On families of self-adjoint operators
Abstract: Suppose M is a compact oriented smooth manifold fibred over S^1, with fibre N; and let M' be its infinite cyclic covering. Let D_\theta denote the Dirac-type operator on \Omega^even(M,\chi_\theta), the space of forms on M with local coefficient system twisted by e^i\theta. Let T denote the induced monodromy action on the homology H(N) of N. Then
(i) there exists a mapping H(N)->Coker D which is an isomorphism when restricted to the parts of H(N) corresponding to eigenvalues of T on the unit circle;
(ii) Jordan blocks of T on H(N) correspond to eigenvalues of D_\theta crossing zerp, and the crossing number at \theta=\theta_0 of an eigenvalue is the signature of the Jordan block corresponding to the eigenvalue e^i\theta_0 of T;
(iii) those Jordan blocks corresponding to eigenvalues of T not on the unit circle, come in pairs, and their signatures cancel out, so not contributing to the total.
Based on unfinished notes by G. Lusztig (1971)
Keywords: twisted Dirac operaors, local index theorem
Length: 80 pages
Reference: Dissertation accepted by Oxford University in support of application to transfer to Advanced Student status. (1987)
Last updated on September 4th, 1996.