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An index theorem on open manifolds. I. II. (English) Zbl 0657.58041
In the first part of the work the author proves the following abstract index theorem for Dirac-type operators on certain noncompact manifolds: Let M be a Riemannian manifold and S a graded Clifford bundle on M, both with bounded geometry. Let D be the Dirac operator on S. Let m be a fundamental class for M associated to a regular exhaustion, and let $$\tau$$ be the corresponding trace on the algebra of uniform operators of order -$$\infty$$. Then D is abstractly elliptic on S, and $$\dim_{\tau}(Ind D)=<I(D),m>,$$ where I(D) is the usual integrand in the Atiyah-Singer formula for D. In the second part of the work he discusses general techniques to de Rham operator, Dirac operator and Dolbeault operator. He also discusses the relationship of his result with $$L^ 2$$ index theorems of M. F. Atiyah and A. Connes.
Reviewer: S.K.Chatterjea

##### MSC:
 58J20 Index theory and related fixed-point theorems on manifolds 58A99 General theory of differentiable manifolds
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