Language learning | English » Sigurdsson Baldur - The Milnor fiber of the singularity

http://www.doksihu The Milnor fiber of the singularity f (x, y) + zg(x, y) = 0. Master’s thesis Baldur Sigurðsson Mathematics department Adviser: Némethi András, Professor Department of Geometry Eötvös Loránd University, Eötvös Loránd University Faculty of Science 2010 http://www.doksihu Contents 1 Introduction 3 1.1 Topological description of hypersurface singularities . 3 1.2 The case of isolated hypersurface singularities . 4 1.3 Methods for non-isolated singularities . 5 1.4 Is F homotopically a bouquet of spheres? . 5 1.5 The present family of germs . 6 2 Preliminaries 7 2.1 The topology of singularities on plane curves . 7 2.2 Handles and boundary-connected sums . 9 2.3 Plumbed manifolds . 9 3 Description of the fiber 10 3.1 The fiber as a subset of a resolution . 10 3.2 Corollaries .

15 4 Proof of theorem (3.2) 16 4.1 Theorem (3.2) restated 16 4.2 The structure of FΓ1 . 18 4.3 Local picture close to the strict transform of f . 19 4.4 The structure of FΓ2 . 21 4.5 Local picture close to the strict transform of g . 23 4.6 Edges connecting Γ1 and Γ2 . 26 4.7 The monodromy . 27 List of Figures 1 The set T1 . 12 2 The set T2 . 12 3 The set Y . 13 4 A homotopy of Xe . 21 2 http://www.doksihu 1 Introduction In this note we give a description of the Milnor fiber of a hypersurface singularity of the form (C3 , 0) (C, 0), (x, y, z) 7 f (x, y) + zg(x, y) for germs f, g : (C2 , 0) (C, 0). We will only require that f and g have no common factors This singularity is not

isolated; the singular set will be the z-axis In fact, we determine the diffeomorphism type of the Milnor fiber in terms of a simultaneous embedded resolution graph of f and g. Singularities of this type play an important role in the investigations of sandwiched singularities, as described in [2]. 1.1 Topological description of hypersurface singularities In this subsection we recall some of the general properties of the Milnor fiber of a holomorphic germ f : (Cn+1 , 0) (C, 0), the monodromy associated to such a germ, and other invariants related to these two. Let f : (Cn+1 , 0) (C, 0) be a hypersurface singularity, denote by Bδ the closed ball with radius δ around the origin in Cn+1 , and by Dǫ the closed disc around the origin in C with radius ǫ. D will denote an arbitrary closed disc in the complex plane. We let Vf = {z ∈ Cn+1 : f (z) = 0} and Sing(Vf ) = {z ∈ Cn+1 : ∂f = 0}. The link of f is defined as K = Vf ∩ ∂Bδ for 0 < δ ≪ 1 The Milnor fiber Ff of f is

by definition the fiber f −1 (ǫ)∩Bδ for 0 < ǫ ≪ δ ≪ 1. Then Ff is a smooth 2n dimensional manifold, and so has the homotopy type of a CW complex. Actually, Ff has the homotopy type of a finite n dimensional CW complex, as proved in [6]. Moreover, if s is the dimension of the singular locus Sing(Vf ), then Ff is (n − s − 1)-connected, as proved in [5]. Let E = f −1 (∂Dǫ ) ∩ Bδ . The function E ∂Dǫ , z 7 f (z) is a locally trivial fiber bundle with fiber Ff . If T = {z ∈ ∂Bδ : |f (z)| < ǫ}, we can define another fiber bundle ∂Bδ T ∂D1 , z 7 f (z)/|f (z). These two fiber bundles are isomorphic. In fact, there is a bundle-isomorphism E ∂B T which restricts to the identity on ∂T . In particular, we have a diffeomorphism ∼ Ff = {z ∈ ∂Bδ T : f (z)/|f (z)| = 1}. (1) The monodromy of this bundle is a diffeomorphism m : F F with the property that this bundle is isomorphic to the bundle given by F × I/((p, 0) ∼ (m(p), 1)) I/(0

∼ 1), (p, t) 7 t. The monodromy is determined by the bundle up to isotopy, and the bundle is determined up to bundle isomorphism by the monodromy. The monodromy induces linear isomorphisms hi : Hi (F ; C) Hi (F ; C). In [4], Griffiths gives a discussion of, and references to four different 3 http://www.doksihu proofs of the theorem stating that the eigenvalues of these monodromy operators are roots of unity. We call the product ζf (t) = n Y i det(I − thi )(−1) . i=0 the zeta function associated with the singularity f . This product is well defnied because F is a finite CW complex, and so dimC H∗ (F ; C) < ∞. The zeta function behaves multiplicatively in the following sense: Let C be a subset of F so that dim H∗ (C; C) < ∞ and mf restricts to a homeomorphism mC : C C. Let us call such a subset good with respect to m. Then mC induces a linear automorphism hC,i on Hi (C; C) and we define ζC (t) = ∞ Y i det(I − thC,i )(−1) . i=0 If A and B are

subsets of F so that the interiors of A and B cover X and A, B and A∩B are good with respect to m, then the identity ζf (t) = ζA (t)ζB (t)ζA∩B (t)−1 follows from the Mayer-Vietoris exact sequnce. The singular fiber of f is defined as Ff,sing = {z : |z| = δ, |f (z)| > 0, f (z)/|f (z)| = 1} ∪ K. Usually, Ff,sing is not a smooth manifold. By the description (1) of Ff we have an inclusion ι : Ff ֒ Ff,sing . If f is an isolated singularity, then ι is a homotopy equivalence, as proved in [6]. For non-isolated singularities this does generally not hold The monodromy mf can be extended to a homeomorphism mf,sing : Ff,sing Ff,sing , which is called the singular monodromy. 1.2 The case of isolated hypersurface singularities A singularity given by f = 0 for some f : (Cn+1 , 0) (C, 0) is called isolated if Sing(Vf ) = {0}. Working with isolated singularities only has proven to give much stronger results than with arbitrary hypersurface singularities. In this case the

singular locus Sing(Vf ) has dimension 0, so by an earlier remark, Ff is n − 1 connected. But a stronger statement holds: If f is isolated, then Ff is homotopically a bouquet of n-spheres. That is, ht Ff ∼ µ 4 Sn. http://www.doksihu The number µ is clearly an important invariant of the singularity, as it determines completely the homotopy type of the Milnor fiber. This number is called the Milnor number of the singularity, and it has several characterizations: - Clearly, µ is the n-th Betti number of Ff . - We have µ = dimC C{z0 , . , zn }/(∂f ), where (∂f ) is the ideal of C{z0 , , zn } spanned by the partial derivatives of f . The algebra dimC C{z0 , , zn }/(∂f ) is called the Milnor algebra of f . - We have µ = deg G where G : S 2n+1 S 2n+1 is the map G= (∂1 f, . , ∂n+1 f ) . k(∂1 f, . , ∂n+1 f )k These characterizations are discussed in [6], [7]. 1.3 Methods for non-isolated singularities The results for isolated singularities

discussed in the previous subsection do not hold in general. The homotopy type of the fiber is not fully understood; even the Betti numbers can be hard to determine. There are however methods which can be used to determine some of the invariants of the fiber in some cases. Let Σ = Sing(Vf ). We can write Σ = Σs ∪ ∪ Σ0 , where Σj Σj−1 is j dimensional and smooth. Here, s = dim Sing(Vf ) as before 1.4 Is F homotopically a bouquet of spheres? In the case of an isolated hypersurface singularity, the Milnor fiber is homotopically a bouquet of n-spheres. For non-isolated singularities this is no longer true, but one can ask if F is a bouquet of spheres in the expected dimension, i.e from n − s to n, where s is the dimension of the singular locus This following theorem is discussed in [11] which also contains a reference to a proof by the same author. Theorem 1.1 Let f : (Cn+1 , 0) (C, 0) be a germ defining such a singularity so that the singular locus is an ICIS, and

the transversal singularity outside the origin has type A1 . Then the homotopy type of Ff is either a bouquet of n-spheres, or a bouquet of n-spheres and a single (n − 1)-sphere. The following results can be found in [8]. Theorem 1.2 Let n ≥ 3 and f : (Cn+1 , 0) (C, 0) be a germ whose singular locus is one dimensional. Then Ff is a bouquet of spheres if and only if H(Ff , Z) is free. 5 http://www.doksihu Note that the condition n ≥ 3 is necessary: If f (x, y, z) = xyz, then Ff is homotopically S 1 × S 1 . The proof of this statement can be found in [8], but it goes along the following lines: Let F be a simply connected CW complex so that H∗ (F, Z) is free and the Hurewicz homomorphism π∗ (F ) H∗ (F, Z) is surjective. Then it is possible to construct a map Si F i≥2 bi which induces an isomorphism on homotopy and homology groups, and so by a theorem of Whitehead, it is a homotopy equivalence (here, bi are the Betti numbers of F ). In particular, if l ≥ 3,

