A SIGNATURE FORMULA FOR MANIFOLDS WITH CORNERS OF CODIMENSION-2

We present a signature formula for compact 4k-manifolds with corners o f codimension two which generalizes the formula of Atiyah et al. for m anifolds with boundary. The formula expresses the signature as a sum o f 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 t he induced signature operators on the boundary hypersurfaces with Atiy ah-Patodi-Singer boundary condition (augmented by the natural Lagrangi an subspace, in the corner null space, associated to the hypersurface) and a third ''corner'' contribution which is the phase of the determi nant of a matrix arising from the comparison of the Lagrangians from t he different hypersurfaces meeting at the corners. To prove the formul a, we pass to a complete metric, apply the Atiyah-Patodi-Singer formul a for the manifold with the corners ''rounded'' and then apply the res ults of our previous work [11] describing the limiting behaviour of th e eta invariant under analytic surgery in terms of the b-eta invariant s of the final manifold(s) with boundary and the eta invariant of a re duced, one-dimensional, problem. The corner term is closely related to the signature defect discovered by Wall [25] in his formula for nonad ditivity of the signature. We also discuss some product formulae for t he b-eta invariant. (C) 1997 Elsevier Science Ltd.