On codimension two and one splitting

According to Cappell and Shaneson, very often the secondary surgery obstruction O2 to codimension two splitting is zero, but examples with O2 ≠ 0 exist. By considering homotopy equivalences with codomain the total space of a two disk bundle, we produce many examples with O2 ≠ 0and other examples with O2 = 0. Our approach is: “compare with codimension one splitting”.