Hi (F, Z) = 0 for i ∈ / {l, l − 1}, H∗ (F, Z) is free and F is simply connected then the Hurewicz map is automatically surjective by a theorem of Hurewicz. Theorem 1.3 Let us now make the assumption that f : (Cn+1 , 0) (C, 0) is a germ whose singular locus Σ is a two dimensional ICIS, and that the transversal singularity is A1 outside a curve in Σ. Then Ff has the same homotopy type as one of the following spaces: Sn ∨ . ∨ Sn, S n−1 ∨ S n ∨ . ∨ S n , S n−2 ∨ S n ∨ . ∨ S n For n = 2, one must replace the last space on this list with S 0 × S 2 ∨ . ∨ S 2 1.5 The present family of germs In this note we will study the particular germs which have the form (C3 , 0) (C, 0), (x, y, z) 7 f (x, y) + zg(x, y), where f , g are plane curve singularities without common factors. In this case, the singular is the z-axis, in particular, it is one dimensional. However, we can not expect to use the theorems from the previous subsection, since the Milnor

fiber is generally not a bouquet of spheres in this case. We find a resolution for this singularity, which can be found by finding a common embedded resolution V C2 for f and g, and then using V × C C3 . The diffeomorphism type of the Milnor fiber is determined and described by the resolution graph associated to the common embedded resolution of f and g, decorated by the multiplicities of f and g. This means in particular that the boundary of the fiber only depends on the resolution graph obtained by resolving f and g. We also find a simple formula for the zeta function and Euler characteristic of the germ. These germs were considered by de Jong and van Straten in [2]. The germs considered in this note include the Ta,∞,∞ germs xa + xyz = 0 and the Ta,b,∞ germs xa + y b + xyz = 0, as well as cylinders of plane curve 6 http://www.doksihu singularities f (x, y) = 0, or singularities of the form zg(x, y) = 0. The boundary of the Milnor fiber of these singularities were all

considered as examples in [9]. 2 Preliminaries 2.1 The topology of singularities on plane curves We will consider the case when n = 1, i.e when f is a plane curve singularity Then f is isolated if and only if f has no repeated factors Write f = f1q1 f2q2 · · · fkqk where f1 , . , fk are the k different factors of f In this case, K is a smooth 1-dimensional submanifold of ∂Bδ , and T is a tubular neigborhood around K. In fact, there exists a projection p : T K which is a trivial D-bundle. To be more precise, we have K = ∪ki=1 Ki , where Ki = {z ∈ ∂Bδ : fi = 0}, and T = ∪ki=1 Ti , where Ti is the component of T which has Ki as a subset. From the definition it is clear that ∂Ff ⊂ ∂T The projection p can be chosen in such a way that the restriction ci = p|Ff ∩∂Ti : Ff ∩ ∂Ti Ki is a covering map. This map can be described in terms of the embedded resolution graph of f as follows. Let Γf = (V, E) be the embedded resolution graph of some fixed embedded

resolution of f . Here V is the set of vertices and E the set of edges Write V = W ∐ Af where Af consists of the arrowhead vertices of Γ, and W consists of the nonarrowhead vertices. The elements of Af correspond to the branches of f , so there is a natural correspondece between the arrowhead vertices of Γ and the components Ki of K, let ai ∈ Af correpond to Ki . For each ai ∈ Af there exists a uniqe wi ∈ W so that (wi , ai ) ∈ E. The map f has multiplicity qi on ai , let mi be its multiplicity on vi . Then Ff ∩ Ti has (qi , mi ) components, and restricting ci to any of these components gives a covering of degree qi /(qi , mi ). The singular fiber Ff,sing of f is homeomorphic to the space Ff / ∼ where the equivalence relations ∼ are given by z1 ∼ z2 if and only if z1 , z2 ∈ Ff ∩ Ti for some i, and ci (z1 ) = ci (z2 ) For convenience we define the map c : ∂Ff K by c|Ff ∩T = ci . The monodromy mf : Ff Ff can be chosen so that it preserves this equivalence

relations, that is, if x1 ∼ x2 , we can assume that m(x1 ) ∼ m(x2 ). Therefore, we get a homeomorphism Ff,sing Ff,sing induced by the monodromy. This homeomorphism coincides with the singular monodromy already constructed. Note that Ff,sing = A ∪ B where A is homotopically eqivalent to Ff and both B and A ∩ B are homotopically equivalent to the disjoint union of copies of S 1 . The monodromy mf,sing restricts to a homeomorphism A A which coincides with the monodromy mf . Also, mf,sing permutes the connencted 7 http://www.doksihu components of B and A ∩ B. (In fact, the map B B, b 7 mf,sing (b) is homotopically equivalent to the identity idB ) If we identify these components with S 1 , we can choose an orientation on them so that the restrictions mf,sing |B and mf,sing |A∩B preserve this orientation. This implies that ζB (t) = ζA∩B (t) = 1, and therefore ζf (t) = ζf,sing (t). Example 2.1 (a) If f (x, y) = xd for d ≥ 2 then the Milnor fiber of f is a disjoint

union of d copies of D. The resolution graph of f consists of a single non-arrowhead vertex v and a single arrowhead vertex a, and the multiplicity of f on this vertex is q = d. The link K has a single component, so we have a covering map c : ∂Ff K. Restriction c to any of the components of ∂Ff gives a covering map of degree 1, i.e a homeomorphism Thus, Ff,sing is homeomorphic to the disjoint union of d copies of closed discs with their boundaries identified. This shows that Ff,sing has the same homotopy type as the wedge of d−1 copies ∼ of the two-sphere, i.e Ff,sing = ∨d−1 S 2 The monodromy mf permutes the connected components of Ff cyclically, and we get ζf (t) = ζf,sing (t) = td − 1. (b) Let f (x, y) = xa + y b with a, b ≥ 2 and (a, b) = 1. Then f is isolated, so Ff,sing = Ff . In [6] §9 it is shown that Ff has the homotopy type of ∨µ S 1 , where µ = (a − 1)(b − 1), and that the characteristic polynomial ∆(t) of the induced isomorphism H1 (F ; C) H1

(F ; C) is given by ∆(t) = (tab − 1)(t − 1) . (ta − 1)(tb − 1) It follows that (1 − ta )(1 − tb ) . 1 − tab (c) Let f (x, y) = (xy)m . Then Ff is the disjoint union of m copies of S 1 × I ζf (t) = (here I = [0, 1]). The resolution graph of f has one non-arrowhead vertice on which f has multiplicity 2m, and two arrowhead vertices both on which f has multiplicity m. The link K consists of two components K1 and K2 For each of the components C of Ff we have ∂C = S 1 ∐ S 1 , and each of the maps c∂C is a homeomorphism. Thus, Ff,sing is a disjoint set of m copies of S 1 × I with the boundaries identified. In particular, if m = 2, then Ff,sing is a torus If m ≥ 2, then Ff,sing is not homotopically equivalent to a bouquet of spheres, not even if we allow spheres of different dimensions. Homotopically, Ff is a disjoint union of m copies of S 1 , and mf permutes these components without reversing orientation. Thus, ζf (t) = ζf,sing (t) = 1 Remark 2.2 In these

examples, the zeta function can also be calculated using A’Campo’s formula, which is described in [1]. 8 http://www.doksihu 2.2 Handles and boundary-connected sums We will use handles to describe the Milnor fiber. More precisely, we will use 2 dimensional 4-handles in our construction. Chapter 4 of [3] gives a presentation of the theory we will need. Let X be a 4-manifold and ψ : (∂D) × D ∂X an embedding. We obtain a new manifold X ∪ψ (D × D) by taking the disjoint union X ∐ (D × D) and then identifying any point x ∈ (∂D) × D with ψ(x) ∈ ∂X. The map ψ induces an isomorphism between the normal bundles of (∂D) × {0} in (∂D) × D and ψ((∂D) × {0}) in ∂X. Since (∂D) × {0} ⊂ (∂D) × D already comes with a canonical framing, this isomorphism can be specified by a framing on ψ((∂D) × {0}). The diffeomorphism type of the resulting manifold is determined by the following data (see for example [3]): ∼ - The embedding ψ|(∂D)×{0} of

