ON THE MILNOR FIBER OF NON-ISOLATED SINGULARITIES

A. HORVÁTH AND A. NÉMETHI

Abstract. We prove that the homotopy type of the Milnor fiber F of a hypersurface singularity f : (ℂ^{n+1}, 0) → (ℂ, 0) (n ≥ 3) with 1-dimensional singular locus is completely determined by its homology H^{∗}(F, ℤ), and, in fact, it has a canonical bouquet decomposition.

1. Introduction

The starting point of the series of ‘bouquet theorems’ is the result of Milnor [6]: the Milnor fiber F of an isolated hypersurface singularity f : (ℂ^{n+1}, 0) → (ℂ, 0) has the homotopy type of a bouquet ⋁ S^{n} of n-spheres. In particular, the homology of F determines completely the homotopy type of F: the number of spheres in the bouquet is exactly the n^{th} Betti number of F (the so called Milnor number).

There are many generalisations of this theorem; they can be grouped basically in two categories.

In the first case one considers germs f : (X, x) → (ℂ, 0) (dim X = n + 1) with X rather general, but with f ‘not very complicated’. For example, Siersma in [10] proved that if both (X, x) and f have an isolated singularity, then the Milnor fiber F of f homotopically has a bouquet decomposition F ∼ F_{0} ∨S^{n} ∨⋯∨S^{n}, where F_{0} is the complex link of X. More complicated bouquet decompositions were determined by Lê in [4] and [5]. Siersma and Lê conjectured a very general result, which was proved by Tibăr in [14]. This says the following: if (X, x) is a stratified space and f defines an isolated singularity (in the sense of stratified spaces) then F ∼ ⋁_{i=1}^{n} F_{i} where each F_{i} is a repeated suspension of the complex link of some stratum of X.

The second category of generalisations deals with hypersurface singularities, like the original result of Milnor, but now the germs may have a large singular locus. In fact, in this case it is also necessary to introduce some restriction for the germ; the right notion which generalises the isolated hypersurface case is the category of germs with fixed singular locus Σ and with finite (extended) codimension c_{e,Σ}(f) with respect to the ideal of Σ (for the details see e.g. [8] and [9]).

Both authors were supported by the Marie Curie Fellowship for the Transfer of Knowledge; the second author was partially supported by OTKA 42769.

Keywords: hypersurface singularities, Milnor fiber AMS Subject Classification: 14B05, 14H20, 14J17, 32S**, 58K**.

In the case when Σ is an isolated complete intersection singularity (ICIS) of dim Σ = 1 and c_{e,Σ}(f) < ∞, Siersma proved that the Milnor fiber F has the homotopy type of a bouquet of spheres (possibly of different dimensions) [11]. In the case dim Σ = 2, (and Σ an ICIS) Zaharia proved the homological analog of the bouquet theorem, cf.

[15] and [16]. The homotopy classification of this case was finished in [7], where F ∼ ⋁_{i}(⋁_{b_{i}} S^{i}) is proved (where b_{i} are the Betti numbers of F).

In the present paper we will consider hypersurface germs f : (ℂ^{n+1}, 0) → (ℂ, 0) with dim Singf^{−1}(0) = 1, but with absolutely no other restriction regarding the singular locus Singf^{−1}(0), or the germ f. We will prove that, in this case too, the homotopy type of F is completely determined by H̃_{∗}(F, ℤ), provided that n ≥ 3.

In order to formulate the result, we recall the following general facts. Since F is an n-dimensional Stein space, it has the homotopy type of a finite n-dimensional CW -complex. In particular, H_{q}(F, ℤ) = 0 for q > n and H_{n}(F, ℤ) is a free ℤ-module. On the other hand, by a theorem of M. Kato and Y. Matsumoto [3], F is (n − 2)-connected. Therefore H̃_{∗}(F, ℤ) is given by the Betti numbers b_{n}, b_{n−1} and the torsion part T of H_{n−1}(F, ℤ).

