A Signature Formula for Manifolds with Corners of Codimension Two

We present a signature formula for compact 4k-manifolds with corners of codimension two which generalizes the formula of Atiyah, Patodi and Singer for manifolds with boundary. The formula expresses the signature as a sum of three terms, the usual Hirzebruch term given as the integral of an L-class, a second term consisting of the sum of the eta invariants of the induced signature operators on the boundary hypersurfaces with Atiyah-Patodi-Singer boundary condition (augmented by the natural Lagrangian subspace, in the corner null space, associated to the hypersurface) and a third ‘corner’ contribution which is the phase of the determinant of a matrix arising from the comparison of the Lagrangians from the different hypersurfaces meeting at the corners. To prove the formula, we pass to a complete metric, apply the AtiyahPatodi-Singer formula for the manifold with the corners ‘rounded’ and then apply the results of our previous work [11] describing the limiting behaviour of the eta invariant under analytic surgery in terms of the b-eta invariants of the final manifold(s) with boundary and the eta invariant of a reduced, one-dimensional, problem. The corner term is closely related to the signature defect discovered by Wall [25] in his formula for nonadditivity of the signature. We also discuss some product formulæ for the b-eta invariant.