Circle Actions Research Articles

A Compactly Supported Formula for Equivariant Localization and Simplicial Complexes of Bialynicki-Birula Decompositions

Let X be a projective scheme carrying a circle action S with isolated fixed points. We associate a simplicial complex Delta(X,S) of closure chains using a refinement of its Morse/Bialynicki-Birula decomposition. If this decomposition is a stratification (e.g. when X is a flag manifold), then Delta(X,S) is just the order complex of the poset of fixed points. For X a toric variety, Delta(X,S) is a triangulation of the moment polytope. We compute some other examples, including a Bott-Samelson manifold and the punctual Hilbert scheme of 4 points in the plane. Summing over the facets of Delta(X,S), we obtain a positive formula for the Duistermaat-Heckman measure on the moment polytope of X, defined for any torus action extending S. We explain how, through brutal use of partial fractions, this can be extended to an AB/BV-type formula for integrating general classes. Throughout we work with equivariant Chow groups, and do not make any smoothness requirements on X.

Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.

Embedding 3-manifolds with circle actions

Constraints on the Seifert invariants of orientable 3-manifolds M M which admit fixed-point free S 1 S^1 -actions and embed in R 4 \mathbb {R}^4 are given. In particular, the generalized Euler invariant of the orbit fibration is determined up to sign by the base orbifold B B unless H 1 ( M ; Z ) H_1(M;\mathbb {Z}) is torsion free, in which case it can take at most one nonzero value (up to sign). An H 2 × E 1 \mathbb {H}^2\times \mathbb {E}^1 -manifold M M with base orbifold B = S 2 ( α 1 , … , α r ) B=S^2(\alpha _1,\dots ,\alpha _r) where all cone point orders are odd embeds in R 4 \mathbb {R}^4 if and only if its Seifert data S S is skew-symmetric.

Cap products in string topology

Chas and Sullivan showed that the homology of the free loop space LM of an oriented closed smooth finite dimensional manifold M admits the structure of a Batalin-Vilkovisky (BV) algebra equipped with an associative product called the loop product and a Lie bracket called the loop bracket. We show that the cap product is compatible with the above two products in the loop homology. Namely, the cap product with cohomology classes coming from M via the circle action acts as derivations on loop products as well as on loop brackets. We show that Poisson identities and Jacobi identities hold for the cap product action, extending the BV structure in the loop homology to the one including the cohomology of M. Finally, we describe the cap product in terms of the BV algebra structure in the loop homology.

A METHOD TO CONSTRUCT 4-DIMENSIONAL SPACETIMES WITH A SPACELIKE CIRCLE ACTION

A general procedure to construct a 4-dimensional spacetime from a 3-dimensional time-oriented Lorentzian manifold and each of its timelike vector fields is exposed. It is based on the construction of the null congruence Lorentzian manifold. As an application, examples of stably causal spacetimes which obey the timelike convergence condition, are semi-symmetric, and admit an isometric spacelike circle action are obtained.

The topology of symplectic circle bundles

We consider circle bundles over compact three-manifolds with symplectic total spaces. We show that the base of such a space must be irreducible or the product of the two-sphere with the circle. We then deduce that such a bundle admits a symplectic form if and only if it admits one that is invariant under the circle action in three special cases: namely if the base is Seifert bered, has vanishing Thurston norm, or if the total space admits a Lefschetz bration.

Rational homotopy of the (homotopy) fixed point sets of circle actions

In this paper, when G is the circle S 1 and M is a G-space, we study the rational homotopy type of the fixed point set M G , the homotopy fixed point set M h G , and the natural injection M G → M h G .

Fixed points of circle actions on spaces with rational cohomology of $$S^{n} \vee S^{2n} \vee S^{3n} {\rm {\bf or}}P^{2}(n) \vee S^{3n}$$

Let $X$ be a finitistic space with its rational cohomology isomorphic to that of the wedge sum $P^2(n)\vee S^{3n} $ or $S^{n} \vee S^{2n}\vee S^{3n}$. We study continuous $\mathbb{S}^1$ actions on $X$ and determine the possible fixed point sets up to rational cohomology depending on whether or not $X$ is totally non-homologous to zero in $X_{\mathbb{S}^1}$ in the Borel fibration $X\hookrightarrow X_{\mathbb{S}^1} \longrightarrow B_{\mathbb{S}^1}$. We also give examples realizing the possible cases.

The volume flux group and nonpositive curvature

We show that every closed nonpositively curved manifold with non-trivial volume flux group has zero minimal volume, and admits a finite covering with circle actions whose orbits are homologically essential. This proves a conjecture of Kedra-Kotschick-Morita for this class of manifolds.

Null congruence spacetimes constructed from 3-dimensional Robertson–Walker spaces

Null congruence spacetimes are constructed from three-dimensional time-orientable Lorentzian manifolds by taking a particular ellipse in the lightcone above every point. Starting from a three-dimensional Robertson–Walker space, new null congruence spacetimes are obtained and several of their curvature properties are deduced. In particular, it is shown that the static Einstein universe is locally isometric to a certain null congruence spacetime. Furthermore, a method is given to construct trapped surfaces which admit an isometric spacelike circle action in null congruence spacetimes.

