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Contact degree and the index of Fourier integral operators. (English) Zbl 0929.58012
Let $$X$$ be a compact contact manifold, and $$L\subset T^*X$$ an oriented contact line bundle on $$X$$. Let $${\mathcal M}(X)$$ be the group of components of the space of contact diffeomorphisms. Using the notion of quantization of the compact structure, the authors construct a contact degree, that is, a homomorphism $$c- \deg: {\mathcal M}(X) \to\mathbb{Z}$$.
The contact degree is reduced to the index of a Dirac operator. Let $$\phi$$ be an oriented contact diffeomorphism, and $$Z_\phi$$ the mapping torus of $$\phi$$, i.e., $$X\times[-1,1]$$ with the ends identified by $$\phi$$. The contact structure on $$X$$ gives $$Z_\phi$$ a natural Spin-$$\mathbb{C}$$ structure. If $$\partial_\phi$$ is the associated Dirac operator then $c-\deg (\phi)= \text{ind} (\partial_\phi).$ The last formula follows from the identification of $$c-\deg (\phi)$$ with the spectral flow of the curve of Dirac operators on the contact manifold associated with an isotropy from any one partial Hermitian structure to its $$\phi$$-conjugate.
The above result leads to a geometric formula for the index of elliptic Fourier integral operators, which gives the answer to the long-open question of Weinstein for operators acting on a fixed manifold. More precisely, the following theorem is proved.
Theorem 3. If $$Y$$ is a compact manifold and $$X=S^*Y$$ is its cosphere bundle, then for any (oriented) contact diffeomorphism, $$\phi$$, of $$X$$, i.e., a canonical diffeomorphism of $$T^*X\setminus 0$$, $\text{ind}(F_\phi)=c-\deg(\phi),$ where $$F_\phi$$ is a Fourier integral operator associated to $$\phi$$ with elliptic symbol corresponding to the positive trivialization of the Maslov bundle.

##### MSC:
 58J40 Pseudodifferential and Fourier integral operators on manifolds 58J20 Index theory and related fixed-point theorems on manifolds
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