Consider an analytic germ f:(Cm, 0)→
(C, 0) (m[ges ]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:(C4, 0)→(C, 0), defined by
f(x1, x2, x3, x4) =
x1x2x3x4
has the homotopy type of
S1×S1×S1. 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∼F0∨Sn∨…∨Sn
(where F0 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∼∨iFi,
where the Fi 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)[les ]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].