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]. Dedicated to Prof. Yasuhiko Kitada on the occasion of his 60th birthday. Krzysztof Pawałowski was supported in part by the KBN Research Grant N 201 008 31/0524.     

Non-symplectic smooth circle actions on symplectic manifolds

We construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets or cyclic isotropy sets. All such actions are not compatible with any symplectic form on the manifold in question. In order to cover the case of non-symplectic fixed point sets, we use two smooth 4-manifolds (one symplectic and one non-symplectic) which become diffeomorphic after taking the products with the 2-sphere. The second type of actions is obtained by constructing smooth circle actions on spheres with non-symplectic cyclic isotropy sets, which (by the equivariant connected sum construction) we carry over from the spheres on products of 2-spheres. Moreover, by using the mapping torus construction, we show that periodic diffeomorphisms (isotopic to symplectomorphisms) of symplectic manifolds can provide examples of smooth fixed point free circle actions on symplectic manifolds with non-symplectic cyclic isotropy sets.     

Geometric quantization on presymplectic manifolds

In this paper, we discuss the possibilities of adapting geometric quantization to presymplectic manifolds, i.e., differentiable manifoldsM ^2n+k (k>0) endowed with a closed 2-form ω of rank2n. We show that such an adaptation is possible in various manners, and that, as a general idea, it reduces the quantization onM to quantization on the symplectic quotientM/V, whereV is the foliation defined by the annihilator of ω.     

Deformation quantization of framed presymplectic manifolds

Theoretical and Mathematical Physics - We consider the problem of deformation quantization of presymplectic manifolds in the framework of the Fedosov method. A class of special presymplectic...     

Semi-free Hamiltonian circle actions on 6-dimensional symplectic manifolds

Assume is a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action, such that the fixed point set consists of isolated points or compact orientable surfaces. We restrict attention to the case . We give a complete list of the possible manifolds, and determine their equivariant cohomology rings and equivariant Chern classes. Some of these manifolds are classified up to diffeomorphism. We also show the existence for a few cases.

     

Free circle actions with contractible orbits on symplectic manifolds

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 the constructions of FGM.     

On the construction of certain 6-dimensional symplectic manifolds with Hamiltonian circle actions

Let be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian action such that the fixed point set consists of isolated points or surfaces. Assume dim . In an earlier paper, we defined a certain invariant of such spaces which consists of fixed point data and twist type, and we divided the possible values of these invariants into six ``types'. In this paper, we construct such manifolds with these ``types'. As a consequence, we have a precise list of the values of these invariants.

     

Polar manifolds and actions

A group action is called polar if there exists an immersed submanifold (a section) which intersects all orbits orthogonally. We show how to construct a manifold admitting a polar group action by prescribing its isotropy groups along a fundamental domain in the section. This generalizes a classical construction for cohomogeneity-one manifolds.We give many examples showing the richness of this class of group actions and relate the topology of the section to the topology of the manifold.     

Proper actions on cohomology manifolds

Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most , then the group is effectively a Lie group.

     

Intersection cohomology of the circle actions

A classical result says that a free action of the circle S¹ on a topological space X is geometrically classified by the orbit space B and by a cohomological class e∈H²(B,Z), the Euler class. When the action is not free we have a difficult open question:

(Π)

“Is the space X determined by the orbit space B and the Euler class?”

The main result of this work is a step towards the understanding of the above question in the category of unfolded pseudomanifolds. We prove that the orbit space B and the Euler class determine:

•

the intersection cohomology of X,

•

the real homotopy type of X.

     

On the action of a circle on manifolds

The bordism groups of actions of a circle are calculated in the complex case and (in particular) in the oriented case; it is proved that certain systems of elements in them are linearly independent.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 511–518, November, 1971.In conclusion the author expresses his gratitude to S. P. Novikov for his assistance.     

Locally free actions on Lorentz manifolds

If a topological group G acts on a topological space M, then we say that the action is orbit nonproper provided that, for some ,the orbit map is nonproper. In this paper we characterize the connected, simply connected Lie groups that admit a locally free, orbit nonproper action by isometries of a connected Lorentz manifold. We also consider a number of variants on this question. Submitted: November 1998, Revised version: April 1999, Final version: April 2000.     

Properly discontinuous actions on Hilbert manifolds

In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic sectional curvature and infinite fundamental group in contrastwith the finite dimensional case and we classify abelian groups acting linearly, isometrically and properly discontinuously on Stiefel manifolds. Finally, we classify homogeneous Hilbert manifolds with constant sectional curvature.     

Orbit nonproper actions on Lorentz manifolds

If a topological group G acts on a topological space X, then we say that the action is orbit nonproper provided that, for some x ? X x \in X , the orbit map g ? gx : G ? X g \mapsto gx : G \to X is nonproper. We consider the problem of classifying the connected, simply connected real Lie groups G such that G admits a locally faithful, orbit nonproper action on a connected Lorentz manifold. In this paper, we describe three collections of groups such that G admits such an action iff G is in one of the three collections. In an earlier paper, we effectively described the first collection. In yet another paper, we describe effectively those groups in the second collection which are not contained in the union of the first and third. Finally, in another paper, we describe effectively the third collection.