We revisit the construction of signature classes in –algebra –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 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).
Nigel Higson. Thomas Schick. Zhizhang Xie. "$C^*$–algebraic higher signatures and an invariance theorem in codimension two." Geom. Topol. 22 (6) 3671 - 3699, 2018. https://doi.org/10.2140/gt.2018.22.3671