Some topological invariants of isolated hypersurface singularities

by András Némethi

The Ohio State University, Columbus, USA

EMS Summer School – Eger (Hungary)

29 July – 9 August, 1996

§0. Introduction.

This manuscript was prepared for the students of the EMS Summer School, Eger (Hungary, 1996). It contains the five lectures of the author, but also a lot of additional material. Each section ends with a list of open problems/conjectures and with a large number of exercises (some of them with hint or with solution). The exercises constitute an important part of the note, many key results of the hypersurface singularities are presented here. The reader can find here some important properties of Brieskorn singularities, quasi–homogeneous singularities and some quotient singularities; some historically crucial examples (like A’Campo’s example for a singularity with non–finite monodromy); the description of the invariants in terms of the resolution graphs; connections with classical topology, realization of the Poincaré sphere, lens spaces, and some exotic spheres; discussions about Casson invariant and some applications of the Dedekind sums; and many others. Many exercises, and especially the last sections, deal with plane curve singularities.

The theorems/remarks/exercises with * are considered difficult, or are presented without any proof. We included them in order to provide a more correct global picture of the theory.

On the other hand, this note is very far to cover all important aspects of the theory of hypersurface singularities. Nevertheless, we hope that it will become a useful guide for beginners.

Contents:

§1. The analytic and topological type. The link. §2. The Milnor fiber, the monodromy and the variation map. §3. The Milnor fibration and the Seifert form. §4. The equivariant signature of plane curve singularities. ϵ–hermitian variation structures. §5. The signature and Dedekind sums.

§1. The analytic and topological type. The link.

1.1. The basic object of our series of lectures is a germ of an analytic function f : (C^{n+1}, 0) → (C, 0). The germ f defines its zero set (V_{f}, 0) = ({f = 0}, 0) ⊂ (C^{n+1}, 0), which is a germ of an analytic set.

In the theory of hypersurface singularities, the most common equivalence relation is the R (=right)-equivalence. 1.2. Definition. Two germs g, f : (C^{n+1}, 0) → (C, 0) are called R-equivalent, if there exists a biholomorphic germ φ : (C^{n+1}, 0) → (C^{n+1}, 0) such that f = g ∘ φ. Notation:

f ∼^{R} g.

1.3. Example. Assume that ∂f(0) := (∂_{1}f(0), …, ∂_{n+1}f(0)) ≠ 0. (Here ∂_{i}f = ∂f/∂z_{i}.) Then, by the submersion theorem, f ∼^{R} (z → z_{1}). These germs are exactly the “smooth germs”, i.e. if ∂f(0) ≠ 0, then 0 is a smooth point of V_{f}.

A germ f with ∂f(0) = 0 is called “singular germ”.

The singular locus of the space–germ (V_{f}, 0) is given by Sing(V_{f}) := {z : ∂f(z) = 0}. The germ f is called isolated singularity if Sing(V_{f}) = {0}. This is the case if and only if dim O/(∂f) < ∞, where O is the local C–algebra of analytic germs (C^{n+1}, 0) → (C, 0), and (∂f) is the ideal generated by the partial derivatives ∂_{i}f, i = 1, …, n + 1. (cf. Exercise 1.2) 1.4. Definition. The algebra M(f) := O/(∂f) is called the Milnor algebra of the isolated singularity f, and its dimension μ(f) is the Milnor number of f. 1.5. Some examples of isolated hypersurface singularities. f : (C^{n+1}, 0) → (C, 0) is called:

a) Brieskorn singularity if f = ∑_{i=1}^{n+1} z_{i}^{a_{i}} (a_{i} ≥ 2);

b) homogeneous singularity of degree d if f = ∑_{α} a_{α}z^{α}, where z^{α} = z_{1}^{α_{1}} ⋯ z_{n+1}^{α_{n+1}}, and ∑ α_{i} = d if a_{α} ≠ 0.

c) quasi–homogeneous singularity of weights w_{1}, …, w_{n+1} (w_{i} ∈ Q, w_{i} ≥ 2) if f = ∑_{α} a_{α}z^{α}, such that if a_{α} ≠ 0 then ∑_{i=1}^{n+1} α_{i}/w_{i} = 1.

Now, we will introduce another equivalence relation. 1.6. Definition. We say that two isolated singularities f and g have the same analytic type if there is a biholomorphic map φ : (C^{n+1}, 0) → (C^{n+1}, 0) such that φ(V_{f}) = V_{g}.

Its topological analogue is the following: 1.7. Definition. We say that two isolated hypersurface germs f and g have the same topological type if there is a homeomorphism φ : (C^{n+1}, 0) → (C^{n+1}, 0) such that φ(V_{f}) = V_{g}; i.e. if (C^{n+1}, V_{f}, 0) is homeomorphic to (C^{n+1}, V_{g}, 0).

Some major questions in the theory of hypersurface singularities are: 1.8. Problems.

a) Classify the germs modulo R–equivalence;

b) Give simple algebraic criterion for (C^{n+1}, V_{f}, 0) to be biholomorphic (respectively, homeomorphic) to (C^{n+1}, V_{g}, 0).

c) (Or, at least) find invariants of the equivalence classes.

1.9. Examples.*

Set A_{1}(f) = M(f), A_{2}(f) = O/(∂f) + (f), and A_{3}(f) = O/integral closure of (∂f). By a theorem of Briançon and Skoda [8], f^{n+1} ∈ (∂f), in particular, A_{1}(f) = M(f) is a C\{t\}/(t^{n+1})–algebra via the multiplication t · ū = \overline{fu}.

a) [Yau] [56], (see also [Scherk] [51]) If A_{1}(f) is isomorphic to A_{1}(g) as C{t}/(t^{n+1})- algebra, then f ∼^{R} g.

b) [Mather and Yau] [25] If A_{2}(f) is isomorphic to A_{2}(g) as C–algebra, then f and g have the same analytic type.

c) [Lê–Ramanujan] [21] If for some i (1 ≤ i ≤ 3) A_{i}(f) is isomorphic to A_{i}(g) as a C–algebra, then f and g have the same topological type.

d) [Lê–Ramanujan] [21] Let f_{t} (t ∈ [0, 1]) be a C^{∞}–family of isolated singularities, such that μ(f_{t}) is constant. Then the topological type of f_{t} is constant, provided that n ≠ 2.

