  We give a proof of the cobordism invariance of the index of elliptic
pseudodifferential operators on sigma-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 K^1(TX),
then its index is zero. This condition is incorporated into the definition of a
cobordism group for non-compact 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 R^n. 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.
