Families Index for Pseudodifferential Operators on Manifolds with Boundary

Details Families Index for Pseudodifferential Operators on Manifolds with Boundary Families Index for Pseudodifferential Operators on Manifolds with Boundary

2003-04-20

arxiv.org/abs/math/0304280v3

Mathematics

Differential Geometry

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

Scientific paper

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 $-\infty$ 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.

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