We give new examples of noncommutative manifolds that are less standard tha
n the NC-torus or Moyal deformations of R-n. They arise naturally from basi
c considerations of noncommutative differential topology and have non-trivi
al 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 noncommut
ative space obtained from this equation as a noncommutative Grassmannian as
well as the corresponding notion of admissible morphisms. This space Gr co
ntains the suspension of a NC-3-sphere S-theta(3) distinct from quantum gro
up deformations SUq (2) of SU(2).
We then construct the noncommutative geometry of S-theta(4) as given by a s
pectral 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 metri
c, the corresponding Dirac operator gives a solution to the following quart
ic 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 g
roup has rank r greater than or equal to 2 admits isospectral deformations
to noncommutative geometries.