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The behaviour of a holomorphic map germ at a critical point has always been an important part of the singularity theory. It is generally known (cf. [5]) that we can associate an integer invariant—called the multiplicity—to each isolated critical point of a holomorphic function of many variables. Several years later it was noticed that similar invariants exist for function germs defined on isolated hypersurface singularities (see [1]). The present paper aims to show a simple approach to critical points of maps defined on the Ak-type singular hypersurfaces. After some changes it can probably be adopted to other isolated hypersurface singularities. Acknowledgements. The author wishes to thank the Mathematical Research Institute in the Netherlands and the University of Utrecht for the invitation and hospitality, and Professor J. H. M.Steenbrink, who supervised this research project, for suggesting the problem and many stimulating discussions. 1. Let On denote the ring of germs of holomorphic functions f : (C, 0)→ (C, 0). Definition. The multiplicity of the critical point of f at zero is μ(f) := dimCOn/Jf , where Jf = 〈∂f/∂z1, . . . , ∂f/∂zn〉On is the Jacobian ideal of f . The number μ(f) is also called the Milnor number of f , because it was first introduced by J. Milnor in 1968 [4]. Proposition. Let f ∈ On have an isolated critical point at zero and let f(0) = 0. Then μ(f) is finite and the preimage of each sufficiently small non-zero complex number intersects a small open disk in a smooth manifold , which is homotopy equivalent to a bouquet of μ(f) (n− 1)-dimensional spheres. (∃ε, δ ∈ R+)(∀t ∈ C) 0 < |t| < ε⇒ f−1(t) ∩Dδ ∼ ∨

The Milnor number of functions on singular hypersurfaces