The first geometrical invariant (in any classification) is the link of f. We will use the following notations: B_{ϵ} = {z ∈ C^{n+1} : ||z|| ≤ ϵ}, and S_{ϵ} = ∂B_{ϵ} (where ϵ > 0). 1.10. Theorem. [Milnor] [26]

Fix a germ f as above which defines an isolated singularity at the origin. Then there exists an ϵ_{0} > 0 such that for any 0 < ϵ ≤ ϵ_{0} one has:

a) The sphere S_{ϵ} meets V_{f} transversely, in particular, K_{f} = S_{ϵ} ∩V_{f} does not depend on the choice of ϵ, it is a (2n−1)–dimensional C^{∞}–manifold. \underline{By definition, K_{f} is the “link of f”.}

b) (B_{ϵ}, B_{ϵ} ∩V_{f}) is homeomorphic to (cone(S_{ϵ}), cone(K_{f})); in particular, the embedding K_{f} ⊂ S_{ϵ} determines the topological type of f completely.

(Above cone(Z) := [0, 1] × Z/(0, z) ∼ (0, z′).) Proof. For simplicity, we will assume that f is a polynomial function (for the analytic case see Exercise 1.10).

Consider the function r : V_{f} → [0, ∞) given by r(z) = ||z||^{2}. Then the set of critical values of r is a real algebraic (constructible) subset of R of dimension zero, hence it is finite. Set ϵ_{0} such that 0 < ϵ_{0}^{2} <the smallest critical value of r. Hence a) follows. For b), we can construct a vector field v(z) on B_{ϵ_{0}} \ {0} (first locally, then gluing the local vector fields by a partition of unity), such that ⟨v(z), z⟩ > 0, and for z ∈ V_{f} \ {0} the vector field v(z) is tangent to V_{f}. Then, we normalize it by w(z) = v(z)/⟨2z, v(z)⟩. Consider the differential equation dz/dt = w(z). Given a solution p(t), one has dp(t)/dt = w(p(t)), hence dr(p(t))/dt = 1, so we can choose the parameter t such that r(p(t)) = t. For a ∈ S_{ϵ_{0}}, let p(a, t) be the solution which satisfies the initial condition p(a, ϵ_{0}^{2}) = a. Then p : S_{ϵ_{0}} × (0, ϵ_{0}^{2}] → B_{ϵ_{0}} \ {0} is a diffeomorphism such that p(K_{f} × (0, ϵ_{0}^{2}]) = V_{f} ∩ B_{ϵ_{0}} \ {0} (K_{f} = S_{ϵ_{0}} ∩ V_{f}). Since p(a, t) tends uniformly to zero as t → 0, p extends to a homeomorphism cone(S_{ϵ_{0}}) → B_{ϵ_{0}} via [s, a] ↦ p(a, sϵ_{0}^{2}) which identifies cone(K_{f}) with V_{f} ∩ B_{ϵ_{0}}. □

1.11. Definition. Two isolated singularities f and g are called link–equivalent if (S_{ϵ}, K_{f}) is homeomorphic to (S_{ϵ′}, K_{g}) for all ϵ and ϵ′ sufficiently small. 1.12. Remark.*

a) By Exercise 1.10.c, if f ∼^{R} g, or if f and g have the same analytic type, then f and g are link–equivalent.

b) Obviously, by theorem 1.10, link–equivalence implies the topological equivalence. Saeki in [49] proved that also the converse is true; in particular, the topological type of f is equivalent to the embedding K_{f} = V_{f} ∩ S_{ϵ} ↪ S_{ϵ}. 1.13. Example.* The case n = 1.

a) Assume that f : (C^{2}, 0) → (C, 0) is an irreducible germ. Let n : (D, 0) → (V_{f}, 0) be the normalization of (V_{f}, 0). With a good choice of the local coordinates t of (D, 0) and (x, y) of (C^{2}, 0) one has: n(t) = (x(t), y(t)) = (t^{n}, ∑_{i>n} a_{i}t^{i}). Therefore, we can write:

y = ∑_{i>n} a_{i}x^{i/n} (the Puiseux expansion of f).

Consider the fractions {i/n}_{i>n,a_{i}≠0}, and rewrite them in an increasing order, in the form:

⋯  < \frac{m_{1}}{n_{1}} < ⋯ < \frac{m_{2}}{n_{1}n_{2}} < · · · · · · < ⋯ < \frac{m_{g}}{n_{1}···n_{g}} < ⋯
integers  type q/n_{1}  type q/n_{1}n_{2}  type q/n_{1} ⋯ n_{g}

where g.c.d.(m_{i}, n_{i}) = 1 and n_{i} > 1 for all i = 1, …, g.

The pairs of integers (m_{i}, n_{i})_{i=1}^{g} are called the Puiseux pairs of f. Let f_{P} be the germ which has the “standard expansion” y(x) = ∑_{i=1}^{g} x^{m_{i}/n_{1}···n_{i}}.

Then, by a result of K. Brauner [7], W. Burau [11] and O. Zariski [57]: f has the same topological type as f_{P}, and f_{P} (i.e. the set of Puiseux pairs) depends only on the topological type of f.

The link K_{f} is diffeomorphic to S^{1}, in particular the abstract manifold K_{f} contains very little information about the topological type. On the other hand, the knot K_{f} ⊂ S_{ϵ} contains all the information about the topological type: K_{f} ⊂ S_{ϵ} is an iterated torus knot (associated with the Puiseux pairs of f).

b) If f has r irreducible components, then K_{f} =disjoint union of r copies of S^{1}’s. By a result of M. Lejeune [22] and O. Zariski: the topological type of f is determined by the topological type of each irreducible component of f, and by all the intersection multiplicities of the pairs of these components.

To complete the discussion, we mention a theorem of J. Reeve [48], which asserts that the intersection multiplicities at the origin of two germs is exactly the linking number (in S_{ϵ}) of their links (which is obviously a topological invariant).

1.14. Resolution graphs.

We generalize the notion of isolated singularity of plane curves f : (C^{2}, 0) → (C, 0) to the following situation. Let (X, x) be a normal surface singularity, and let f : (X, x) → (C, 0) be the germ of an analytic function which defines a one–dimensional isolated singularity. We consider an embedded resolution ϕ : (Y, D) → (X, f^{−1}(0)) of (f^{−1}(0), x) ⊂ (X, x). This means that the restriction ϕ : Y \ ϕ^{−1}(x) → X \ {x} is biholomorphic and the divisor ϕ^{−1}(f^{−1}(0)) in Y has only normal crossing singularities. Let E = ϕ^{−1}(x) be the exceptional divisor and E = ∪_{w∈W}E_{w} be its decomposition in irreducible divisors. Similarly, let ∪_{a∈A}S_{a} be the irreducible decomposition of the strict transform S of f^{−1}(0). Then D = E ∪ S. Let G_{f} be the resolution graph of f (associated with ϕ), i.e. its vertices V = W ∐ A consist of the “nonarrowhead” vertices W (corresponding to E_{w}’s), and “arrowhead” vertices A (corresponding to S_{a}’s). We will assume that no E_{w} has self intersection, W ≠ ∅, and #E_{α} ∩ E_{β} ≤ 1 for any α, β ∈ W, α ≠ β. If two irreducible divisors, corresponding to v_{1}, v_{2} ∈ V have an intersection point, then (v_{1}, v_{2}) = (v_{2}, v_{1}) is an edge of G_{f} connecting v_{1} and v_{2}. The set of edges connecting two nonarrowheads is denoted by E. For any w ∈ W, we denote by V_{w} the set of vertices v ∈ V adjacent to w.

