Singularities with critical locus a 1-dimensional complete intersection and transversal type A₁

In this paper we study germs of holomorphic functions f: (Cⁿ⁺¹,0)→C with the following two properties: (i) the critical set Σ of f is a 1-dimensional isolated complete intersection singularity (icis); (ii) the transversal singularity of f in points of Σ-{0} is of type A₁. We first compute the homology of the Milnor fibre F of f in terms of numbers of special points in certain deformations. Next we show that the homotopy type of the Milnor fibre F of f is a bouquet of spheres. There are two cases: (a) general case Sⁿ v⋯v Sⁿ (b) special case S^n-1 v sⁿ v⋯v Sⁿ.