(∂D) × {0} = S 1 into ∂X. - The framing of the normal bundle of ψ|(∂D)×{0} . It will be convenient to be able to specify an alternative framing on (∂D) × {0} rather then on ψ((∂D) × {0}). For j ∈ Z, let us call (∂D) × D with the framing S 1 × D (∂D) × D, (eiθ , z) 7 (eiθ , eijθ z) a 4 dimensional 2-handle with the j-th framing. With this notation, the canonical framing on (∂D) × {0} is the 0-th framing. 2.3 Plumbed manifolds The description that we will provide will rely on plumbed manifolds associated with plumbing graphs. A plumbing graph Γ is a graph whose vertices v are decorated with an Euler number ev and a genus gv . A 4 dimensional plumbed manifold associated to the graph Γ is constructed as follows: For each vertex v ∈ V(Γ) we have a compact (real) 2-dimensional surface Ev and a locally trivial disc bundle bv : Tv Ev with Euler number (or first chern-class) ev . For each edge (v, w) ∈ E(Γ) we choose a small closed disc-shaped

neigborhood U1 and U2 in Ev and Ew . Choose trivializations tv : −1 D × D b−1 v (U1 ) and tw : D × D bw (U2 ) and glue together the points tv (x, y) and tw (y, x). The plumbed 4-manifold associated to the graph Γ is the disjoint union of the bundles Tv , glued together according to the rules defined by the edges. A 3 dimensional plumbed manifold associated to the graph Γ is the boundary of the corresponding 4 dimensional plumbed manifold. The graph Γ can also have arrowhead vertices. An arrowhead vertex a must be the end-vertex of exactly one edge, the other end vertex being a non- 9 http://www.doksihu arrowhead vertex w. Then a represents a generic fiber over Ew , the fiber being a disc in the case of a 4 dimensional plumbed manifold, but an S 1 in the case of a 3 dimensional manifold. The graph Γ can have dash-edges. Such an edge should have one nonarrowhead end vertex w, and an arrowhead end-vertex The dash-edge indicates that the preimage b−1 w (U ) of a small

closed disc-shaped neigborhood should be removed. In [10] the vertex w is decorated with a new number rw , the number of dash-edges with w as end-vertex, rather then presenting this date with new edges. A 3 dimensional plumbed manifold whose graph has dash-edges will have boundary. For a precise discussion of plumbed manifolds see [10], [9]. 3 Description of the fiber 3.1 The fiber as a subset of a resolution Let f, g : (C2 , 0) (C, 0) be any plane curve singularities without common factors and define Φ(x, y, z) = f (x, y) + zg(x, y). Now consider a fixed common embedded resolution φ : V C2 of f and g. The resolution graph of this embedded resolution will be denoted by Γ. For the graph Γ we have the set of vertices V(Γ) and the set of edges E(Γ). We have V(Γ) = W(Γ) ∐ A(Γ) where W(Γ) is the set of non-arrowhead vertices and A(Γ) the set of arrowhead vertices. We decompose A(Γ) further as A(Γ) = Af (Γ) ∐ Ag (Γ), where elements of Af (Γ) and Ag (Γ) correspond

to conponents of the strict transform of f and g respectively. A vertex v ∈ V(Γ) corresponds to a component Ev of the exceptional divisor φ−1 (0), or the strict transform of f or g. In each case, we denote by mv the multiplicity of f on Ev , and lv the multipilcity of g on Ev . In particular, mv = 0 if and only if v ∈ Ag and lv = 0 if and only if v ∈ Af . Let f ′ = f ◦ φ, g ′ = g ◦ φ and Ff′ = (f ′ )−1 (ǫ) ∩ φ−1 (Bδ ) = φ−1 (Ff ). The map V φ−1 (0) C2 {(0, 0)}, r φ(r) is a diffeomorphism. In particular, it restricts to a diffeomorphism Ff′ Ff . We have a map φ × idC : V × C C3 which restricts to a diffeomorphism (V φ−1 (0)) × C C3 {(0, 0, z) : z ∈ C}. We set Φ′ = Φ ◦ (φ × idC ), and F ′ = (φ × idC )−1 (F ). Clearly, F ′ is diffeomorphic to F For each w ∈ W we may choose a small tubular neigborhood Tw around 10 http://www.doksihu Ew in V and a map bw : Tw Ew which is a smooth disc bundle. These

neigborhoods can be chosen so that they satisfy the following property: If w, w′ ∈ W and (w, w′ ) ∈ E, then we have b−1 w (Ew ∩ Ew′ ) = Ew′ ∩ Tw and −1 b−1 w (Ew ∩Tw′ ) = Tw ∩Tw′ . If w ∈ W, a ∈ A and (w, a) ∈ E, then bw (Ew ∩Ea ) = Ea ∩ Tw . Then the set T = ∪w∈W Tw is the plumbed 4-manifold with plumbing graph Γ. If w, w′ ∈ W(Γ) and e = (w, w′ ) ∈ E(Γ), then we let Te = Tw ∩ Tw′ . If w ∈ W(Γ) and a ∈ A(Γ) so that e = (w, a) ∈ E(Γ), then we pick a small disc-shaped neigborhood Ua in Ew around Ew ∩ Ea and let Te = b−1 w (Ua ). The Milnor fiber F can be described in terms of the embedded resolution graph Γ, with the additional arrowhead vertices, and all vertices decorated by the multiplicities of f ′ and g ′ . This description will rely on which of the two functions f ′ and g ′ has higher multiplicities on the exceptional divisors. The following definition makes this more precise. Definition 3.1 (a) For a

subgraph Γ′ of Γ, let VΓ (Γ′ ) = {v ∈ V(Γ) V(Γ′ ) : ∃v ′ ∈ V(Γ′ ), (v, v ′ ) ∈ E(Γ)}, EΓ (Γ′ ) = {(v ′ , v) ∈ E(Γ) : v ′ ∈ V(Γ′ ), v ∈ VΓ (Γ′ )}. (b) Split Γ into two parts, Γ1 and Γ2 . The graph Γ1 is the subgraph of Γ generated by the non-arrowhead vertices w ∈ W(Γ) for which mw ≤ lw , and Γ2 is generated by the non-arrowhead vertices w ∈ W(Γ) for which mw > lw . This means that v ∈ V(Γ) is a vertex of Γ1 (resp, Γ2 ) if and only if v ∈ W(Γ) and mv ≤ lv (resp, mv > lv ). Two vertices in Γi are adjacent in Γi if and only they are adjacent in Γ for i = 1, 2. Let T1 be the closure of the set    [  Tv   v∈V(Γ1 ) [ v∈VΓ (Γ1 )Ag (Γ)  Tv  . Essentially, T1 is the plumbed 4-manifold with plumbing graph Γ1 , but we must leave out a neigborhood of the strict transform of f , and the exceptional divisors which belong to Γ1 . Let FΓ1 = F ′ ∩ (T1 × C) In the following

picture, the pair (m, l) indicates that f (resp, g) has multiplicity m (resp, l) on the divisor. Similarly, let T2 be the closure of the set    [ [  Tv   v∈V(Γ2 ) v∈VΓ (Γ2 )Af (Γ) 11  Tv  . http://www.doksihu (1, 0) (4, 3) (3, 4) (0, 1) 1 0 01111111111 1 0000000000 0 1 0 1 0000000000 1111111111 0 1 0 1 0000000000 1111111111 0 1 0 1 0000000000 0 1 01111111111 1 0000000000 1111111111 0 1 0 1 0000000000 1111111111 00 1 11 0001 111 0 0 0000000000 1111111111 (3, 3) 00 11 000 111 0 0 0000000000 1 1 00 1 11 0001 0111 01111111111 0000000000 1111111111 0 1 0 1 0000000000 0 1 01111111111 1 0000000000 1111111111 0 1 0 1 0000000000 1111111111 0 0 0000000000 1 1 0 1 01111111111 1 0000000000 1111111111 0 1 01111111111 1 0000000000 Figure 1: The set T1 . Again, T2 is similar to the plumbed 4-manifold with plumbing graph Γ2 , but this time we must leave out a neigborhood of the strict transform of g, and the exceptional divisors which belong to Γ1 . (1,

0) (3, 3) 1 0 0 1 0 1 0 1 0 1 0 1 00000 11111 0 1 00000 11111 0 000001 11111 0 1 0 1 0 1 0 1 0 1 0 1 0 1 (5, 4) (0, 1) 1111111 0000000 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 000 111 0000000 1111111 (4, 3) 000 111 0000000 1111111 000 111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 Figure 2: The set T2 . Let FΓ2 = F ′ ∩ (T2 × C). (c) Let Y2 be the closure of the set  (∂T2 )  [ v∈VΓ (Γ2 )Af (Γ)  Tv  . Then Y2 is a 3-manifold with boundary. Alternatively, Y2 is the plumbed 3manifold with boundary, obtained from the graph Γ2 , with the following modification: For any (v, v ′ ) ∈ EΓ (Γ2 ) with v ∈ V(Γ2 ) and v ′ ∈ / V(Γ2 ) ∪ Af (Γ) we remove a tubular neigborhood of a generic fiber over Ev . In [10] this is indicated by decorating the vertex v with the numbers [g, r], where g is the genus of Ev , and r is the number of

neigborhoods removed. In [9] this is indicated by adding 12 http://www.doksihu a dash-edge to the graph, with end vertices v and an arrowhead vertex v ′ . (1, 0) (3,1 3) 0 (5, 4) 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 (0, 1) (4, 3) Figure 3: The set Y . The diffeomorphism type of F can be explicitly described by the graph Γ as follows: Let T1′ = T1 ∪ [ e Te ! . where the union ranges over edges e ∈ E(Γ) which have one end-vertex in V(Γ1 ) and the other end-vertex in V(Γ2 ). Let W ′ be a closed tubular neigborhood around Ff ∩T1′ in T1′ so that W ′ ∩∂T1′ is a tubular neigborhood around Ff ∩ ∂T1′ in ∂T1′ . Then W ′ is diffeomorphic to (Ff ∩ T1′ ) × D, because the normal bundle of Ff ∩ T1′ is oriented and Ff ∩ T1′ is a 2 dimensional surface with boundary, so it has the homotopy type of a 1 dimensional CW complex. For an edge e = (w, a) ∈ E(Γ) where w ∈ V(Γ1 ) and a ∈ Af (Γ), let ue , ve be