The graph G_{f} is decorated by the self intersection (or Euler–) numbers e_{w} := E_{w} · E_{w} for any w ∈ W. The genus of E_{w} is denoted by g_{w} (w ∈ W).

The intersection matrix (A_{αβ}) (α, β ∈ W) defined by A_{αβ} = e_{α} if α = β, and = 1 if α ≠ β, (α, β) ∈ E; and = 0 otherwise. By a result of Grauert [17]:

(1.15) (A_{αβ}) is negative definite (in particular, non-degenerate).

For any v ∈ V, let m_{v} be the order of vanishing of f ∘ ϕ along the irreducible divisor corresponding to v. In particular, m_{a} = 1 for any a ∈ A. These multiplicities, for any w ∈ W, satisfy:

(1.16) e_{w} · m_{w} + ∑_{v∈V_{w}} m_{v} = 0.

By (1.15) and (1.16), the Euler numbers {e_{w}}_{w} determine the multiplicities completely.

The resolution graph G(X) of (X, x) (without a map–germ f) is obtained from (any) G_{f} by deleting its arrows and the multiplicity structure {m_{w}}_{w}. 1.13’. Example.* The case n = 1 (continued).

The following pieces of data, concerning of a plane curve singularity, are equivalent:

(a) the topological type – characterized by the iterated torus knots corresponding to the irreducible components of f, and their linking numbers;

(b) the Puiseux pairs of the irreducible components and the intersection multiplicities of the irreducible components with each other;

(c) any embedded resolution graph of f.

For plane curve singularities the resolution graph is special, e.g.: each E_{w} is rational, G_{f} is a tree, and (A_{αβ}) is unimodular (cf. Exercise 1.16).

1.17. Example.* The case n = 2.

The germ (V_{f}, 0) ⊂ (C^{3}, 0) is a normal surface singularity (cf. E.1.14), in particular has a resolution graph G(V_{f}). By a topological construction, the link K_{f} can be recovered from the graph G(V_{f}): K_{f} can be constructed as a (connected) plumbing manifold where G(V_{f}) is exactly its plumbing graph. Conversely, by a result of W. Neumann [42], the oriented homeomorphism type of K_{f} determines G(V_{f}) (up to blowing ups). (Actually, by another result of W. Neumann [loc.cit.], already π_{1}(K_{f}) determines G(V_{f}), excepting some cases which are classified. This generalizes a result of D. Mumford [30] which says that ∂f(0) = 0 implies π_{1}(K_{f}) ≠ 1.)

On the other hand, the graph G(V_{f}) says very little about the embedding K_{f} ⊂ S_{ϵ} (cf. exercises E.1.15 and E.3.3.c).

For an algorithm of construction of the graph G(V_{f}) for germs of type f(x, y, z) = g(x, y) + z^{n}, see Appendix.

1.18.* K_{f} in the general case.

By a result of J. Milnor [26], the (2n − 1)–manifold K_{f} is (n − 2)–connected. In particular, K_{f} is a (rational) homology sphere if and only if H_{n−1}(K_{f}, Z) = 0 (resp. H_{n−1}(K_{f}, Q) = 0). If n ≥ 3, then π_{1}(K_{f}) = 1, hence H_{n−1}(K_{f}, Z) = 0 implies that K_{f} is a homotopy sphere, and by a result of S. Smale follows that K_{f} is homeomorphic to S^{2n−1} (but maybe not diffeomorphic) (cf. E.3.11).

Open problems and conjectures. (see also (1.8))

1.19. [58] The Zariski’s multiplicity problem: The multiplicity m(f) of f is determined by the topological type of f.

Here, if f = f_{d} + f_{d+1} + ⋯ is the decomposition of f in homogeneous terms f_{i} of degree i, (with f_{d} ≠ 0), then m(f) = d. In the case n = 1, the conjecture has a positive answer. Indeed, for irreducible f with Puiseux pairs (m_{i}, n_{i})_{i=1}^{g}, the multiplicity is m(f) = n_{1} ⋯ n_{g}. But already the case n = 2 appears to be a very difficult problem. 1.20. Two–dimensional singularities. [H. Laufer]

Let G(X) be the resolution graph of a normal surface singularity (X, x).

(a) Give necessary conditions on G(X) for (X, x) to be a hypersurface singularity (i.e. (X, x) = (V_{f}, 0) for some f).

(b) Give sufficient conditions on a graph G for a hypersurface singularity f to exist with G = G(V_{f}).

Exercises for the first section.

E.1.1. (Morse Lemma) A singular germ (i.e. ∂f(0) = 0) f : (C^{n+1}, 0) → (C, 0) is called non–degenerate if its Hessian det(∂_{ij}^{2}(f)(0)) ≠ 0. For such a germ f prove that

f ∼^{R} (z ↦ ∑_{i=1}^{n+1} z_{i}^{2}).

Hint: First write f = ∑_{i,j} a_{ij}(z)z_{i}z_{j}, where a_{ij}(z) = a_{ji}(z), e.g. a_{ij}(z) = − ∫_{0}^{1} ∂_{j}∂_{i}f(tz)(t− 1)dt. Then use the diagonalization method similarly as for quadratic forms. E.1.2. Let f ∈ O. Prove that the following facts are equivalent:

(a) there is a neighborhood 0 ∈ U ⊂ C^{n+1} such that V_{f} ∩ U \ {0} is smooth;

(b) (∂f) ⊃ m^{k} for some k;

(c) (∂f) + (f) ⊃ m^{k} for some k;

(d) dim O/(∂f) < ∞;

(e) dim O/(∂f) + (f) < ∞. Hint (cf. e.g. [24], page 2): Use the local analytic Nullstellensatz.

E.1.3. Let f and g be isolated singularities. Then the following statements are equivalent:

(a) f and g are analytically equivalent, i.e. (C^{n+1}, V_{f}, 0) and (C^{n+1}, V_{g}, 0) are biholomorphic.

(b) The germs (V_{f}, 0) and (V_{g}, 0) are isomorphic as analytic sets, i.e. there is a C– algebra isomorphism O/(f) → O/(g).

