An index theorem on open manifolds. I. II. (English) Zbl 0657.58041

Roe, John

J. Differ. Geom. 27, No. 1, 87-113, 115-136 (1988).

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

Keywords:

open manifolds; index theorem for Dirac-type operators; de Rham operator; Dirac operator; Dolbeault operator