coordinates on Te so that ve is constant on the fibers of bw , Te is presented in these coordinates as {(ue , ve ) : |ue |, |ve | ≤ 1} and φ(r) = 0 if and only if ue (r) = 0 for r ∈ Te . Choose η so small that |ue (r)| > η for any r ∈ W ′ ∩ Te Let We′ be the set {r ∈ Te : η ≤ |ue (r)|}. Then let A′ = W ′ ∪ (∪e We′ ) Here, the union ranges over edges e for which We′ is defined. Let W ′′ be obtained by removing a small tubular neighborhood around φ−1 (0) from T2 . This means that we remove a small open tubular neighborhood around each irreducible component of the exceptional divisor, and we assume that the union of these small neighborhoods form a small plumbed 4-manifold (without the boundary) as is constructed from the graph Γ2 . Let e′ = (w, a) ∈ E(Γ) where w ∈ V(Γ2 ) and a ∈ Ag (Γ). Let u, v be coordinates on Te so that the map r 7 (u(r), v(r)) takes Te to D1 × D1 Assume 13 http://www.doksihu that v is constant on the fibers

of bw , and that W ′′ ∩ Te ⊂ ∂W ′′ is given by {r ∈ ∂T2 ∩ Te : η ≤ |u| ≤ 1, |v| = 1} for some η ≪ ǫ. Let Ce ⊂ W ′′ ∩ Te be described in coordinates as Ce = {(u, v) : umw = ǫ}. For any c ∈ Ce we have a direct sum decompostion Tc ∂W = Tc Ce ⊕ h∂1 , ∂2 i where ∂1 and ∂2 are the tangent vectors which differentiate with respect to the real and imaginary part of u. This gives a framing on Ce , which allows us to attach handles along the components of Ce . Let A′′ be the the manifold obtained by attaching a 4 dimensional 2-handle to each of the components of Ce (with the framing described in the previous paragraph) for all edges e = (w, a) with w ∈ V(Γ2 ) and a ∈ Ag (Γ). The handles should have the (−k)-th framing, as defined in (2.2), where k is the multiplicity of g on a. The set A is defined as A′ ∪ W ′′ with the handles glued to each of the sets Ce as already described. More abstractly, we can define the space A as

follows: The set A′ ∩ W ′′ is a closed tubular neigborhood around a submanifold of ∂A′ or ∂W ′′ . The union A′ ∪ W ′′ is a manifold with boundary (assuming that we smooth out some corners on the boundary) which can clearly be constructed as A′ ∐ W ′′ / ∼ where the equivalence relation identifies points in A′ ∩ W ′′ in the obvious way. This gives A′ ∪ W ′′ as the boundary connected sum of A′ and W ′′ as abstract manifolds. The attaching spheres (attaching circles) of the handles which are attached to W ′′ to obtain A′′ do not intersect the set of points of W ′′ which are glued to points of A′ in this construction. Therefore, we can construct a boundary connected sum of A′ and A′′ using essentially the same gluing as when constructing A′ ∪ W ′′ as a boundary connected sum. This construction yields the same space A. The main result in this note is that this construction gives the Milnor fiber of Φ.

Theorem 3.2 (i) The Milnor fiber F of the singularity Φ = f + zg and A have the same diffeomorphism type. (ii) The monodromy mΦ can be chosen so that the subsets W ′ , W ′′ and We′ are invariant (for edges e where We′ is defined). Its restriction to W ′ = (Ff ∩ T1 ) × D sends (r, z) to (mf (r), z) where mf is the monodromy of f , chosen so that Ff ∩ T1 is an invariant subset of Ff . The restrictions mΦ |We′ and mΦ |W ′′ are isotopic to the identity. For an edge e = (w, a) with w ∈ V(Γ2 ) and a ∈ Ag (Γ), the monodromy permutes the mw handles corresponding to e cyclically. 14 http://www.doksihu 3.2 Corollaries Corollary 3.3 (a) If mw ≤ lw for all w ∈ W(Γ), then F and Ff,sing have the same homotopy type. (b) If mw > lw for all w ∈ W(Γ), then F has the same homotopy type as P the plumbed 3-manifold with plumbing graph Γ2 with a∈Ag (Γ2 ) mwa open balls removed from it. Proof. (a) In this case, W ′ = Ff × D has the same homotopy type

as Ff To construct A′ we glue (∂Ff ) × D to ∐Af (Γ) D1 × (D1 Dη ). Up to homotopy, this is equivalent to gluing ∂Ff to ∐Af (Γ) S 1 . It is easy to see that this gluing is precisely the construction of Ff,sing . (b) In this case, W ′′ = Y2 × I where Y2 is the plumbed 3-manifold with plumbing graph Γ2 with a dash-edge added in place of any edge in Ag (Γ). In particular, W ′′ has the same homotopy type as this plumbed manifold. Up to homotopy, A′′ is obtained by adding 2-cells to this plumbed 3-manifold along mwa disjoint meridians. This is equivalent to starting with the plumbed 3-manifold whose plumbing graph is Γ2 without any dash-arrows, and then removing mw small balls for each edge (w, a) ∈ E(Γ) with w ∈ V(Γ2 ) and a ∈ Ag (Γ). Corollary 3.4 (a) The Euler characteristic of F is given by the formula χ(F ) = X mw (2 − δw ) + w∈V(Γ1 ) X mwa . a∈Ag (Γ2 ) (b) The zeta function associated to Φ is given by the formula  

 Y Y ζΦ (t) = ζ(t) =  (tmw − 1)2−δw   (tmwa − 1) w∈V(Γ1 ) a∈Ag (Γ2 ) where δw is the number of edges in Γ having w as one of its end-vertices. Proof. It is enough to prove (b), since χ(F ) = deg ζ(t) For e = (w, a) with w ∈ V(Γ1 ) and a ∈ Af (Γ) We have ζWe′ (t) = 1 because the monodromy acts trivially on H∗ (We′ ) and We′ has the same homotopy type as S 1 . The set We′ ∩W ′ has the homotopy type of a disjoint union of copies of S 1 , and the monodromy permutes these components, preserving their orientation. Thus ζWe′ ∩W ′ (t) = 1. A similar argument shows that ζA′ ∩A′′ (t) = 1 Using the result of [1] we get ζA′ (t) = Y (tmw − 1)2−δw . w∈V(Γ1 ) ′′ The monodromy acts trivially on H∗ (W ′′ ) so ζW ′′ (t) = (1 − t)χ(W ) where χ(W ′′ ) is the Euler characteristic of W ′′ . But W ′′ has the homotopy type of a 15 http://www.doksihu closed 3-manifold from which subsets of

the type S 1 × D have been removed. Hence, we have χ(W ′′ ) = 0. For each e = (w, a) ∈ E(Γ) where w ∈ V(Γ2 ) and a ∈ Ag (Γ) we attach mw 2-handles, the intersection of these handles with W ′′ is up to homotopy a disjoint union of copies of S 1 , which the monodromy permutes, preserving their orientation. The monodromy permutes the handles cyclically, so we get Y ζA′′ (t) = ζW ′′ (t) · (tmwa − 1). a∈Ag (Γ2 ) Now we get ζΦ (t) = ζA′ (t)ζA′′ (t)ζA′ ∩A′′ (t) = ζA′ (t)ζA′′ (t) which is the desired result. Example 3.5 Let f (x, y) = xd and g = y d where d ≥ 2 Then we can choose the resolution V so that V has a single element, say V = {v}. Then mv = lv = d, so we can apply (3.3)(a) The Milnor fiber F associated to Φ has the same homotopy type as Ff,sing , which is up to homotopy a bouquet of d − 1 twospheres. Note that in spite of this, Φ is not isolated The zeta function of this singularity is ζ(t) = td − 1. 4 Proof

of theorem (3.2) 4.1 Theorem (3.2) restated The following theorem is a more technical reformulation of theorem (3.2) The set W ′ is replaced with the set FΓ1 , the set W ′′ is replaced with FΓ2 and A′ and A′′ are obtained by enlargeing W ′ and W ′′ appropriately. Theorem 4.1 (a) The set FΓ1 is diffeomorphic to (Ff ∩ T1 ) × D In fact, the map FΓ1 Dδ , induced by the projection V × C ⊃ V × Dδ Dδ , (r, z) 7 z is a trivial fiber bundle with fiber Ff ∩ T1 . (b) Let w ∈ V(Γ1 ) and a ∈ Af (Γ) so that e = (w, a) ∈ E(Γ). Let (u, v) be some coordinates on Te so that v is constant on the fibers of bw , we have φ(r) = 0 if and only if u(r) = 0 for r ∈ Te , and that the image of the map (u, v) is D1 × D1 . Then, for some small η > 0, there exists a diffeomorphism σe : F ′ ∩ (Te × C) (D1 Dη◦ ) × D1 so that σe (r, z) = (u(r), v(r)) whenever |v(r)| = 1 (here, Dη◦ is the open disc with center 0 and radius η). (c) The set FΓ2 is