(c) There is a biholomorphic isomorphism φ : (C^{n+1}, 0) → (C^{n+1}, 0) such that φ^{∗}((g)) = (f). E.1.4. Let f and g be isolated singularities. Then the following statements are not equivalent (give examples!):

(a) f and g are topological equivalent,

(b) (V_{f}, 0) and (V_{g}, 0) are homeomorphic. E.1.5. (a) The local degree of a germ G : (C^{n+1}, 0) → (C^{n+1}, 0) (with G^{−1}(0) = 0), by definition, is the number of solutions of the equation G(z) = a for a generic. Prove that the local degree of G is exactly the dimension of O/(G_{1}, …, G_{n+1}) where G_{i} is the i^{th}–component of G.

(Notice that this fact in the case of real analytic functions is not true; take e.g. G : (R^{2}, 0) → (R^{2}, 0) given by (x, y) ↦ (x^{3} + xy^{2}, y).)

(b) If G is as above, and G_{i} is a homogeneous polynomial of degree d_{i}, then the local degree of G is ∏_{i} d_{i}. E.1.6. The topological degree of ∂f/||∂f||.

Fix an isolated singularity f. Choose ϵ > 0 with {z ∈ B_{ϵ} : ∂f(z) = 0} = {0}. Define G = ∂f/||∂f|| : S_{ϵ}^{2n+1} → S_{1}^{2n+1}. Prove that μ(f) = deg G (where deg G is the topological degree of G), in the following steps:

(a) First prove the assertion for non–degenerate germs (cf. E.1.1).

(b) Let f_{a} be a deformation of f (morsification) such that f_{a}(z) = f(z) + ∑_{i} a_{i}z_{i}, where {a_{i}}_{i=1}^{n+1} are generic, small coefficients such that all the critical points of f_{a} (in B_{ϵ}) are non-degenerate. Prove that deg G = deg G_{a}, where G_{a} = ∂f_{a}/||∂f_{a}||.

(c) The critical points {z_{i}}_{i} of f_{a} satisfy the equation ∂f(z) + a = 0, hence their number is exactly μ(f) (cf. E.1.5).

(d) Fix small balls B_{i} around z_{i}, and notice that G_{a} is well–defined in B_{ϵ} \ ∪_{i}B_{i}, hence by a property of the degree, one has: deg G_{a}|B_{ϵ} = ∑_{i} deg G_{a}|B_{i}. Now, apply (a) for the points z_{i}. E.1.7. Assume that f : (C^{n+1}, 0) → (C, 0) defines an isolated singularity.

(a) If f = ∑ z_{i}^{a_{i}}, then μ(f) = ∏_{i}(a_{i} − 1).

(b) If f is homogeneous of degree d, then μ(f) = (d − 1)^{n+1}.

(c)* [Milnor–Orlik] If f is quasi–homogeneous with weights {w_{i}}_{i}, then μ(f) = ∏_{i}(w_{i}− 1).

(d) Construct rational numbers w_{1}, …, w_{n+1} (w_{i} ≥ 2) such that there is no quasi– homogeneous isolated singularity of weights {w_{i}}_{i}. Hint (cf. [28]): Use E.1.5.b. In the case (c), construct a covering G : C^{n+1} → C^{n+1} such that f ∘ G has homogeneous components, and use the multiplicativity of the degree with respect to the composed maps. E.1.8. Assume that f is a quasi–homogeneous isolated singularity of weights w_{1}, …, w_{n+1}. Then:

(a) M(f) = O/(∂f) has a base {m̂_{i}}_{i=1}^{μ(f)}, where m_{i} are monomials.

(b) Let l(α) = ∑_{i}(α_{i} + 1)/w_{i}. Then O has a grading O = ⊕_{l}O_{l} such that O_{l} = C⟨z^{α} : l(α) = l⟩. This induces a grading M(f) = ⊕_{l}M(f)_{l}.

(This grading is the shifted version of the “canonical grading” provided by l′(α) = ∑_{i} α_{i}/w_{i}. The disadvantage of our grading provided by l is that M_{l} · M_{l′} ⊄ M_{l+l′}. Nevertheless, we prefer this one bacause of the special form of P_{M(f)}(t), cf. E.2.14*.)

(c) The Poincaré series P_{A}(t) of a graded C–algebra A = ⊕_{l}A_{l} is P_{A}(t) := ∑_{l} dim A_{l}·t^{l}. Prove that the Poincaré series P_{O}^{w}(t) of O (with the above grading) is:

P_{O}^{w}(t) = ∏_{i} P_{C\{z\}}^{w_{i}}(t) = ∏_{i} \frac{t^{1/w_{i}}}{1 − t^{1/w_{i}}} .

(d) Let l_{0} = ∑_{i} 1/w_{i}, and assume that f_{i} ∈ O_{l_{i}+l_{0}}, i = 1, …, s. Then, using the exact sequence:

0 → O/(f_{1}, …, f_{i−1})(−l_{i}) →^{·f_{i}} O/(f_{1}, …, f_{i−1}) → O/(f_{1}, …, f_{i}) → 0,

show that the Poincaré series of A = O/(f_{1}, …, f_{s}) is

P_{A}(t) = P_{O}^{w}(t) · ∏_{i=1}^{s}(1 − t^{l_{i}}).

(e) Using ∂_{i}f ∈ O_{1+l_{0}−1/w_{i}}, show that

P_{M(f)}(t) = ∏_{i=1}^{n+1} \frac{t^{1/w_{i}} − t}{1 − t^{1/w_{i}}} .

E.1.9. Using the result (1.9.d) of Lê–Ramanujan, show that:

(a) the topological type of a quasi–homogeneous isolated singularity depends only on its weights;

(b) if f is homogeneous of degree d, then it has the same topological type as ∑_{i=1}^{n+1} z_{i}^{d}.

In the following exercise the following lemma is useful: Curve Selection Lemma:* [Milnor] [26] Let V be an open neighbourhood of p ∈ R^{m} and let f_{1}, …, f_{k}, g_{1}, …, g_{l} be real analytic functions on V such that p is in the closure of Z := {x ∈ V : f_{i}(x) = 0, i = 1, …, k; g_{j} > 0, j = 1, …, l}. Then there exists a real analytic curve γ : [0, δ) → V with γ(0) = p and γ(t) ∈ Z for t > 0. E.1.10.* (see, e.g. [24].) Let (X, x) ⊂ (C^{N}, 0) be an analytic space–germ such that X \ {x} is smooth. (E.g. (V_{f}, 0) ⊂ (C^{n+1}, 0) for some isolated singularity f.)

(a) Let r : X → [0, ∞) be the restriction of a real analytic function r̃ defined on a neighbourhood of x ∈ X such that r^{−1}(0) = {x}. Then 0 is not an accumulation point of the critical values of r|X \ {x}.

