The authors obtain a signature formula for compact manifolds (of dimension \(4k)\) with corners of codimension 2 by rounding the corners and using the Atiyah-Patodi-Singer index formula for manifolds with boundary. The approximation procedure uses the \(b\)-calculus that has been developed by the third named author; cf. R. Melrose [‘The Atiyah-Patodi-Singer index theorem’, A. K. Peters, Wellesly (1993; Zbl 0796.58050)].
As a corollary of the signature formula they deduce Wall’s formula of the nonadditivity of signatures, thereby giving a more detailed account of its informal treatment by U. Bunke [J. Differ. Geom. 41, 397-448 (1995; Zbl 0821.58037)]. Finally, product formulae for the \(b\)-eta invariant are given which, however, have also been obtained independently by W. Müller [J. Differ. Geom. 44, 97-177 (1996)].