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 $\Theta$-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 $\Theta$-handle glues Liouville foliations, while the spherical handle generates (pre)quantized connected sums of symplectic manifolds. In this way, nonorientable manifolds may arise.