Large Fredholm triples

We give several equivalent condition for Busby extensions of a given algebra to be absorbing, considerably improving our earlier results [G.A. Elliott, D. Kucerovsky, An abstract Brown-Douglas-Fillmore absorption theorem, Pacific J. Math. 198 (2001) 385-409], and establish sufficient conditions for Fredholm triples to be absorbing in a suitable sense. As an application of one of our criteria, we prove a multivariable Brown-Douglas-Fillmore type theorem.