Noncommutative manifolds, the instanton algebra and isospectral deformations

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-theta(4). We construct the noncommutative algebras A = C-infinity(S-theta(4)) of functions on NC-spheres as solutions to the vanishing, ch(j) (e) = 0, j < 2, of the Chem character in the cyclic homology of A of an idempotent e is an element of M-4(A), e(2) = e, e = e*. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmannian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere S-theta(3) distinct from quantum group deformations SUq (2) of SU(2). We then construct the noncommutative geometry of S-theta(4) 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(muv) on S-4 whose volume form rootg 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, [(e - 1/2) [D, e](4)] = gamma (5), where [] is the projection on the commutant of 4 x 4 matrices. We shall show how to construct the Dirac operator D on the noncommutative 4-spheres S-theta(4) 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 greater than or equal to 2 admits isospectral deformations to noncommutative geometries.