  The Atiyah-Singer index theorem is investigated on various compact manifolds
which admit finite matrix approximations (``fuzzy spaces'') with a view to
applications in a modified Kaluza-Klein type approach in which the internal
space consists of a finite number of points. Motivated by the chiral nature of
the standard model spectrum we investigate manifolds that do not admit spinors
but do admit $Spin^c$ structures. It is shown that, by twisting with
appropriate bundles, one generation of the electroweak sector of the standard
model, including a right-handed neutrino, can be obtained in this way from the
complex projective space $\CP^2$. The unitary Grassmannian $U(5)/(U(3)\times
U(2))$ yields a spectrum that contains the correct charges for the Fermions of
the standard model, with varying multiplicities for the different particle
states.
