Exotic Cohomology and Index Theory

The main result of this paper is a version of the Atiyah-Singer index theorem for Dirac-type operators on (noncompact) complete Riemannian manifolds. The statement of the theorem involves a novel "cohomology" theory for such manifolds. This theory, called exotic cohomology, depends on the structure at infinity of a space; more precisely, it depends on the way that large bounded sets fit together. For each cohomology class in this theory, we define a "higher index" of a Dirac-type operator, enjoying the stability and vanishing properties of the usual Atiyah-Singer index; these higher indices are analogous to the Novikov higher signatures. Our main theorem will compute these higher indices in terms of standard topological invariants. Applications of this index theorem include a different approach to some of the results of Gromov and Lawson [10] on topological obstructions to positive scalar curvature. The concept of index that we will use involves the A^-theory functors K0 and Kx for operator algebras [3, 14]. Suppose that B is an ideal in a unital algebra C, and let T e C be invertible modulo B. (In the classical Atiyah-Singer index theorem, one takes C to be the bounded operators on the L space of some compact manifold, B the compact operators, and T an elliptic pseudodifferential operator of order zero.) Then T has an "index" in the ^-theory group KQ(B) (in the classical case this is just Z , and one recovers the usual Fredholm index). Now let M be a complete Riemannian manifold, possibly noncompact. In [18] I introduced an algebra %?(M) which is defined as follows: Sf(M) consists of all bounded operators A on L(M) that have a kernel representation