Quantum characteristic classes and the Hofer metric

Given a closed monotone symplectic manifold $M$, we define certain characteristic cohomology classes of the free loop space $L \text {Ham}(M, \omega)$ with values in $QH_* (M)$, and their $S^1$ equivariant version. These classes generalize the Seidel representation and satisfy versions of the axioms for Chern classes. In particular there is a Whitney sum formula, which gives rise to a graded ring homomorphism from the ring $H_{*} (L\text {Ham}(M, \omega), \mathbb{Q})$, with its Pontryagin product to $QH_{2n+*} (M)$ with its quantum product. As an application we prove an extension of a theorem of McDuff and Slimowitz on minimality in the Hofer metric of a semifree Hamiltonian circle action, to higher dimensional geometry of the loop space $L \text {Ham}(M, \omega)$.

On the formality and strong Lefschetz property of symplectic manifolds

AbstractThe main aim of this paper is to give some non-trivial results that exhibit the difference and similarity between Kähler and symplectic manifolds. To be precise, it is known that simply connected symplectic manifolds of dimension greater than 8, in general, do not satisfy the formality satisfied by all Kähler manifolds. In this paper we show that such non-formality of simply connected symplectic manifolds occurs even in dimension 8. We do this by some complicated but explicit construction of a simply connected non-formal symplectic manifold of dimension 8. In this construction we essentially use a variation of the construction of a simply connected symplectic manifold by Gompf. As a consequence, we can give infinitely many simply connected non-formal symplectic manifolds of any even dimension no less than 8.Secondly, we show that every compact symplectic manifold admitting a semi-free Hamiltonian circle action with only isolated fixed points must satisfy the strong Lefschetz property satisfied by all Kähler manifolds. This result shows that the strong Lefschetz property for the symplectic manifold admitting Hamiltonian circle actions is closely related to their fixed point set, as expected.

Semifree circle actions, Bott towers and quasitoric manifolds

A Bott tower is the total space of a tower of fibre bundles with base and fibres . Every Bott tower of height n is a smooth projective toric variety whose moment polytope is combinatorially equivalent to an n-cube. A circle action is semifree if it is free on the complement to the fixed points. We show that a quasitoric manifold over a combinatorial n-cube admitting a semifree action of a 1-dimensional subtorus with isolated fixed points is a Bott tower. Then we show that every Bott tower obtained in this way is topologically trivial, that is, homeomorphic to a product of 2-spheres. This extends a recent result of Il'inskiĭ, who showed that a smooth compact toric variety admitting a semifree action of a 1-dimensional subtorus with isolated fixed points is homeomorphic to a product of 2-spheres, and makes a further step towards our understanding of Hattori's problem of semifree circle actions. Finally, we show that if the cohomology ring of a quasitoric manifold is isomorphic to that of a product of 2-spheres, then the manifold is homeomorphic to this product. In the case of Bott towers the homeomorphism is actually a diffeomorphism. Bibliography: 18 titles.

Goldman flows on a nonorientable surface

Given an embedded cylinder in an arbitrary surface, we give a gauge theoretic definition of the associated Goldman flow, which is a circle action on a dense open subset of the moduli space of equivalence classes of flat SU(2)-connections over the surface. A cylinder in a compact nonorientable surface lifts to two cylinders in the orientable double cover, and the composite flow is the composition of one of the associated flows with the inverse flow of the other. Providing explicit descriptions, we relate the flow on the moduli space of the nonorientable surface with the composite flow on the moduli space of the double cover. We prove that the composite flow preserves a certain Lagrangian submanifold.

Rigidity and vanishing theorems for almost quaternionic manifolds

We prove the rigidity under (compatible) circle actions of several twisted Dirac operators on almost quaternionic manifolds, and the vanishing of the indices of some of them as a consequence.

The first homology of the group of equivariant diffeomorphisms and its applications

Let V be a representation space of a finite group G. We determine the group structure of the first homology of the equivariant diffeomorphism group of V. Then we can apply it to the calculation of the first homology of the corresponding automorphism groups of smooth orbifolds, compact Hausdorff foliations, codimension one or two compact foliations and a locally free circle action on the 3-sphere.

Smooth circle actions on highly symmetric manifolds

We construct for the first time smooth circle actions on highly symmetric manifolds such as disks, spheres, and Euclidean spaces which contain two points with the same isotropy subgroup whose representations determined on the tangent spaces at the two points are not isomorphic to each other. This allows us to answer negatively a question of Hsiang and Hsiang [Some Problems in Differentiable Transformation Groups, Springer, Berlin, Problem 16, p. 228, 1968].

SU(3)-manifolds of cohomogeneity one

In this article, we classify 7- and 8-dimensional manifolds M admitting an SU(3) action of cohomogeneity one such that (i) M is simply connected and the orbit space M/G is isomorphic to [0, 1], and (ii) \({M/G\cong S^{1}}\) and the principal orbits are simply connected. We discuss applications to the study of the group manifold SU(3) and to 8-dimensional quaternion-Kahler spaces, and links between dimension 7 and 8 given by circle actions.

Equivariant Cohomology of S1-Actions on 4-Manifolds

AbstractLet M be a symplectic 4-dimensional manifold equipped with a Hamiltonian circle action with isolated fixed points. We describe a method for computing its integral equivariant cohomology in terms of fixed point data. We give some examples of these computations.

Proper Holomorphic Maps from Domains in ℂ2 with Transverse Circle Action

The authors consider proper holomorphic mappings between smoothly bounded pseudoconvex regions in complex 2-space, where the domain is of finite type and admits a transverse circle action. The main result is that the closure of each irreducible component of the branch locus of such a map intersects the boundary of the domain in the union of finitely many orbits of the group action.