Mathematical Research Letters

Volume 5, Issue 3, May 1998 pp. 363-381.

Contact degree and the index of Fourier integral operators

Authors: Charles Epstein and Richard Melrose
Author institution: University of Pennsylvania, and Massachusetts Institute of Technology

Summary: An elliptic Fourier integral operator of order $0,$ associated to a homogeneous canonical diffeomorphism, on a compact manifold is Fredholm on $L^2.$ The index may be expressed as the sum of a term, which we call the contact degree, associated to the canonical diffeomorphism and a term, computable by the Atiyah-Singer theorem, associated to the symbol. The contact degree is shown to be defined for any oriented-contact diffeomorphism of a contact manifold and is then reduced to the index of a Dirac operator on the mapping torus, also computable by the theorem of Atiyah and Singer. In this case, of an operator on a fixed manifold, these results answer a question of Weinstein in a manner consistent with a more general conjecture of Atiyah.

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