Kostant Prequantization of Symplectic Manifolds with Contact Singularities

The relationship between the Bohr-Sommerfeld quantization condition and the integrality of the symplectic structure in Planck constant units is considered. Constructions of spherical and toric Θ-handles are proposed which allow one to obtain symplectic manifolds with contact singularities, preserve Kostant-Souriau prequantization, and expect interesting topological applications. In particular, the toric Θ-handle glues Liouville foliations, while the spherical handle generates (pre)quantized connected sums of symplectic manifolds. In this way, nonorientable manifolds may arise.