Abstract. A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H -equivariantly projective (faithfully flat) over the coaction-invariant subalgebra P coH . We prove that principality is a piecewise property: given N comodule-algebra surjections P ! P i whose kernels intersect to zero, P is principal if and only if all P i 's are principal. Furthermore, assuming the principality of P , we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with P coH . Finally, assuming the above distributivity property, we obtain a flabby sheaf of principal comodule algebras over a certain space that is universal for all such N -families of surjections P ! P i and such that the comodule algebra of global sections is P . (2010). 58B32. Mathematics Subject Classification

Extending work of Budzyński and Kondracki, we investigate coverings and gluings of algebras and differential algebras. We describe in detail the gluing of two quantum discs along their classical subspace, giving a C * -algebra isomorphic to a certain Podleś sphere, as well as the gluing of U q 1/2 (sl 2 )-covariant differential calculi on the discs. MSC: 81R50, 46L87

We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C * -algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C * -algebras. Then we employ this result to show that the K-groups of our family of noncommutative 3-spheres coincide with their classical counterparts.

The irreducible * -representations of the polynomial algebra O(S 3 pq ) of the quantum 3-sphere introduced by Calow and Matthes are classified. The K-groups of its universal C * -algebra are shown to coincide with their classical counterparts. The U(1)-action on O(S 3 pq ) corresponding for p = 1 = q to the classical Hopf fibration is proven to be Galois (free). The thus obtained locally trivial Hopf-Galois extension is shown to be equivariantly projective (admitting a strong connection) and non-cleft. The latter is proven by determining an appropriate pairing of cyclic cohomology and K-theory. (2000): 16W30, 46L87. Mathematics Subject Classifications