(b) Define X_{r≤ϵ} = {y ∈ X : r(y) ≤ ϵ} and similarly X_{r=ϵ}. Let r : X → [0, ∞) be as above and ϵ be such that X_{r≤ϵ} is compact and r|X \ {x} has no critical values in (0, ϵ].

Then X_{r=ϵ} is a compact real analytic submanifold of X and there exists a homeomorphism H of the cone of X_{r=ϵ} onto X_{r≤ϵ} such that r ∘ H/ϵ is the projection onto [0, 1].

(c) The diffeomorphism type of X_{r=ϵ} (ϵ small) does not depend on the choice of the function r. In particular, the diffeomorphic type of the link X_{r=ϵ} of X at x depends only on the abstract analytic space–germ (X, x).

(d) Extend the above results for the pair (C^{n+1}, V_{f}, 0) instead of (X, x) = (V_{f}, 0). E.1.11. If the isolated singularity f : (C^{2}, 0) → (C, 0) has r irreducible components then its link K_{f} is the disjoint union of r copies of S^{1}’s. E.1.12. Find the link K_{f} ⊂ S_{ϵ}^{3} of

(a) f = x^{2} + y^{2}; f = x^{2} + y^{3}; f = x^{a} + y^{b};

(b) y = x^{3/2} + x^{7/4}. E.1.13. Find a link K ⊂ S^{3} which is not algebraic (i.e. does not exist f such that (S^{3}, K) is homeomorphic to (S_{ϵ}^{3}, K_{f})). E.1.14. If f : (C^{3}, 0) → (C, 0) is an isolated singular germ, then (V_{f}, 0) is a normal surface space–germ. Hint: Use Serre’s criterion. E.1.15. Find the resolution graph G(V_{f}) of V_{f}, where:

(a) f is a homogeneous isolated singularity of degree d (cf. E.5.6.c);

(b) Using the Appendix, prove that K_{f} is diffeomorphic to K_{h}, where f = x^{2} +y^{7} +z^{14} and h = x^{3} + y^{4} + z^{12}. Notice that μ(f) ≠ μ(h). Hint: G(V_{f}) = G(V_{h}) contains only one vertex with e = −1 and g = 3. E.1.16. Let G(V_{f}) be the resolution graph of (V_{f}, 0) (n = 2). Then:

(a) H_{1}(K_{f}, Q) = H_{1}(E, Q). In particular, K_{f} is a rational homology sphere if and only if G(V_{f}) is a tree and E_{w} is rational for all w ∈ W.

(b) The order of the torsion part of H_{1}(K_{f}, Z) is | det(A_{αβ})|. (Obviously, H_{2}(K_{f}, Z) is free.) Hint: Consider a resolution (U, E) → (V_{f}, 0). Then ∂U = K_{f}, and U and the exceptional curve E have the same homotopy type. In particular, H_{3}(U) = 0. Now consider the exact sequence of the pair (U, K_{f}):

H_{2}(U) → H_{2}(U, K_{f}) → H_{1}(K_{f}) → H_{1}(U) → H_{1}(U, K_{f}).

By duality: H_{q}(U, K_{f}) = H_{4−q}(U)^{∗}, and by this identification, the first arrow of the sequence is exactly A : H_{2}(U) → H_{2}(U)^{∗} (which has trivial kernel because A is nondegenerate, cf. 1.15). Therefore, one has the exact sequence:

0 → cokerA → H_{1}(K_{f}, Z) → H_{1}(E, Z) → 0.

E.1.17. Some quotient singularities.

(a) Consider the subgroup Z_{a} ⊂ SU(2), given by the diagonal matrices diag(ζ, ζ^{−1}), where ζ runs through all a^{th}–roots of unity. If u and v are the coordinates on C^{2}, then x = uv, y = u^{a} and z = −v^{a} are generators of the invariant subalgebra C[u, v]^{Z_{a}}, and satisfy the relation x^{a} + yz = 0.

The map q : C^{2} → C^{3} defined by q(u, v) = (x, y, z) induces a topological covering S^{3} → K_{f}, where f = x^{a} + yz with Galois group Z_{a}. Deduce that π_{1}(K_{f}) = Z_{a}.

(b)* Extend the above computation for all finite subgroups of the special unitary group SU(2) computing π_{1}(K_{f}) for all quotient hypersurface singularities f : (C^{3}, 0) → (C, 0) (which are exactly the “simple singularities” in the sense of Arnold). E.g. the binary icosahedral group (the inverse image of the icosahedral group under the surjection SU(2) → SO(3)) is the fundamental group of V_{f}, where f = x^{2} + y^{3} + z^{5}. This group has 120 elements. (cf. Exercise 3.10). E.1.18. Consider a homogeneous polynomial f : C^{n+1} → C of degree d, which defines an isolated singularity at the origin. We can construct two spaces from f. First, the zero set Z(f) of f in the projective space CP^{n}, and also the link K_{f}. What is the relation between these two spaces? Hint: There is a natural projection K_{f} → Z(f) which is an S^{1}-fiber bundle. Actually, this is the S_{1}-bundle associated with O(−d). E.1.19. Prove that f ∼^{R} g ⇒ f ∼^{an} g ⇒ f ∼^{top} g. Constructing examples, show that these equivalence relations are not identical.

Our last exercise has an informative role, it illustrates that any analytic germ is R– equivalent to an algebraic germ. E.1.20.* Finite determinacy property. Let f ∈ O such that m^{k} ⊂ m · (∂f), then f ∼^{R} j_{k}f, where j_{k}f is the k^{th}–jet of f (namely if f = ∑ f_{i}, where f_{i} is homogeneous of degree i, then j_{k}f = ∑_{i≤k} f_{i}). Hint (for details, see [Gibson] [16], page 116):

We have to show that j_{k}f = j_{k}g implies f ∼ g. Take F (z, t) = f_{t}(z) = (1 − t)f(z) + tg(z) (t ∈ R). It is enough to prove that f_{t} ∼ f_{s} for s fixed and t closed to s. Hence, it is enough to construct a germ H : (C^{n+1} × R, (0, s)) → (C^{n+1}, 0) such that (a) H(z, s) = z,

(b) H(0, t) = 0, and (c) F (H(z, t), t) = F (z, s). Notice that (c) is automatically satisfied for t = s, so we can replace (c) by the relation that its left hand side does not depend on t: i.e. by (c’): ∑_{i} ∂_{t}H_{i}(z, t) · ∂_{i}F (H(z, t), t) + ∂_{t}F (H(z, t), t) = 0. If we succeed to construct ξ : (C^{n+1} × R, (0, s)) → (C^{n+1}, 0) such that (d) ∑_{i} ξ_{i}∂_{i}F = −∂_{t}F and (e) ξ_{i}(0, t) = 0, then the flow associated with ξ satisfies (a-b-c’). Since ∂_{t}F ∈ m^{k+1}, for (d-e) we need to show m^{k+1} ⊂ m · (∂F). Since ∂_{i}F − ∂_{i}f ∈ t · m^{k}, one has m^{k+1} ⊂ m · (∂F) + t · m^{k+1}, hence Nakayama’s Lemma (in the variables (z, t)) finishes the proof.

