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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.

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