Recall that T can be written in a unique way as T = ⊕ℤ_{p^{α}}, a direct sum of cyclic p-groups. We will show that for each cyclic group ℤ_{d} and integer l ≥ 2, there exists a space T (l, d) – unique up to homotopy equivalence, – such that H̃_{q}(T (l, d), ℤ) = 0 for q ≠ l, and H_{l}(T (l, d), ℤ) = ℤ_{d}. Then our bouquet theorem is the following:

Theorem 1.1. Consider a germ of an analytic function f : (ℂ^{n+1}, 0) → (ℂ, 0) (n ≥ 3) with dim Singf^{−1}(0) = 1. Write the torsion group T of H_{n−1}(F, ℤ) as a direct sum ⊕_{i}G_{i} of cyclic p-groups. Then one has the following homotopy equivalence

F ∼ (⋁_{b_{n}} S^{n}) ∨ ( ⋁_{b_{n−1}} S^{n−1}) ∨ (⋁_{i} T (n − 1, |G_{i}|)) ,

where |G_{i}| is the order of the group G_{i}.

Remark 1.2. In fact, for any direct sum decomposition ⊕_{i}G_{i} of T into cyclic groups G_{i} the above formula is correct (i.e. it is not necessary to have p-groups only). This follows from the homotopy equivalence T (n − 1, d) ∨ T (n − 1, e) ∼ T (n − 1, de) valid for (d, e) = 1. If we consider the p-group decomposition of T as in the theorem above, then the bouquet decomposition of F is unique.

Remark 1.3. (1) In some cases the torsion group really appears. E.g, if

f(x, y, z, w_{1}, ⋯, w_{n−2}) = x^{2}y^{2} + y^{2}z^{2} + z^{2}x^{2} − 2xyz(x + y + z) + ∑ w_{i}^{2}

then its Milnor fiber F satisfies H_{n−1}(F, ℤ) = ℤ_{3}, cf. [12], Example 7.4 or [1], Example 4.1.

(2) On the other hand, in some cases the whole group H_{n−1}(F, ℤ) can vanish (even if dim Singf^{−1}(0) = 1). E.g., for the Whitney umbrella

f(x, y, z, w_{1}, ⋯, w_{n−2}) = x^{2} − zy^{2} + ∑ w_{i}^{2},

one has F ∼ S^{n}.

The proof of Theorem 1.1 is purely topological, and it is presented in the next section.

Acknowledgement. The first author is grateful for the warm hospitality of the Alfréd Rényi Institute of Mathematics (Budapest, Hungary) where he spent two fruitful months in an ideal mathematical environment.

2. The space T (l, m)

For any pair (l, d), (l ≥ 1, d ≥ 2), we fix a map f_{l,d} : S^{l} → S^{l} of degree d. Consider B^{l+1}, the (l + 1)-dimensional real unit ball, with ∂B^{l+1} = S^{l}. The inclusion S^{l} = ∂B^{l+1} ↪ B^{l+1} is denoted by i. Then, by definition, T (l, d) is the co-cartesian product of f_{l,d} and i. In other words, T (l, d) = B^{l+1} ⨆ S^{l}/ ∼, where x ∈ ∂B^{l+1} is identified with f_{l,d}(x) ∈ S^{l}.

It is not difficult to see that the homotopy type of T (l, d) does not depend on the choice of f_{l,d} (cf. 2.3).

Lemma 2.1. The space T (l, d) (l ≥ 2, d ≥ 2) has the following properties:

(a) H̃_{q}(T (l, d), ℤ) = 0 for q ≠ l,

(b) H̃_{q}(T (l, d), ℤ) = ℤ_{d} for q = l.

Lemma 2.2. Consider an arbitrary topological space X and fix an element [a] ∈ π_{l}(X), (l ≥ 2) of order d. Then there is a continuous map g : T (l, d) → X, such that π_{l}(g)[S^{l}] = [a].

