  Relative index theorems, which deal with what happens with the index of
elliptic operators when cutting and pasting, are abundant in the literature. It
is desirable to obtain similar theorems for other stable homotopy invariants,
not the index alone. In the spirit of noncommutative geometry, we prove a
full-fledged "relative index" type theorem that compares certain elements of
the Kasparov KK-group KK(A,B).
