#### Room 217

#### Homology Smale-Barden manifolds with K-contact and Sasakian structures

#### Aleksy Tralle

##### University of Warmia and Mazury, Olsztyn

*The talk is devoted to a problem of finding 1-connected manifolds*

with K-contact structures which do not admit Sasakian structures in

dimension 5. This question (posed by Boyer and Galicki) should be considered

as an odd-dimensional version of the topic "symplectic versus Kaehler". The

answer to the question is positive in dimensions greater or equal to 7 (the

works of the author together with Biswas, Fernandez, Hajduk and Munoz). In

dimension 5 the problem is much more complicated, since the methods of

homotopy theory do not work anymore. We prove a weaker version of the result

constructing closed manifold with the vanishing first homology group, which

admits K-contact structures but no Sasakian structures. Our approach uses

methods of symplectic topology and algebraic geometry. Joint work with

Vicente Munoz and Juan Angel Rojo.

Done