**On families of self-adjoint operators**

**R.J. Lawrence**

**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.*

ruthel@ma.huji.ac.il