Cobordism Invariance of the Index

We give a proof of the cobordism invariance of the index of elliptic pseudodifferential operators on σ-compact manifolds, where, in the non-compact case, the operators are assumed to be multiplication outside a compact set. We show that, if the principal symbol class of such an elliptic operator on the boundary of a manifold X has a suitable extension to K1(TX), then its index is zero. This condition is incorporated into the definition of a cobordism group for noncompact manifolds, called here “cobordism of symbols”. Our proof is topological, in that we use properties of the push-forward map in K-theory defined by Atiyah and Singer, to reduce it to Rn. In particular, we generalize the invariance of the index with respect to the push-forward map to the non-compact case, and obtain an extension of the K-theoretical index formula of Atiyah and Singer to operators that are multiplication outside a compact set. Our results hold also for G-equivariant operators, where G is a compact Lie group.