Fibre singularities of functions on principal S1-bundles

Let π : E → B denote a smooth principal S1-bundle with total space E and base space B of dimension m. In C.Z.W. Hassel Sweatman, 1993, it was noted that, as a consequence of the general results of K. Igusa, 1990, the set of smooth real-valued functions on E whose singularities on fibres are only of types A1,…,Am + 1 is open and dense in the Whitney C∞ topology and further results on the existence of good functions were proved there. In this article, we prove the existence of a smooth real-valued function on E with no fibre singularities of type Ak for k > [m/2] + 1.