diffeomorphic to Y2 × I. 16 http://www.doksihu (d) If v ∈ V(Γ2 ) and a ∈ Ag (Γ) so that e = (w, a) ∈ E(Γ), then F is diffeomorphic to F ′ (Te × C) with mv copies of 4-dimensional 2-handles attached along Ff′ ∩∂Te ⊂ F ′ (Te × C). If u, v are coordinates on Te so that v is constant on the fibers of bw , then Ff′ ∩ ∂Te has the framing induced by these coordinates (similarly as in the discussion before (3.2)) The handle should have the (−k)-th framing where k is the vanishing order of g on Ea . (e) Let e = (v1 , v2 ) ∈ E(Γ), where v1 ∈ V(Γ1 ) and v2 ∈ V(Γ2 ). Then F ′ is diffeomorphic to F ′ (Te × C), after identifying a tubular neigborhood around (Ff′ ∩ (∂Te ∩ ∂Tv1 )) × {0} in ∂(F ′ (Te × C)) with a tubular neigborhood of (Ff′ ∩ (∂Te ∩ ∂Tv2 )) × {0} in ∂(F ′ (Te × C)). The identification is obtained from a trivialisation of the normal bundle (or a tubular neigborhood) around (Ff × {0}) ∩ (Te × C)

⊂ (F ′ ∩ (Te × C)) which is a disjoint union of copies of S 1 × I. (f ) The monodromy mΦ restricts to a diffeomorphism FΓ1 FΓ1 . Identifying FΓ1 with (Ff ∩T1 )×D, this diffeomorphism is given by (r, z) 7 (mf (r), z), where the monodromy mf of f is chosen so that it maps the set (Ff′ ∩ T1 ) to itself. (g) The monodromy mΦ restricts to a diffeomorphism FΓ2 FΓ2 which is isotopic to the identity map. (h) If (w, a) ∈ E(Γ) with a ∈ Ag and w ∈ V(Γ2 ), then the monodromy permutes the mw copies of 2-handles constructed in (d) cyclically. The following paragraphs describe general guidelines for notation used throughout the proof. Let w ∈ V(Γ). Pick a small disc-shaped neigborhood U ⊂ Ew and let m = mw , l = lw . If U ∩ Ev 6= ∅ for some v ∈ Vw = {v ∈ V : (w, v) ∈ E}, then let n = mv k = lv , otherwise let n = k = 0. By assuming that U and Tw are chosen small, there exist coordinates (u, v) ′ m n ′ l k and a function αp on b−1 w (U ) so that f

= u v and g = αp u v . Since α does not vanish, there exist positive real constants α1 , α2 so that 0 < α1 < |α| < α2 on b−1 w (Up ). These constants are independent of our choice of δ and ǫ Since ∪w∈W(Γ) Ew is compact, it is covered by finitely many such discs, say φ −1 (0) = ∪p∈P Up for some finite index set P . Let the coordinates on Up be n l k up , vp so that f = um p vp and g = αp up vp for the appropriate values of m, n, l, k. Since P is finite, we may assume that the functions αp , p ∈ P are all bounded by the same constants α1 , α2 . We choose 0 < ǫ ≪ δ ≪ 1. For the proof of (41)(a), let S1 = F ′ ∩ (T1 × C) We start with a lemma. 17 http://www.doksihu 4.2 The structure of FΓ1 Lemma 4.2 Let r ∈ T1 and z ∈ C with |z| ≤ δ and Φ′ (r, z) = ǫ Then |f ′ (r)| < 2ǫ and |x′ (r)|, |y ′ (r)| < δ. Proof. We may assume that r ∈ b−1 w (Up ) for some p ∈ P and w ∈ V(Γ1 ). We also assume that if

Up intersects some Ev with v 6= W then v ∈ / Af , V(Γ2 ). Write (u, v) = (up (r), vp (r)) and α = αp (r). In coordinates we have Φ′ (r, z) = um v n + zαul v k where m ≤ l and n, k = 0 or n ≤ k. Then the equation Φ′ (r, z) = ǫ gives um v n − ǫ = |z| ≤ δ. αul v k If |f ′ (r)| = |um v n | ≥ 2ǫ then the triangle inequality gives |um v n | − 12 |um v n | 1 um v n − ǫ ≥ ≥ |um−l v n−k | > δ αul v k |αul v k | 2α2 because |u|, |v| < 1, m ≤ l, n ≤ k and δ is chosen small independent of α2 . This shows that |f ′ (r)| = |um v n | < 2ǫ. The lemma is proven as soon as we prove the following claim: Claim: We can choose ǫ small enogh so that if |f ′ (r)| < 2ǫ then |x′ (r)| < δ and |y ′ (r)| < δ. To see this, note that the sets {t ∈ T1 : |f ′ (t)| > 2ǫ0 }, for all ǫ0 > 0, form an open cover of the compact set {t ∈ T1 : |x′ (t)| ≥ δ}. This implies that for ǫ small enough, we have {t ∈ T1 : |x′

(t)| ≥ δ} ⊂ {T ∈ T1 : |f ′ (t)| > 2ǫ}. The same holds if we replace x′ with y ′ . What the lemma says is that FΓ1 = {(r, z) ∈ T1 × C : Φ′ (r, z) = ǫ, |z| ≤ δ}. Since the function Z : T1 × C C, Z(r, z) = z behaves well when restricted to Φ′−1 (ǫ), it will give information on the set FΓ1 . Lemma 4.3 The derivative of the map Z : FΓ1 C, (r, z) 7 z is surjective at every point, so is the derivative of Z|F ′ ∩∂T1 . Proof. Let q ∈ FΓ1 Then q ∈ b−1 w (Up ) for some p ∈ P and w ∈ V(Γ1 ). On this neigborhood we have the coordinates (u, v) = (up , vp ). We will show that the map (r, z) 7 (v(r), z) will give coordinates on a neigborhood around q in FΓ1 . This is clearly enough to prove the first statement of the lemma. For this, we only need to show that ∂u Φ′ (q, z) 6= 0, since the result will then follow from the implicit function theorem. We get ∂u Φ′ (q, z) = ∂u (um v n + zαul v k ) = mum−1 v n + z((∂u α)ul v k +

αlul−1 v k ) = um−1 v n (m + zul−m v k−n (∂u αu + αl)). 18 http://www.doksihu We have u 6= 0, and we can have v = 0 only if q is on the strict transform of g. Since f and g have no common factors, this would mean that n = 0, and we get ∂u Φ′ (q, z) = mum−1 6= 0. Since m ≤ l and n ≤ k we have |zul−m v k−n (∂u αu + αl)| ≤ δ(|∂u α| + |α|) < m. The last inequality is valid because δ is chosen small with respect to |∂u α| and |α|. This shows that m + zul−m v k−n (∂u αu + αl) 6= 0, and thus ∂u Φ′ (q, z) 6= 0 This proves the first statement of the lemma. For the second statement, if q ∈ F ′ ∩ ∂T1 , then q ∈ b−1 w (∂Up ) for some p ∈ P1 , and we may assume that for some coordinate v in an neigborhood of bw (q) in Up , we have v(bw (q)) = 0 and that the image of v is {v : Rev ≥ 0, |v| < 1}. Then the previous argument shows that the map (r, z) 7 (Rev(r), z) provides coordinates in a neigborhood around q in F ′

∩ ∂T1 . This proves the second statement in the theorem Proposition 4.4 The map Z : FΓ1 D is a trivial fibration, whose fiber is diffeomorphic to Ff ∩ T1 . Proof. The map Z is clearly a proper map Therefore, lemma (43) and Ehresmann’s fibration theorem imply together that the map is a locally trivial fibration In fact this fibration is trivial, since D is contractible The fiber Z −1 (0) is the subset of T1 × {0} given by f ′ = ǫ, so it is diffeomorphic to Ff′ ∩ T1 . The proposition proves (4.1)(a) 4.3 Local picture close to the strict transform of f Let (w, a) ∈ E(Γ) for some w ∈ V(Γ1 ) and a ∈ Af . We have Te = b−1 w (Ue ) for some disc shaped neigborhood Ue around Ew ∩Ea in Ew . Choose a function u on Te so that ul = g ′ . This is possible since g ′ vanishes with multiplicity l = lw on Ew , but does otherwise not vanish on Te . Then it is possible to coose a function v so that f ′ = um v n on Te . We assume that Ue = {v : |v| < ρ} for some

