We formulate a more conceptual interpretation of the Cappell-Lee-Miller glueing/splitting theorem using the new language of asymptotic maps and asymptotic exactness. Additionally, we present an asymptotic description of the Mayer-Vietoris sequence naturally associated to the Cech cohomology of the sheaf of local solutions of a Dirac type operator. We discuss applications to eigenvalue estimates, approximation of obstruction bundles and glueing of determinant line bundles frequently arising in gauge theoretic problems. The operators involved in all these results need not be translation invariant.

Latex 2.09 with macro sec.sty (included), 26 pages, 1 figure