Aleksy Tralle

On simply connected K-contact non-Sasakian manifolds

We solve the problem posed by Boyer and Galicki about the existence of simply connected K-contact manifolds with no Sasakian structure.We prove that such manifolds do exist using the method of fat bundles developed in the framework of symplectic and contact geometry by Sternberg, Weinstein and Lerman.

Koszul–Sullivan Models and the Cohomology of Certain Solvmanifolds

In this paper we develop a technique of working with graded differential algebra models of solvmanifolds, overcoming the main difficulty arising from the ‘non-nilpotency’ of the corresponding Mostow fibrations. A graded differential model for solvmanifolds of the form G/Γ with G=R⋊ϕN is presented (N is a nilpotent Lie group, G is a semi-direct product). As an application, we prove the Benson–Gordon conjecture in dimension four.

Homotopic Properties of Kähler Orbifolds

We prove the formality and the evenness of odd-degree Betti numbers for compact Kahler orbifolds, by adapting the classical proofs for Kahler manifolds. As a consequence, we obtain examples of symplectic orbifolds not admitting any Kahler orbifold structure. We also review the known examples of non-formal simply connected Sasakian manifolds, and produce an example of a non-formal quasi-regular Sasakian manifold with Betti numbers b 1 = 0 and b 2 > 1.

Proper SL(2,R)-actions on homogeneous spaces

We study the existence problem of proper actions of SL(2, ℝ) on homogeneous spaces G/H of reductive type. Based on Kobayashi’s properness criterion [T. Kobayashi, Proper action on a homogeneous space of reductive type, Math. Ann. 285 (1989) 249–263.], we show that G/H admits a proper SL(2, ℝ)-action via G if a maximally split abelian subspace of Lie H is included in the wall defined by a restricted root of Lie G. We also give a number of examples of such G/H.

New constructions of symplectically fat fiber bundles

This work is devoted to new constructions of symplectically fat fiber bundles. The latter are constructed in two ways: using the Kirwan map and expressing the fatness condition in terms of the isotropy representation related to the G-structure over some homogeneous spaces.

On symplectically fat twistor bundles

This paper deals with the question when are twistor bundles over homogeneous spaces symplectically fat? It shows that twistor bundles over even-dimensional Grassmannians of maximal rank have this property.

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.

Exotic smooth structures and symplectic forms on closed manifolds

In this paper we discuss relations between symplectic forms and smooth structures on closed manifolds. Our main motivation is the problem if there exist symplectic structures on exotic tori. This is a symplectic generalization of a problem posed by Benson and Gordon. We give a short proof of the (known) positive answer to the original question of Benson and Gordon that there are no Kähler structures on exotic tori. We survey also other related results which give an evidence for the conjecture that there are no symplectic structures on exotic tori.

On solvable compact Clifford-Klein forms

In the present article we show that there is a large class of homogeneous spaces G/H of reductive type which cannot be a local model for any compact manifold M with solvable fundamental group. Another way of expressing this is: we prove that under certain assumptions, a reductive homogeneous space G/H does not admit a solvable compact Clifford-Klein form. This generalizes the well known non-existence theorem of Benoist for nilpotent Clifford-Klein forms. This generalization works for a particular class of homogeneous spaces determined by a ?very regular? embeddings of H into G.