ρ > 0, and that in these coordinates, Te is the polydisc {(u, v) : |u|, |v| ≤ ρ}. The number ρ should be thaught of as small, since the tubular neighborhood Tw is choosen small. However, ρ does not depend on the choice of δ, ǫ Thus, we have 0 ≪ ǫ ≪ δ ≪ ρ ≪ 1. If r is a point in Te so that g ′ (r) = 0, then u(r) = 0, so we have f ′ (r) + zg ′ (r) = 0. Otherwise, if g ′ (r) 6= 0, then the equation f ′ (r) + zg ′ (r) = ǫ can be solved for z. This means that the map F ′ ∩ (Te × C) Te , (r, z) 7 r is injective 19 http://www.doksihu Since this map is continous and its domain is compact, it is a homeomorphism onto its image Xe ⊂ Te . In fact, if we solve for z, we see that Xe = {r : |u|, |v| ≤ ρ, |um v n − ǫ| ≤ δ}. |u|l (2) We will show that Xe is diffeomorphic to I × S 1 × I. First, consider the set Xe1 = Xe ∩ {|v|n ≤ ǫ/ρm }. Let r be a point in Xe1 with coordinates (u, v), and denote by z(r) the unique point in C for which

(r, z) ∈ F ′ ∩ (Te × C). We get   ∂ um v n − ǫ ∂u z(r) = = (m − l)um−l v n + lǫu−l−1 ∂u ul = ((m − l)um+1 v n + lǫ)u−l−1 6= 0 since (assuming ρ ≤ l/|m − l|) |(m − l)um+1 v n | ≤ |(m − l)ρm+1 v n | ≤ |lρm v n | < lǫ This means that the function z : Xe1 C has surjective derivative everywhere, and so by the implicit function theorem, this means that the level set {|z| = δ} is a submanifold of Xe1 , and that the projection of this submanifold onto the v-axis is a submersion. Now the relative Ehresmann theorem implies that this projection Xe1 Dǫ/ρm is a locally trivial fiber bundle. But this bundle is trivial because Dǫ/ρm is contractible. The fiber of this fiber bundle is Xe1 ∩ {v = 0} = {(u, v) : |u| ≤ ρ, ǫ ≤ δ}, |ul | ∼ so there is a diffeomorphism Xe1 = {u : (ǫ/δ)1/l ≤ |u| ≤ ρ} × Dǫ/ρm . Now consider the set Xe2 = Xe ∩ {|v|n ≥ ǫ/ρm , |u| ≥ ρ/2}. If (u, v) are in this set, then we get: If t ∈

[0, 1] is such that |tv|n ≥ ǫ/ρ, then |um (tv)n − ǫ| |um v n − ǫ| ≤ . |u|l |u|l This follows from the a simple geometrical fact: If a and b are vectors (in C or Rn ), t ∈ [0, 1] and |tb| ≥ |a|, then |ta − b| ≤ |a − b|. To apply this we need to check: |um (tv)m | ≤ ρm ǫ/ρm = ǫ. This means that the map Xe2 Xe1 ∩ X2e , (u, v) (u, ǫ1/l v) ρm/l |v| is a strong homotopy retraction. We can extend this to a strong homotopy retraction Xe1 ∩ Xe2 X 1 . It is easy to check that the homotopy Xe × I Xe given by ( (t−n u, t−m v) if |t−n u| ≤ ρ, |tm v|n ≥ ǫ/ρm (u, v) 7 (u, v) otherwise 20 http://www.doksihu yields a strong homotopy retraction Xe Xe1 ∪ Xe2 . We have now constructed a strong retraction ∼ ◦ Xe Xe1 = (Dρ D(ǫ/δ) l ) × D(ǫ/ρm )1/l . It is easy to see from this retraction that Xe can be constructed by gluing I × I × S 1 to Xe1 by an embedding I × S 1 × {0} ∂Xe1 , and then gluing another copy of I × I × S 1 to

this space (Xe1 ∪ Xe2 ), again by an embedding I × S 1 × {0} ∂(Xe1 ∪ Xe2 ). We can therefore replace Xe by the set {(u, v) : η ≤ |u| ≤ ρ, |v| ≤ ρ}. This proves (32)(b) u ρ v η ρ Figure 4: A homotopy of Xe . 4.4 The structure of FΓ2 We have a map π1 : V × C V given by π1 (r, z) = r. The set F ′ can be given by explicitly the equations Φ′ (r, z) = ǫ, N (r) = k(x, y)k ≤ δ, |z| ≤ δ, where k · k is a norm on C2 . We may essentially choose any (smooth) norm for this purpouse. In particular, we may multiply the norm with a positive real constant which is fixed before choosing δ and ǫ. The multiplicity of g is always less then that of f on any divisor which intersects TΓ2 , and strictly less on φ−1 (0). Therefore, the function f /g is holomorphic on TΓ2 , and it vanishes on φ−1 (0) ∩ TΓ2 . That is, the order of vanishing of the function f /g is 1 or higher on the components of φ−1 (0) ∩ TΓ2 . For technical purposes, we scale the

norm function N (r) = k(x, y)k with a large enough constant so that the following holds: There exists a δ ∗ > 0 so that for any r ∈ TΓ2 with N (r) < δ ∗ we have |f ′ (r)/g ′ (r)| < N (r)/2. Afterwards, we assume that δ < δ∗. Since g does not vanish on TΓ2 φ−1 (0), the equation Φ(r, z) = ǫ can be solved for z with r ∈ TΓ2 φ−1 (0). That is, the restriction π1 |FΓ′ ′ is injective 21 http://www.doksihu Since its domain is compact, it is a homeomorphism onto its image. This shows that FΓ′ ′ is homeomorphic to the set {r ∈ TΓ′ : φ(r) 6= 0, N (r) ≤ δ, |z(r)| ≤ δ}. Here z(r) is the unique point z(r) ∈ C for which Φ(r, z(r)) = ǫ. We know that {r ∈ TΓ′ : N (r) ≤ δ} (3) is a tubular neighborhood around φ−1 (0) ∩ TΓ2 in TΓ2 . What we need to show is that {r ∈ TΓ′ : φ(r) = 0 or |z(r)| > δ} (4) is a tubular neighborhood around φ−1 (0) inside (3). Around any point r in φ−1 (0) ∩ TΓ2 , the

function |z|−1 can be described in real coordinates (x1 , x2 , x3 , x4 ) as (x21 + x22 )n (x21 + x22 )k for some integers n > 0, k ≥ 0. If r ∈ φ−1 (0) ∩ ∂TΓ2 , then we may assume that r has coordinates (0, 0, 0, 0) and that ∂TΓ2 is given as x4 = 0. Using Morse theory, the following claim will prove that (4) is in fact a tubular neighborhood around φ−1 (0) inside (3), proving (3.2)(c) Claim. For any point r ∈ TΓ2 , we have either |z(r)|−1 ≥ δ −1 , or that the derivative of |z|−1 (and |z|−1 |∂TΓ2 if r ∈ ∂TΓ2 ) at r is surjective. Proof. We choose the constants δ, ǫ as before, ie 0 ≪ ǫ ≪ δ ≪ 1 Let r ∈ TΓ2 . Choose a neighborhood around r with coordinates u, v so that f = um v n , g = αul v k , where α is a non-vanishing holomorphic function (as before, we can start by choosing finitely many such charts covering TΓ2 , allowing us to have a universal lower bound for |α|, and an upper bound for |∂u α|). The integers m, n, l, k

satisfy m > l > 0 and n > k ≥ 0 or n = k = 0. If r ∈ ∂TΓ2 , then we assume that the boundary is given in coordinates as Rev = 0. First consider the case when |g ′ (r)| < 2ǫ/δ. This gives |ul v k α| < 2ǫ/δ In case that k = 0, we get |u|l |α| < 2ǫ/δ. If however k 6= 0, then we have n 6= 0, and we can reverse the roles of u and v if neccesary and get |ul+m α| ≤ 2ǫ/δ|α|. In any case, we assume that |ul+m α| ≤ 2ǫ/δ (the main point is to bound |u| by something which depends on ǫ, making the bound small with respect to any other variable). This yields in particular |mum | ≤ δ|(∂u α)u − αl| (5) if ǫ is small enough with respect to δ and |α|. Notice also that since we insist on |u| being small, the magnitude of |(∂u α)u − αl| is essentially |αl|, which is bounded below by a constant which does not effect the choice of ǫ. 22 http://www.doksihu Now, assuming that |z(r)|−1 < δ −1 (and |z(r)| 6= ∞), we want to show

