Resolution except for minimal singularities II. The case of four variables

In this sequel to Bierstone and Milman [4], we find the smallest class of singularities in four variables with which we necessarily end up if we resolve singularities except for normal crossings. The main new feature is a characterization of singularities in four variables which occur as limits of triple normal crossings singularities, and which cannot be eliminated by a birational morphism that avoids blowing up normal crossings singularities. This result develops the philosophy of [4], that the desingularization invariant together with natural geometric information can be used to compute local normal forms of singularities.