Hypersurface Singularities with 2‐Dimensional Critical Locus

Consider an analytic germ f :( C m , 0)→( C , 0) ( m ⩾3) whose critical locus is a 2‐dimensional complete intersection with an isolated singularity (icis). We prove that the homotopy type of the Milnor fiber of f is a bouquet of spheres, provided that the extended codimension of the germ f is finite. This result generalizes the cases when the dimension of the critical locus is zero [ 8 ], respectively one [ 12 ]. Notice that if the critical locus is not an icis, then the Milnor fiber, in general, is not homotopically equivalent to a wedge of spheres. For example, the Milnor fiber of the germ f :( C 4 , 0)→( C , 0), defined by f ( x 1 , x 2 , x 3 , x 4 ) = x 1 x 2 x 3 x 4 has the homotopy type of S 1 × S 1 × S 1 . On the other hand, the finiteness of the extended codimension seems to be the right generalization of the isolated singularity condition; see for example [ 9 – 12 , 17 , 18 ]. In the last few years different types of ‘bouquet theorems’ have appeared. Some of them deal with germs f :( X , x )→( C , 0) where f defines an isolated singularity. In some cases, similarly to the Milnor case [ 8 ], F has the homotopy type of a bouquet of (dim X −1)‐spheres, for example when X is an icis [ 2 ], or X is a complete intersection [ 5 ]. Moreover, in [ 13 ] Siersma proved that F has a bouquet decomposition F ∼ F 0 ∨ S n ∨…∨ S n (where F 0 is the complex link of ( X , x )), provided that both ( X , x ) and f have an isolated singularity. Actually, Siersma conjectured and Tibăr proved [ 16 ] a more general bouquet theorem for the case when ( X , x ) is a stratified space and f defines an isolated singularity (in the sense of the stratified spaces). In this case F ∼∨ i F i , where the F i are repeated suspensions of complex links of strata of X . (If ( X , x ) has the ‘Milnor property’, then the result has been proved by Lê; for details see [ 6 ].) In our situation, the space‐germ ( X , x ) is smooth, but f has big singular locus. Surprisingly, for dim Sing f −1 (0)⩽2, the Milnor fiber is again a bouquet (actually, a bouquet of spheres, maybe of different dimensions). This result is in the spirit of Siersma's paper [ 12 ], where dim Sing f −1 (0) = 1. In that case, there is only a rather small topological obstruction for the Milnor fiber to be homotopically equivalent to a bouquet of spheres (as explained in Corollary 2.4). In the present paper, we attack the dim Sing f −1 (0) = 2 case. In our investigation some results of Zaharia are crucial [ 17 , 18 ].