We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of R n . They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features.The new examples include the instanton algebra and the NC-4-spheres S 4 θ . We construct the noncommutative algebras A = C ∞ (S 4 θ ) of functions on NCspheres as solutions to the vanishing, ch j (e) = 0 , j < 2, of the Chern character in the cyclic homology of A of an idempotent e ∈ M 4 (A) , e 2 = e , e = e * . We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmanian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere intimately related to quantum group deformations SU q (2) of SU(2) but for unusual values (complex values of modulus one) of the parameter q of q-analogues, q = exp(2πiθ).We then construct the noncommutative geometry of S 4 θ as given by a spectral triple (A, H, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric g µν on S 4 whose volume form √ g d 4 x is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation,where is the projection on the commutant of 4 × 4 matrices. We shall show how to construct the Dirac operator D on the noncommutative 4-spheres S 4 θ so that the previous equation continues to hold without any change. Finally, we show that any compact Riemannian spin manifold whose isometry group has rank r ≥ 2 admits isospectral deformations to noncommutative geometries.

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We construct noncommutative principal fibrations S 7 θ → S 4 θ which are deformations of the classical SU (2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU (2) as modules of coequivariant maps and construct corresponding projections. The index of Dirac operators with coefficients in the associated bundles is computed with the Connes-Moscovici local index formula. The algebra inclusion A(S 4 θ ) ֒→ A(S 7 θ ) is an example of a not trivial quantum principal bundle.