Contact degree and the index of Fourier integral operators

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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Published 1 January 1998