that   ∂z ∂ um v n − ǫ (r) = 6= 0. (6) ∂u ∂u αul v k From this it follows that the derivative of |z|−1 at r is surjective. We have   ∂ um v n − ǫ mum−1 v n αul v k − (um v n − ǫ)((∂u α)ul − αlul−1 )v k = . ∂u αul v k (αul v k )2 In order to prove that the right hand side of this equation does not vanish, we will show that mum−1 v n αul v k > (um v n − ǫ)((∂u α)ul − αlul−1 )v k . (7) By assumption we have |z|−1 < δ −1 , that is, |um v n − ǫ| > δ|αul v k |. In order to prove (7), it is therefore enough to prove |mum−1 v n αul v k | > |δαul v k ((∂u α)ul − αlul−1 )v k |. By cancellation, this is eqivalent to |mum−l v n−k | ≤ δ|(∂u α)u−αl| which follows from (5) (since |v|n−k ≤ 1). This finishes the case when |g ′ (r)| < 2ǫ/δ In the case when |g ′ (r)| ≥ 2ǫ/δ we use the inequality |z(r)| = f ′ (r) − ǫ ǫ f ′ (r) ≤ − . g ′ (r) g ′ (r) g ′ (r) In the

beginning of this section we concluded that unless N (r) > δ, we had |f ′ (r)/g ′ (r)| < δ/2, and by assumption we have |ǫ/g ′ (r)| < δ/2. This gives |z(r)| < δ, and the claim has been proved. 4.5 Local picture close to the strict transform of g Let w ∈ V(Γ2 ) and a ∈ Ag (Γ) so that (w, a) ∈ E(Γ). We pick a disc shaped coordinate neigborhood Ue around Ew ∩ Ea in Ew , and look closely at the set Te = b−1 w (Up ). We may assume that we have coordinates (u, v) on Te with the properties f = um and g = ul v k , where m = mw , l = lw and k = la . We have ma = 0 since f does not vanish on Ea . For convenience, we will identify points in Te with their coordinates (u, v). Via this correpondance, we may assume that Te is precisely the set {(u, v) : |u|, |v| ≤ ρ} for some positive real number ρ. As in the previous subsections, we consider the coordinate map F ′ ∩ (Te × C) Te , (r, z) 7 r. Call this map π1 Unfortunately, this map will not be

injective. However, if we restrict π1 to some subset where v 6= 0, we will get an injective map. Let 0 < η < ρ The map F ′ ∩ ((Te {|v| ≥ η}) × C) Te , 23 http://www.doksihu (r, z) 7 r is injective, in fact it is a diffeomorphism onto its image. The set F ′ ∩ (Te × C) is the union of two subsets: F ′ ∩ (Te × C) ∼ = ({(u, v, z) ∈ F ′ : η ≤ |v| ≤ ρ}) ∪ ({(u, v, z) ∈ F ′ : |v| ≤ η}) = Aη ∪ Hη . Here, Aη is mapped diffeomorphically onto its image in Te by π1 . In coordinates, this image is determined by the equations um + zul v k = ǫ and k(x′ , y ′ )k < δ. The first equation translates to |um − ǫ|/|ul v k | ≤ δ, and the second equation can be replaced by |u| ≤ δ. We get   um − ǫ π1 (Aη ) = (u, v) : |u| ≤ δ, ≤ δ, η ≤ |v| ≤ ρ . ul v k (8) Let (u, v) ∈ π1 (Aη ) ∩ Te . For any v ′ with |v| ≤ |v ′ | ≤ ρ it is clear from (8) that (u, v ′ ) ∈ π1 (Aη ). This shows that π1

({(u, v, z) ∈ F ′ : |v| = ρ} is a strong deformation retract of π1 (Aη ). In fact, this shows that there is a diffeomorphism F ′ π1−1 (Hη ) (F ′ Te ) × C which takes (u, v) ∈ F ′ with |v| = η to (u, ρv/|v|). Thus, for any η ≤ ρ, we have F ′ = (Hη ∐ F ′ (Te × C))/λ, where λ is the map λ : {(u, v) ∈ Hη : |v| = η} F ′ (Te × C), (u, v) 7 (u, ρv/η, z(u, v)). Here, z(u, v) is the unique z ∈ C so that (u, v, z) ∈ F ′ . If η is small, then the set Hη has a particularly nice description. We will show that the map Hη Dη × Dδ , (u, v, z) 7 (v, z) is an m-covering. The map is obviously proper, and it maps ∂Hη to ∂(Dη × Dδ ). Therefore, we only need to show that it has surjective derivatives at all points, and that the same holds for points on the boundary. For this, it is enough to show that ∂u Φ′ 6= 0 on Hη . From this, the implicit function theorem implies that (v, z) will locally give coordinates on Hη , (v, arg z)

will give local coordinates on {|z| = δ} and (arg v, z) will give local coordinates on {|v| = η}. We get ∂u Φ′ = ∂u (um + zul v k ) = mum−1 + zlul−1 v k . Since Te is compact, and u does not vanish on F ′ ∩(Te ×C), the function mum−1 is bounded below by a positive constant. Similarly, the function zlul−1 is bouded above. We can therefore choose η so small that |mum−1 | > |zlul−1 v k | whenever |v| < η. From this we get ∂u Φ′ 6= 0 whenever |v| ≤ η This proves that the map (v, z)|Hη is a covering map. Now we only need to observe that ((v, z)|Hη )−1 (0) = {(u, v, z) : v = 0, z = 0, um + zul v k = ǫ, |z| ≤ δ} = {(u, 0, 0) : um = ǫ, |z| ≤ δ}. 24 http://www.doksihu This shows that the fibers of (v, z) consist of m points. The cover is trivial because the disc Dη × Dδ is contractible. Therefore, Hη = ∐m (D × D) The set Hη is glued to F ′ (Te × C) by a diffeomorphism ∐m (D × ∂D) ∂(F ′ (Te × C)). In other words,

we attach m copies of 2-handles to F ′ (Te × C) Let us fix one handle h, which has coordinates given by v, z, i.e we have ∼ h = {(v, z) : |v| ≤ η, |z| ≤ δ}. Let λh : {(v, z) : |v| = η, |z| ≤ δ} {(u, v) : |u| ≤ δ, |v| = ρ} be the attaching map. Then λ({(v, 0) : |v| = η}) = {(u0 , v) : |v| = δ} for some fixed u0 for which um 0 = ǫ. The function z gives a parametrization of the fibers of the tubular neigborhood (ηS 1 ) × Dδ of the submanifold (ηS 1 ) × {0} (thought of as a submanifold of ∂h). This function induces the canonical framing on the handle h. The function u−u0 parametrizes the fibers of a tubular neigborhood of the image (ρS 1 ) × {u0 } which is thought of as a submanifold of ∂(F ′ Te ). This function induces the framing of the attaching circle which was discussed in section (3). For some small η ′ > 0, the section (v, 0) 7 (v, η ′ v −k ) of the tubular neigborhood of (ηS 1 ) × {0} ⊂ (ηS 1 ) × Dδ corresponds to a

non-vanishing section of the corresponding normal bundle. It can be thought of as the first element of a basis of each fiber, i.e a framing This framing will be the (−k)-th framing of the handle as discussed in (2.2) Write λ(v, η ′ v −k ) = (u0 + u1 (v), δv/η) Then we have the formula (u0 + u1 (v))m − ǫ = −v −k ((u0 + u1 )l v k ) = −(u0 + u1 )l (9) because the coordinates (u0 +u1 , v, v −k ) correspond to an element of F ′ . Since η ′ is chosen small, we may assume that |u1 | is small with respect to to |u0 | = ǫ1/m . In particular, we may assume that for any fixed choice of κ > 0 we have Arg (u0 + u1 (v))m − ǫ −(u0 + u1 (v))l = Arg ∈ (−κ, κ) ul0 ul0 mod 2π for all v ∈ ηS 1 (here, Arg : C∗ R/(2π)Z is the standard argument function). But we have for u1 = u1 (v)   m   m−2 X X m j m−j−2 m j m−j  2 (u0 + u1 )m − ǫ =  u0 u1 − ǫ = mum−1 u + u u0 u1 1 1 0 j j j=0 j=0 (10) because um 0 = ǫ. Assuming again

that |u1 (v)| is small with respect to |u0 | we Pm−2
j m−j−2 can make |u21 j=0 m | small with respect to |mum−1 u1 |. Therefore, 0 j u0 u1 we may assume that Arg (u0 + u1 (v))m − ǫ ∈ (−κ, κ) mum−1 u1 (v) 0 25 mod 2π (11) http://www.doksihu Combining (10) and (11) we get Arg mu0m−1 u1 (v) ∈ (−2κ, 2κ) ul0 mod 2π which means that the variable u1 (v) does not wind around the origin. Phrased differently, the loop I C∗ , t 7 u1 (e2πit ) is null-homotopic. In the language of framings, this means that the if we complete the section corresponding to u0 + u1 (v) in the normal bundle of ρS 1 × {u0 } ⊂ ∂(F ′ Te ) to a (positive) basis on all fibers, then we get a framing which is equivalent to the framing already constructed. This means that the handle h is in fact attached with the (−k)-th framing. 4.6 Edges connecting Γ1 and Γ2 Let v1 ∈ V(Γ1 ) and v2 ∈ V(Γ2 ) so that e = (v1 , v2 ) ∈ E(Γ). We want to describe F ′ ∩ (Te ×

