On Carvalho's $K$-theoretic formulation of the cobordism invariance of   the index
  We give an analytic proof of the fact that the index of an elliptic operator on the boundary of a compact manifold vanishes when the principal symbol comes from the restriction of a $K$-theory class from the interior. The proof uses noncommutative residues inside the calculus of cusp pseudodifferential operators of Melrose. 
Sergiu Moroianu