Abstract
We formulate a more conceptual interpretation of the Cappell-Lee-Miller glueing/splitting theorem in terms of asymptotic maps and asymptotic exact sequences. Additionally, we show this gluing result is equivalent to a MayerVietoris-type long exact sequence. We also present applications to eigenvalue estimates, approximation of obstruction bundles and glueing of determinant line bundles arising frequently in gauge theory. All these results are true in a slightly more general context than in [6]. We work with operators which differ from translation invariant ones by exponentially decaying terms.
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