Families Index for Pseudodifferential Operators on Manifolds with Boundary

2003/04/20 by Richard Melrose, Melrose, Richard, Frédéric Rochon +2
Mathematics · #Advanced Operator Algebra Research #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics #math.AP #math.DG #msc:58J20 #msc:58J40

paper · pdf · doi:10.48550/arxiv.math/0304280

21 pages; corrected typos, changed the abstract, added a paragraph in the introduction

arxiv created 2003/11/20 · arxiv updated 2009/11/30

Abstract

An analytic index is defined for a family of cusp pseudodifferential operators, P_b, on a fibration with fibres which are compact manifolds with boundaries, provided the family is elliptic and has invertible indicial family at the boundary. In fact there is always a perturbation Q_b by a family of cusp operators of order -∞ such that each P_b+Q_b is invertible. Thus any elliptic family of symbols has a realization as an invertible family of cusp pseudodifferential operators, which is a form of the cobordism invariance of the index. A crucial role is played by the weak contractibility of the group of cusp smoothing operators on a compact manifold with non-trivial boundary and the associated exact sequence of classifying spaces of odd and even K-theory.

Related

• Index in K-theory for families of fibred cusp operators
• Boundaries, eta invariant and the determinant bundle
• Homology of pseudodifferential operators I. Manifolds with boundary
• Pseudodifferential operators on manifolds with foliated boundaries
• Cobordism invariance of the family index