Proof. Let a : S^{l} → X be a representative of [a]. Then a ∘ f_{l,d} : S^{l} → X is trivial in π_{l}(X), hence there exists H : B^{l+1} → X, such that H|_{S^{l}=∂B^{l+1}} = a ∘ f_{l,d}. By universal property of the co-cartesian product, one has the diagram

[figure: B^{l+1}; T (l, d); S^{l}; S^{l}; i; α; f_{l,d}; X; g; H; a]

hence there exists g, with the required property. □

Lemma 2.3. Assume that a space X is a CW-complex and has the property of T (l, d) given in the Lemma 2.1 (a) and (b). Then X and T (l, d) have the same homotopy types.

Proof. By Lemma 2.2 there is a map g : T (l, d) → X which induces isomorphism at the level of homology group. Then the statement follows from Whitehead theorem, see e.g. [13], p. 187. □

Notice that 2.3 proves the assertion of remark 1.2 as well.

Proposition 2.4. Consider an n-dimensional CW-complex which is (n − 2)-connected (n ≥ 3). Let T be the torsion group of H_{n−1}(X, ℤ) and write T as a direct sum of cyclic groups, T = ⊕_{i}G_{i}. Let b_{i} be the i^{th} Betti number of X. Then X has the homology type of the bouquet:

B := (⋁_{b_{n}} S^{n}) ∨ ( ⋁_{b_{n−1}} S^{n−1}) ∨ (⋁_{i} T (n − 1, |G_{i}|)) .

Proof. First we construct a map g : B → X. Recall that the Hurewicz homomorphism h_{n−1} : π_{n−1}(X) → H_{n−1}(X, ℤ) = ℤ^{b_{n−1}} ⊕ (⊕_{i}G_{i}) is an isomorphism (by Hurewicz theorem, cf. [13], page 185). Then, by definition, {g|_{S^{n−1}}}_{i=1}^{b_{n−1}} represent a h_{n−1}–lifted base of the free part of H_{n−1}(X, ℤ). Similarly, Lemma 2.2 gives a map g|_{⋁_{i} T (n−1,|G_{i}|)}. Using the fact that h_{n} : π_{n}(X) → H_{n}(X, ℤ) is onto (cf. [loc. cit.]), we define g|_{⋁ S^{n}} in such a way that {g|_{⋁ S^{n}}}_{i=1}^{b_{n}} represent a h_{n}–lifted base of H_{n}(X, ℤ). By construction, g_{∗} : H_{∗}(B, ℤ) → H_{∗}(X, ℤ) is an isomorphism. Since X is simply connected, the result follows from Whitehead theorem. □

Corollary 2.5. Assume that for a germ f : (ℂ^{n+1}, 0) → (ℂ, 0) one has dim Singf^{−1}(0) = 1 (n ≥ 3). Then F has the homotopy type of a bouquet of spheres if and only if H_{n−1}(F, ℤ) = 0 is free.

Remark 2.6. In the above proposition and its proof it was crucial that all the Hurewicz homomorphisms h_{q} : π_{q}(X) → H_{q}(X, ℤ) were onto (for q > 0). For an (n − 2)-connected space (where n ≥ 3), one has automatically the surjectivity for q = n − 1 and q = n. But, in general, this is not the case for the other q’s, – even if there are only two non-trivial groups H_{q}(F, ℤ). For example, one can consider the complex projective plane X = ℂℙ^{2}. It is simply connected, its homology is free, and obviously h_{2} is an isomorphism. But π_{4}(X) = 0 while H_{4}(X, ℤ) = ℤ, hence h_{4} is not onto. Moreover, X is not a bouquet of type ⋁_{i} ⋁_{b_{i}(X)} S^{i}, since ℂℙ^{2} ≁ S^{2} ⋁ S^{4}.

It is natural to ask about the validity of theorem 1.1 in the case n = 2. In the next paragraph we collect some facts about this case.

Remark 2.7. (1) According to a classical result of Whitehead [18], every connected finite 2-dimensional CW -complex with fundamental group π_{1} has the homotopy type of a cellular model associated with some finite presentation of π_{1}. In particular, the fundamental group determines completely the homotopy type.

