The Dirac operator on nilmanifolds and collapsing circle bundles (Q1389887)

The spectrum of the Dirac operator on 3-dimensional Heisenberg manifolds is computed and the behaviour under collapse to the 2-torus is studied. The Dirac eigenvalues on complex projective space including the multiplicities are determined by using the Hopf fibration and spin structures. It is shown that there are 1-parameter families of Riemannian nilmanifolds such that the Laplacian on functions and the Dirac operator for certain spin structures have a constant spectrum while the Laplacian on 1-forms and the Dirac operator for the other spin structures have a nonconstant spectrum.