Cutting and Pasting of Elliptic Operators

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Abstract
IfA : C∞E → C∞Fis an elliptic operator between sections of vector bundlesE, Fover a closed smoothn‐manifoldX, Ya smooth (n– 1)‐submanifold ofXwith trivial normal bundle, and Ψ = (ΨE, ΨF) a pair of automorphisms ofE|YandF|Yinducing a diffeomorphismfofYand commuting with the principal symbol σAofAoverY, then an elliptic operatorAΨis (uniquely up to operators of lower order) defined between sections of vector bundlesEΨ,FΨover a closed manifoldXf, all obtained by cutting and pasting the respective objects alongY. The difference indexAΨ– indexAis investigated and the relations with additivity properties of topological invariants and with classical transmission problems are explained.

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