(2) In the case n = 2, theorem 1.1 is not true anymore: e.g. S^{1} × S^{1} is not a bouquet. In fact, S^{1} × S^{1} ≁ S^{1} ∨ S^{1} ∨ S^{2} also shows that the homology, in general, does not determine the homotopy type of a connected 2-dimensional CW -complex. Notice that S^{1} × S^{1} can be realized as a Milnor fiber, e.g. it is the Milnor fiber of f(x, y, z) = xyz. Notice that if we take the ‘suspension’ singularity in four variables g = xyz + w^{2}, then its Milnor fiber is the suspension S(S^{1} × S^{1}) of the torus, which has the homotopy type of S^{2} ∨ S^{2} ∨ S^{3}.

(3) On the other hand, in some cases, a 2-dimensional CW -complex X can be a bouquet, even if it is not simply connected. This happens e.g. if π_{1}(X) is a free group (in this case X ∼ ∨_{b_{1}}S^{1}∨∨_{b_{2}}S^{2}) by a result of C.T.C. Wall [17]; or, if π_{1}(F) = ℤ_{d} (in this case X ∼ T (1, d)∨∨_{b_{2}}S^{2}), by a result of Dyer and Sieradski [2].

References

[1] Dimca, A. and Némethi, A.: On the monodromy of complex polynomials, Duke Math. Journal 108 (2) (2001), 199-209.

[2] Dyer, M.N. and Sieradski, A.J.: Trees of homotopy types of 2-dimensional CW -complexes, Commentarii Math. Helv. 48 (1973), 31-44.

[3] Kato, M. and Matsumoto, Y.: On the connectivity of the Milnor fiber of a holomorphic function at a critical point, Manifolds Tokyo 1973, University of Tokyo Press, 1975, 131-136.

[4] Lê, D.T.: Complex analytic functions with isolated singularities, J. Alg. Geom. 1 (1992), 83-100.

[5] Lê, D.T.: Le concept de singularité isolée de fonction analytique, Adv. Studies Pure Math. 8 (1986), 213-227.

[6] Milnor, J.: Singular points of complex hypersurfaces, Annals of Math. Studies 61, Princeton University Press 1968.

[7] Némethi, A.: Hypersurface singularities with 2-dimensional critical locus, J. of London Math. Soc. 59 (2) (1999), 922-938.

[8] Pellikaan, G.R.: Hypersurface singularities and resolutions of Jacobi modules, Thesis, Rijksuniversiteit Utrecht, 1985.

[9] Pellikaan, G.R.: Finite determinacy of functions with non-isolated singularities, Proc. London Math. Soc. 57 (3) (1988), 357-382.

[10] Siersma, D.: A bouquet theorem for the Milnor fiber, J. Alg. Geom. 4 (1995), 51-66.

[11] Siersma, D.: Singularities with critical locus a 1-dimensional complete intersection and transversal type A_{1}, Topology and its Applications 27 (1987), 51-73.

[12] Siersma, D.: Variation mappings on singularities with a 1-dimensional critical locus, Topology 30 (1991), 445-469.

[13] Switzer, R.M.: Algebraic Topology – Homology and Homotopy, Classics in Mathematics, Springer Verlag 1975.

[14] Tibăr, M.: Bouquet decomposition of the Milnor fiber, Topology 35 Vol. 1, (1996), 227-241.

[15] Zaharia, A.: A study about singularities with non-isolated critical locus, Thesis, Rijksuniversiteit Utrecht, 1993.

[16] Zaharia, A.: Topological properties of certain singularities with critical locus a 2-dimensional complete intersection, Topology and its Applications 60 (1994), 153-171.

[17] Wall, C.T.C.: Finiteness conditions for CW -complexes I, Ann. Math. 81 (1965), 56-69.

[18] Whitehead, J.H.C.: Simplicial spaces, nuclei, and m-groups, Proc. London Math. Soc. 45 (1939), 243-327.

Petru Maior University, Tg-Mureş, Dept. of Math. and Comp. Sci., str. N. Iorga 1, 4300 Tg-Mureş, România

E-mail address: shorvath@ms.sapientia.ro

Rényi Mathematical Institute Budapest, Hungary; and Ohio State University, Columbus OH, USA

E-mail address: nemethi@renyi.hu and nemethi@math.ohio-state.edu