arXiv research

We give examples of symplectic actions of a cyclic group, inducing a trivial action on homology, on four-manifolds that admit Hamiltonian circle actions, and show that they do not extend to Hamiltonian circle actions. Our work applies holomorphic methods to extend combinatorial tools developed for circle actions to stu…

The paper identifies conditions for free circle actions on specific 7-manifolds.

The paper identifies manifolds with free circle actions.

Study free circle actions on specific 7-manifolds with positive Ricci curvature.

Study of symplectomorphisms on ruled surfaces under circle actions.

The paper calculates a specific invariant for a manifold with a circle action.

We show that recent results of Friedl-Vidussi and Chen imply that a symplectic manifold admits a fixed point free circle action if and only if it admits a symplectic circle action and we give a complete description of the symplectic cone in this case. This then completes the characterisation of symplectic 4-manifolds t…

Classifies circle actions on 6D manifolds with 4 fixed points.

We show that almost complex circle actions with exactly three fixed points do not exist in dimension 8 and present an infinite series of 6-dimensional manifolds possessing an almost complex circle action with exactly two fixed points.

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 of $(P^1)^n$.

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 provide a necessary and sufficient condition for a pair of integers to arise as …

We define a notion of semi-conjugacy between orientation-preserving actions of a group on the circle, which for fixed point free actions coincides with a classical definition of Ghys. We then show that two circle actions are semi-conjugate if and only if they have the same bounded Euler class. This settles some existin…

Paper finds at least 6 fixed points for a specific circle action on a 10D manifold.

We prove that closed symplectic four-manifolds do not admit any smooth free circle actions with contractible orbits, without assuming that the actions preserve the symplectic forms. In higher dimensions such actions by symplectomorphisms do exist, and we give explicit examples based on a construction of Fernandez, Gray…

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 this result permit us to show that products of Kazhdan groups and certain latti…

The paper classifies manifolds with free torus actions and positive Ricci curvature.

Study circle actions on 4-manifolds, deriving formulas and graphs.

New actions found on exotic spheres using group theory.

Suspensions of manifolds by circle surgeries are key in free action constructions.

Finite index subgroups of certain groups cannot act faithfully on the circle.

We obtain a topological and equivariant classification of closed, connected three-dimensional Alexandrov spaces admitting a local isometric circle action. We show, in particular, that such spaces are homeomorphic to connected sums of some closed 3-manifold with a local circle action and finitely many copies of the susp…

A new homomorphism connects group actions on circles to Euler classes.

Study the geometry of twistor spaces with rotating circle action.

Study of spectral invariants on CR contact manifolds with circle action.

The main results of this paper describes a formula for the Seiberg-Witten invariant of a 4-manifold which admits a nontrivial free S^1-action. We use this theorem to produce a nonsymplectic 4-manifold with a free circle action whose orbit space fibers over S^1. We also describe a 3-manifold which is not the orbit space…

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 many identities concerning the weights on the fixed points. In particular, it giv…

In this paper we show that the Seiberg--Witten invariant is zero for all smooth 4--manifolds with $b_+{>}1$ which admit circle actions that have at least one fixed point. Furthermore, we show that all symplectic 4--manifolds which admit circle actions with fixed points are rational or ruled, and thus admit a symplectic…

Thurston showed that the fundamental group of a close atoroidal 3-manifold admitting a co-oriented taut foliation acts faithfully on the circle by orientation-preserving homeomorphisms. This action on the circle is called a universal circle action due to its rich information. In this article, we first review Thurston's…

In this paper, it is shown that non-isomorphic effective linear circle actions yield non-diffeomorphic differential structures on the corresponding orbit spaces.

Kawakubo and Uchida showed that, if a closed oriented $4k$-dimensional manifold $M$ admits a semi-free circle action such that the dimension of the fixed point set is less than $2k$, then the signature of $M$ vanishes. In this note, by using $G$-signature theorem and the rigidity of the signature operator, we generaliz…

We present some methods to construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets or non-symplectic cyclic isotropy point sets. All such actions are not compatible with any symplectic form.

Let $Γ$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $Γ$ on the circle is either trivial or semi-conjugate to a unique minimal action on the so-called simple circle.

The author proved that if the circle acts symplectically on a compact, connected symplectic manifold $M$ with three fixed points, then $M$ is equivariantly symplectomorphic to some standard action on $\mathbb{CP}^2$. In this paper, we extend the result to a circle action on an almost complex manifold; if the circle act…

We obtain a topological and weakly equivariant classification of closed three-dimensional Alexandrov spaces with an effective isometric circle action. As an application of the classification we prove a version of the Borel conjecture for closed three-dimensional Alexandrov spaces with circle symmetry.

We define a "circle Euler characteristic" of a circle action on a compact manifold or finite complex X. It lies in the first Hochschild homology group of ZG where G is the fundamental group of X. It is analogous in many ways to the ordinary Euler characteristic. One application is an intuitively satisfying formula for …

Study circle actions on unitary manifolds with discrete fixed points.

New findings on mapping class group actions on the circle, improving critical regularity.

New geometric interpretation of a group class using circle action and rotation numbers.

Minimal sphere dimension for equivariant embedding of circles.

We show that a complete simply-connected hyperkaehler 4-manifold with an isometric triholomorphic circle action is obtained from the Gibbons-Hawking ansatz with some suitable harmonic function.

The paper classifies circle actions on 6D manifolds with isolated fixed points.

If M is an atoroidal 3-manifold with a taut foliation, Thurston showed that pi_1(M) acts on a circle. Here, we show that some other classes of essential laminations also give rise to actions on circles. In particular, we show this for tight essential laminations with solid torus guts. We also show that pseudo-Anosov fl…

We prove a true bootstrapping result for convergence groups acting on a Peano continuum. We give an example of a Kleinian group H which is the amalgamation of two closed hyperbolic surface groups along a simple closed curve. The limit set Lambda H is the closure of a `tree of circles' (adjacent circles meeting in pairs…

Study circle actions with exactly three fixed points on specific manifolds.

Study rigidity of PSU(1,1) actions on circle via harmonic measures.

We classify positively curved Alexandrov spaces of dimension 4 with an isometric circle action up to equivariant homeomorphism, subject to a certain additional condition on the infinitesimal geometry near fixed points which we conjecture is always satisfied. As a corollary, we also classify positively curved Riemannian…

Paper distinguishes 2-knots with circle actions using fundamental groups.

The paper explores universal circles for Anosov foliations and their uniqueness.