A cohomological formula for the Atiyah-Patodi-Singer index on manifolds   with boundary
  We give a cohomological formula for the index of a fully elliptic pseudodifferential operator on a manifold with boundary. As in the classic case of Atiyah-Singer, we use an embedding into an euclidean space to express the index as the integral of a cohomology class depending in this case on a noncommutative symbol, the integral being over a $C^\infty$-manifold called the singular normal bundle associated to the embedding. The formula is based on a K-theoretical Atiyah-Patodi-Singer theorem for manifolds with boundary that is drawn from Connes' tangent groupoid approach. 
Paulo Carrillo Rouse (IMT), Jean-Marie Lescure, Bertrand Monthubert   (IMT)
math.OA math.DG math.KT