§2. The Milnor fiber, the monodromy and the variation map.

2.1. Fix an isolated singularity f : (C^{n+1}, 0) → (C, 0), and fix ϵ_{0} > 0 such that for any 0 < ϵ ≤ ϵ_{0}, the sphere S_{ϵ} meets V_{f} transversely (cf. 1.10). (This means that at any point z ∈ S_{ϵ} ∩ V_{f} the tangent spaces satisfy: T_{z}S_{ϵ} + T_{z}V_{f} = T_{z}C^{n+1}. Notation: S_{ϵ} ∩| V_{f}.)

Consider the restriction of f on B_{ϵ_{0}}. Since {c ∈ C : f^{−1}(c) has singular points} ⊂ C is a finite (analytic) set, there is δ_{1} > 0 such that for any c ∈ D_{δ_{1}} \ {0} (where D_{δ} is the disc {c : |c| ≤ δ}):

(2.2) f^{−1}(c) ∩ B_{ϵ_{0}} is smooth.

Moreover, since the transversality is an open property, there is a δ_{2} > 0 such that for any c ∈ D_{δ_{2}}:

(2.3) f^{−1}(c) ∩| S_{ϵ_{0}},

(or equivalently, f|f^{−1}(D_{δ_{2}}) ∩ S_{ϵ_{0}} has no critical points). Set δ = min(δ_{1}, δ_{2}). Now, we recall: 2.4. The relative Ehresmann’s Theorem. Let (E, ∂E) be a C^{∞}–manifold with boundary and f : (E, ∂E) → B a proper C^{∞}–map. Assume that f (on E) and f|∂E : ∂E → B have no critical points. Then f is a C^{∞} locally trivial fibration of pair of spaces.

(2.2), (2.3) and Ehresmann’s theorem give:

2.5. Theorem.

(a) For 0 < δ << ϵ_{0} << 1:

(f^{−1}(D_{δ} \ {0}) ∩ B_{ϵ_{0}}, f^{−1}(D_{δ} \ {0}) ∩ S_{ϵ_{0}}) →^{f} D_{δ} \ {0}

is a C^{∞} locally trivial fibration of pair of spaces (called the “local fibration of f”). Its fiber (F, ∂F) is called the Milnor fiber of f (actually, it is the local “nearby” fiber of the local “central fiber” f^{−1}(0)).

(b) The fibration f : f^{−1}(D_{δ} \ {0}) ∩ S_{ϵ_{0}} → D_{δ} \ {0} extends to a C^{∞} fibration f : f^{−1}(D_{δ}) ∩ S_{ϵ_{0}} → D_{δ} (with fiber ∂F). Since D_{δ} is contractible, this latter fibration is a C^{∞} trivial fibration.

An immediate consequence of (b) is that for any c ∈ D_{δ}, ∂(f^{−1}(c) ∩ B_{ϵ_{0}}) is diffeomorphic to ∂(f^{−1}(0) ∩ B_{ϵ_{0}}) = K_{f}; i.e.:

(2.6) ∂F = K_{f}.

A crucial result of Milnor asserts:

2.7. Theorem. [Milnor] [26] F has the homotopy type of a bouquet of (μ copies of) n–spheres ⋁_{μ} S^{n}, where μ is the Milnor number (cf. 1.4).

2.8. Remark. By elementary algebraic topology, it is not difficult to prove that H_{∗}(F, Z) = H_{∗}(⋁_{μ} S^{n}, Z) (cf. E.2.4). But for n ≥ 2, the vanishing of π_{1}(F) needs some Morse theory (the core of Milnor’s proof). These two facts imply Milnor’s theorem (2.7) via Whitehead theorem (for Whitehead theorem, see, e.g. [54], page 399).

By (2.5), F is a (2n)–dimensional real manifold with boundary, with a natural orientation. Therefore, by Lefschetz duality H_{q}(F, ∂F, Z) = H^{2n−q}(F, Z) (see, e.g. [54], page 298). Hence, via (2.7), one has:

(2.9)  H̃_{q}(F, Z) = H̃_{q}(F, ∂F, Z) = \{ \begin{matrix}Z^{μ} & if q = n \\ 0 & otherwise. \end{matrix}

Notational Convention: In order to simplify the notations, we write H_{n}(F, Z) instead of H̃_{n}(F, Z). This is important only if n = 0 (e.g. in E.2.11 or E.3.6). For singularities f : (C, 0) → (C, 0), the reader must replace H_{n}(F, Z) by H̃_{n}(F, Z).

2.10. Pairings. The perfect pairing <, >: H_{n}(F, ∂F, Z) ⊗ H_{n}(F, Z) → Z is induced by the algebraic intersections of type < α, β >, where α (resp. β) is a relative (resp. absolute) oriented n–cycle. It is convenient to introduce its “opposite” intersection pairing as well: <, >′: H_{n}(F, Z) ⊗ H_{n}(F, ∂F, Z) → Z induced by < β, α >. Obviously, < α, β >= (−1)^{n} < β, α >′. The non–degenerate pairing <, > identifies H_{n}(F, ∂F, Z) with H_{n}(F, Z)^{∗} = Hom_{Z}(H_{n}(F, Z), Z) via [α] ↦< α, · >. Then the natural inclusion j : H_{n}(F, Z) → H_{n}(F, ∂F, Z) is identified with b : H_{n}(F, Z) → H_{n}(F, Z)^{∗}, b(β) = (β, ·), where (,) : H_{n}(F, Z) ⊗ H_{n}(F, Z) → Z is the intersection form on H_{n}(F, Z) (induced by the algebraic intersection of absolute n–cycles). Notice that (,) is (−1)^{n}–symmetric, i.e. (a, b) = (−1)^{n}(b, a).

2.11. ker(,). In general (,) is degenerate. In fact, by the homology long exact sequence of the pair (F, ∂F), by (2.9), and by the above identification (2.10), one has (cf. also (1.18)):

(2.12) 0 → H_{n}(K_{f}, Z) → H_{n}(F, Z) →^{b} H_{n}(F, Z)^{∗} → H_{n−1}(K_{f}, Z) → 0.