C). The function g ′ does not vanish on this set As in the previous subsections, we can project it down to a subset of Te . Thus, F ′ ∩ (Te × C) is diffeomorphic to {r ∈ Te : |x′ (r)|2 + |y ′ (r)|2 ≤ δ, |z(r)| ≤ δ}. Here, z is the function which takes r in the projection of F ′ ∩ (Te × C) to the unique point z ∈ C so that Φ′ (r, z(r)) = ǫ. We use the notation m = mv1 , l = lv1 , n = mv2 , k = lv2 . By our choice of v1 and v2 we have m ≤ l and n > k. Lemma 4.5 There exist coordinates (u, v) on Te so that in these coordinates f ′ = um v n and g ′ = ul v k . n Proof. We can find coordinates (u0 , v0 ) on Te so that f = um 0 v0 . In these co- ordinates, we have g = αul0 v0k for some function α, which does not vanish on Te . Since mk − nl < 0, there exits a function β1 on Te so that β1mk−nl = αn Let β2 be a function so that β2n = β1−m . Let u = β1 u0 and v = β2 v0 Then n m n m n f = um = um v n , and g = αul0 v0k = αβ1l β2k ul

v k . We have 0 v0 = β1 β2 u v (β1l β2k )n = β1nl β2nk = β1nl β1−mk = αn . Thus, αβ1l β2k is a constant function, whose value is an n-th root of unity. By choosing our functions β1 , β2 carefully, we may assume that this constant is 1. This proves the lemma From lemma (4.5), we see that the projection of F ′ ∩(Te ×C) to Te is precisely the set Fe =   um v n − ǫ (u, v) : |u|, |v| ≤ ρ, |x′ (u, v)|2 + |y ′ (u, v)|2 ≤ δ, ≤ δ . ul v k We need to describe this set, and how it glues to to F ′ (Te × C). From the theory of plumbed 4-manifolds, and how the Milnor fiber of plane curves can be 26 http://www.doksihu ∼ embedded into these manifolds, we have Ff′ ∩ Te = ∐(m,n) S ′ × I. Also, ∂Te is the union of two solid tori. For any of the connected components L ⊂ Ff′ ∩ Te we have ∂L = S 1 ∐ S 1 . One of these circles lies inside each of the tori Choose a tubular neigborhood N around Ff′ ∩ Te in Te so that N ∩ ∂Te is a

tubular neigborhood around Ff′ ∩ ∂Te in ∂Te . Clearly, N is a trivial bundle, ie we have a diffeomorphism N ∐(m,n) S ′ × I × D. Let κ : N ∐(m,n) S 1 × D be the projection which simply disregards the second coordinate. Then any fiber of κ contains exactly two points in ∂Te , one is in ∂Tv1 and the other in ∂Tv2 , corresponding to the partition of ∂Te to two tori. The following proposition reformulates the statement (4.1)(e): Proposition 4.6 The Milnor fiber F is diffeomorphic to F ′ (Te × C)/κ In other words, we may diregard the interior of F ′ ∩ (Te × C), and simply glue together the two points (r, z), (r′ , z ′ ) ∈ F ′ ∩ (∂Te × C) if κ(r) = κ(r′ ). This procedure will not alter the diffeomorphism type of F ′ Proof. Let us consider the set Fe,θ,φ = {(u, v) ∈ Fe : Argu = θ, Argv = φ}, Then Fe,θ,φ can be identified with a subset of [0, ρ] × [0, ρ], via (t, s) ↔ (|u|, |v|) for (u, v) ∈ Fe,θ,φ . Using simple

calculus it is easy to see that Fe,θ,φ has exactly one component which is simply connected, and that {(u, v) ∈ Fe,θ,φ : |v| = ρ} is a closed interval i.e a one dimensional compact connected manifold with nonempty boundary. Thus, the pair (Fe,θ,φ , {(u, v) ∈ Fe,θ,φ : |v| = ρ}) is topologycally the same as ([0, 1], {1} × [0, 1]) We can extend this description to all of C, namely (C, {(u, v) ∈ C : |v| = ρ}) = (S 1 × S 1 × [0, 1] × [0, 1], S 1 × S 1 × [0, 1] × {1}). The inclusion F ′ ∩ ((Tv2 Te ) × C) F ′ ∩ (Tv2 × C) is therefore isotopic to a diffeomorphism h : F ′ ∩ ((Tv2 Te ) × C) F ′ ∩ (Tv2 × C). We may even assume that if h(r, z) = (r′ , z ′ ), and r ∈ N , then κ(r) = r′ , and that h is the identity outside some small neigborhood around Te × C. Thus, h extends to a diffeomorphism which satisfies the required properties. 4.7 The monodromy Let E = {(x, y, z, λ) ∈ C3 × S 1 : k(x, y, z)k ≤ δ, Φ(x, y, z) = λǫ}. We want

to describe the monodromy of the bundle µ : E S 1 , (x, y, z, λ) 7 λ. We have a map p : E V , (x, y, z, λ) 7 φ−1 (x, y). Here, φ : V C2 is the 27 http://www.doksihu common embedded resolution of f and g as before, and φ−1 (x, y) is well defined because (x, y) 6= (0, 0). We split E into three parts: Let E1 be the preimage of the union of T1 and Te for any e = (v, a) ∈ E(Γ) with v ∈ V(Γ1 ) and a ∈ Af . Let E2 be the preimage of the union of T2 and Te for any e = (v1 , v2 ) ∈ E(Γ) with v1 ∈ V(Γ1 ) and v2 ∈ V(Γ2 ). Let E2 be the preimage of the union of all Te where e = (v, a) ∈ E(Γ) with v ∈ V(Γ2 ) and a ∈ Ag . Explicitly, we have E1 = p−1 (T1 ∪ (∪e∈EΓ (Γ1 )EΓ (Γ2 ) Te )) E2 = p−1 (T2 ∪ (∪e∈EΓ (Γ1 )∩EΓ (Γ2 ) Te )) E3 = p−1 (∪e∈EΓ (Γ1 )∩EΓ (Γ2 ) Te ). Restricting µ to any of these will yield a subbundle of E, whose fibers are described in (4.2)-(46) We have F = µ−1 (1) = ∪3j=1 (µ−1 (1) ∩ Ej )

The sets Ej are actually subbundles of E, so we can assume that the monodromy mΦ of the bundle E acts on each of the smaller fibers µ−1 (1) ∩ Ej individually, that is, these sets are invariant under mΦ . By (4.2) we have µ−1 (eit ) ∩ p−1 (T1 ) = f −1 (ǫeit × D From this description it is clear that up to homotopy, we have mΦ (r, z) = (mf (r), z) for any r ∈ FΓ1 , where mf is chosen to map Ff ∩ T1 to it self. For any edge e = (w, a) ∈ E(Γ) with w ∈ V(Γ1 ) and a ∈ Af (Γ), consider p −1 ∼ (Te ). Using the description of F ′ ∩ (Te × C) from (43), we have p−1 (Te ) = (Dρ Dη◦ ) × Dρ ) × S 1 , so the monodromy acts (up to isotopy) trivially on F ′ ∩ (Te × C). To describe the monodromy action of the bundle K2 , it is enough to look at p −1 (T2 ), since the inclusion F ′ ∩ (T2 × C) ֒ F ′ ∩ ((T2 ∪ (∪e∈EΓ (Γ1 )∩EΓ (Γ2 ) Te )) × C) is a homotopy equivalence. From the description of F ′ ∩ T2 in (44) we see

that for any t the set there is a diffeomorphism F ′ ∩ (T2 × C) {r ∈ T2 : δ/2 ≤ k(x′ (r), y ′ (r))k ≤ δ}. Using this diffeomorphism on each fiber we obtain a trivialization K2 = (F ′ ∩ (T2 × C)) × S 1 . This means that restricting mΦ to F ′ ∩ (T2 × C) we get a map which is isotopic to the identity. We have seen that the complement F ′ (∪e Te × C), where the union ranges over edges e = (w, a) ∈ E(Γ) with w ∈ V(Γ2 ) and a ∈ Ag (Γ), is invariant under mΦ . Therefore, F ′ ∩ (∪e Te × C) is also invariant under mΦ In fact, it is easy to see that the set Hη , defined in (4.5) is invariant under mΦ The set 28 http://www.doksihu {(r, z) ∈ Hη : z = 0} = ∐mw D is naturally identified with a subset of Ff , and it is a strong homotopy retract of Hη . The monodromy mf permutes these discs cyclically, and restricting mΦ to the set {V ×{0} ⊂ V ×C we get the monodromy of f . This shows that mΦ must permutes the handles Hη = ∐mw

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59(2):922–938, 1999 [9] A. Némethi and Á Szilárd The boundary of the Milnor fibre of non-isolated hypersurface surface singularities, 2009. arxiv:09090354 [10] W. D Neumann A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves. Trans of Amer Math Soc., 268(2):299–344, 1981 [11] D. Siersma The vanishing topology of non isolated singularities New Developments in Singularity Theory (Cambridge 2000), pages 447–472, 2001 29