  We revisit the construction of signature classes in C*-algebra K-theory, and
develop a variation that allows us to prove equality of signature classes in
some situations involving homotopy equivalences of noncompact manifolds that
are only defined outside of a compact set. As an application, we prove a
counterpart for signature classes of a codimension two vanishing theorem for
the index of the Dirac operator on spin manifolds (the latter is due to Hanke,
Pape and Schick, and is a development of well-known work of Gromov and Lawson).