2.13. Corollary. [Milnor] [26]

(a) H_{n}(K_{f}, Z) = ker(,) ⊂ H_{n}(F, Z), in particular, K_{f} is a rational homology sphere if and only if (,) is non–degenerate;

(b) K_{f} is an integer homology sphere if and only if (,) is unimodular. 2.14. Monodromy. Now, we return to the theorem (2.5), and we consider the locally trivial fibration (f^{−1}(∂D_{δ}) ∩ B_{ϵ_{0}}, f^{−1}(∂D_{δ}) ∩ S_{ϵ_{0}}) → ∂D_{δ} = S_{δ}^{1}. Any C^{∞} fibration with fiber (F, ∂F) over S^{1} is given (modulo an isomorphism) by a ”characteristic map” (= geometric monodromy) m : (F, ∂F) → (F, ∂F) (well–defined modulo an isotopy), such that the fibration identifies with

(F, ∂F) × [0, 1]/_{(x,1)∼(m(x),0)} → [0, 1]/_{0∼1} = S^{1}.

(2.5.b) assures that in our case, the geometric monodromy m can be chosen with:

(2.15) m|∂F = id_{∂F} .

The algebraic monodromies h : H_{n}(F, Z) → H_{n}(F, Z) and h^{r} : H_{n}(F, ∂F, Z) → H_{n}(F, ∂F, Z) are induced by m.

Since m preserves the intersection of the cycles, one has:

(2.16) < h^{r}α, hβ >=< α, β >, (hβ, hβ′) = (β, β′).

The identification H_{n}(F, ∂F, Z) = H_{n}(F, Z)^{∗} (cf. 2.10), and (2.16) give:

(2.17) h^{r} = (h^{∗})^{−1}.

(In matrix notation, h^{∗} is the transpose of h.) The characteristic polynomial det(t − h) of h is denoted by Δ(t).

Since our local fibration is provided by algebraic (or analytic) maps, it satisfies the following: 2.18. Monodromy Theorem* [Grothendieck, Brieskorn] [10]

(a) The eigenvalues of the monodromy operator h are roots of unity.

(b) The nilpotent part h_{nil} of h satisfies (h_{nil})^{n+1} = 0, i.e. the size of any Jordan block is ≤ n + 1.

(c) the size of any Jordan block of h_{1} := h|H_{n}(F, Z)_{1} is ≤ n.

(Here H_{n}(F, Z)_{1} is the generalized 1–eigenspace of H_{n}(F, Z), see below).

The above bounds for the Jordan blocks are optimal. On the other hand, in the case of some special singularities, h can have only small Jordan blocks (cf. E.2.2). For example: 2.19. Theorem. [Lê] [20] (For a generalization, see [Némethi–Steenbrink] [41].)

If f : (C^{2}, 0) → (C, 0) is irreducible, then h_{nil} = 0.

2.20. The equivariant signature.

If A : R^{μ} ⊗ R^{μ} → R is a symmetric bilinear form, then let μ_{+} (resp. μ_{−}) be the dimension of a maximal subspace of R^{μ}, where A is positive(resp. negative) definite. The signature of A is defined by σ(A) := μ_{+} − μ_{−}. There is a similar definition for (complex) hermitian forms (i.e. if Ā^{t} = A).

If A is a complex form with Ā^{t} = −A, then we define σ(A) via σ(iA) (because: \overline{iA}^{t} = iA).

In our situation, consider the generalized eigenspace decomposition ⊕_{λ}H_{n}(F, C)_{λ} of H_{n}(F, C) (i.e. H_{n}(F)_{λ} = {v ∈ H_{n}(F, C) : (h ⊗ 1_{C} − λ)^{k}v = 0 for some k}). By (2.16), the decomposition is compatible with (,). Extend (,) to an (−1)^{n}–hermitian form on H_{n}(F, C). Then:

(H_{n}(F, C), (,)) = ⊕_{λ}(H_{n}(F, C)_{λ}, (,)_{λ}),

and (a, b)_{λ} = (−1)^{n}\overline{(b, a)_{λ}}. We define the equivariant signature σ_{λ}(f) (respectively the signature σ(f)) by σ((,)_{λ}) (resp. σ(,)). Obviously σ(f) = ∑_{λ} σ_{λ}(f).

If n is odd then σ(f) = 0, but even in this case the equivariant signatures are non-zero in general. They satisfy:

σλ̄(f) = (−1)^{n}σ_{λ}(f).

2.21. The variation map.

(2.15) guarantees the existence of another operator V : H_{n}(F, ∂F, Z) → H_{n}(F, Z), called the variation map. Its construction follows.

If α is a relative n–cycle (with ∂α ∈ ∂F = K_{f}), then m(α) satisfies ∂(m(α) − α) = m(∂α) − ∂α = 0, hence m(α) − α is an absolute n–cycle. The correspondence [α] ∈ H_{n}(F, ∂F, Z) →^{V} [m(α)−α] ∈ H_{n}(F, Z) is a well–defined map, and by its very construction, it makes the following diagram commutative:

[figure: H_{n}(F, Z); H_{n}(F, Z); H_{n}(F, ∂F, Z); H_{n}(F, ∂F, Z); j; j; h − 1; h^{r} − 1; V; (2.22)]

Moreover, V satisfies also the following relations. 2.23. Lemma. For any x_{1}, x_{2} ∈ H_{n}(F, ∂F, Z) one has:

(a) (V x_{1}, V x_{2})+ < x_{1}, V x_{2} > + < V x_{1}, x_{2} >′= 0,

(b) < h^{r}x_{1}, V x_{2} > + < V x_{1}, x_{2} >′= 0. Proof. (a) Let α_{i} be relative n–cycles representing x_{i} (i = 1, 2), and with ∂α_{1} ∩ ∂α_{2} = ∅. Recall that V x_{i} = [mα_{i}−α_{i}]. Therefore, the left hand side of (a) is (where now the bracket denotes the intersection of the cycles): < m(α_{1})−α_{1}, m(α_{2})−α_{2} > + < α_{1}, m(α_{2})−α_{2} > + < m(α_{1})−α_{1}, α_{2} >=< m(α_{1}), m(α_{2}) > − < α_{1}, α_{2} >= 0. For (b) we use h^{r} = id+j∘V (cf. 2.22), therefore its left hand side is: < α_{1} + j(m(α_{1}) − α_{1}), m(α_{2}) − α_{2} > + < m(α_{1}) − α_{1}, α_{2} >= 0. □

2.24. Remark. In the next section we will prove that V is an isomorphism. Then (a) (resp. (b)) will provide (,) (resp. h and h^{r}) in terms of V. It turns out that the (integer) variation map is an extremly strong invariant of f.

Open problems and conjectures.

