)…