2.25. Given an isolated singularity f, find an algorithm for the computation of the (integer) operators h and/or V. 2.26. [Durfee] [14] Assume that n = 2k. Prove that (−1)^{k}σ(f) > 0. 2.27. [Durfee] [14] Assume that n = 2. Prove that σ(f) ≤ −μ(f)/3.

2.28. [Durfee] [14] If the germ f degenerates to the germ g, then |σ(f)| ≤ |σ(g)|. (Remark: the Milnor numbers satisfy μ(f) ≤ μ(g).)

Exercises for the second section.

E.2.1. Let f_{i} : (C^{2}, 0) → (C, 0) (i = 1, 2) be two isolated singularities such that V_{f_{1}}∩V_{f_{2}} = {0}. Set f = f_{1} · f_{2}. Prove that:

μ(f) = μ(f_{1}) + μ(f_{2}) + 2 · i_{0}(f_{1}, f_{2}) − 1.

(Here i_{0} denotes the intersection multiplicity at the origin.) E.2.2. Let f be a quasi–homogeneous isolated singularity of weights w_{1}, …, w_{n+1}. Set E = f^{−1}(∂D_{δ}) ∩ (B_{ϵ} \ S_{ϵ}) and F = f^{−1}(δ) ∩ (B_{ϵ} \ S_{ϵ}).

(a) Using the natural C^{∗}–action, find the geometric monodromy m : F → F (without the property (2.15) !) as follows. Consider the map F × [0, 1] →^{T} E defined by T (z, t) = (z_{1}e^{2πit/w_{1}}, …, z_{n+1}e^{2πit/w_{n+1}}). Then f(T (z, t)) = δe^{2πit}, hence m = T (·, 1).

(b) Conclude that h has a finite order.

(c) The (open) Milnor fiber F of f is diffeomorphic to F_{1} := {z ∈ C^{n+1} : f(z) = 1}. Moreover, the local fibration f^{−1}(∂D_{δ})∩(B_{ϵ} \S_{ϵ}) → ∂D_{δ} is isomorphic (as a fiber bundle)

to E := {z ∈ C^{n+1} : |f(z)| = 1} →^{f} ∂D_{1}.

E.2.3. The non-degenerate singularity (see [24], pages 36-41).

Assume that f = ∑ z_{i}^{2}. Write z = x + iy. Then the Milnor fiber f^{−1}(δ) ∩ B_{ϵ} is F = {x+iy ∈ C^{n+1} : |x|^{2}+|y|^{2} ≤ ϵ^{2}, |x|^{2}−|y|^{2} = δ, (x, y) = 0}. With the parametrization u = x/|x|, v = c · y (c = 2/(ϵ^{2} − δ), c > 0), F can be identified with the unit disk bundle E of the tangent bundle of S^{n}:

E = {u + iv ∈ C^{n+1} : |u| = 1, |v| ≤ 1, (u, v) = 0}.

This diffeomorphism h_{0} : E → F is given by: h_{0}(u + iv) = (|v|/c + δ)u + iv/c. Now, consider the family of homeomorphisms g_{θ} : E → E (θ ∈ R):

g_{θ}(u + iv) = [u cos(π|v|θ) + \frac{v}{|v|} sin(π|v|θ)] + i[−u|v| sin(π|v|θ) + v cos(π|v|θ)].

Set h_{θ} := e^{πiθ}h_{0}∘g_{θ} : E → f^{−1}(D_{δ}∩B_{ϵ}. Then h_{θ} is a homeomorphism onto f^{−1}(δe^{2πiθ})∩B_{ϵ}. Notice that h_{0} and h_{1} coincide on ∂E, thus defining a trivialization on f^{−1}(D_{δ}) ∩ S_{ϵ} (note that this trivialization extends over D_{δ}). So, the automorphism h_{1} ∘ h_{0}^{−1} of F represents the geometric monodromy m (with m|∂F = id_{∂F}). Under the identification of F with E via h_{0}, this corresponds to h_{0}^{−1} ∘ h_{1} = −g_{1} on E.

(b) H_{n}(E) = Z is generated by [S^{n}], where S^{n} = {v = 0}. We orient S^{n} by letting (e_{1}, …, e_{n}) be an orientation basis of T_{e_{0}}S^{n}, where (e_{0}, …, e_{n}) is the standard basis in R^{n+1}. Notice that −g_{1}|S^{n} : S^{n} → S^{n} is −id_{S^{n}}. Therefore, h = (−1)^{n+1}. The intersection ([S^{n}], [S^{n}])_{E}, with respect to the tangent bundle orientation, is the Euler– number e(S^{n}) = 1+(−1)^{n}. Since the complex orientation differs from the tangent bundle orientation by the sign of the permutation (e_{1}, …, e_{n}, ie_{1}, …, ie_{n}) ↦ (e_{1}, ie_{1}, …, e_{n}, ie_{n}), one has: ([S^{n}], [S^{n}])_{F} = (−1)^{n(n−1)/2}(1 + (−1)^{n}).

(c) Fix u^{0} ∈ S^{n} = {v = 0} ⊂ E, and let D^{n} be the fiber in E over u^{0}, i.e. D^{n} = {u^{0} + iv : (u^{0}, v) = 0, |v| ≤ 1}. It is oriented by (ie_{1}, …, ie_{n}). Then [D^{n}, ∂D^{n}] is a base for H_{n}(E, ∂E). Moreover < [D^{n}, ∂D^{n}], [S^{n}] > is the sign of the permutation (ie_{1}, …, ie_{n}, e_{1}, …, e_{n}) → (e_{1}, ie_{1}, …), which is (−1)^{n(n+1)/2}. Take δ^{∗} = (−1)^{n(n+1)/2}[D^{n}, ∂D^{n}], and δ = [S^{n}], then < δ^{∗}, δ >= 1.

(d) Prove that V (δ^{∗}) = (−1)^{(n+1)(n+2)/2}δ.

(e) Consider the case n = 1. Then (,) = 0, h = id_{Z} (i.e. both invariants are trivial). But V is a non-trivial endomorphism, in fact it is an isomorphism. E.2.4. With the notations of (2.5), prove that F = f^{−1}(δ) ∩ B_{ϵ} satisfies: H_{∗}(F, Z) = H_{∗}(⋁_{μ} S^{n}, Z) (cf. 2.8) in the following steps:

(a) For 0 < δ << ϵ << 1 prove that B_{ϵ,δ} := f^{−1}(D_{δ}) ∩ B_{ϵ} is contractible. Hint: Using the Curve Selection Lemma, prove that the differentials of r(z) = ||z||^{2} and of z ↦ |f(z)| do not point in opposite direction. Hence, there is a vector field whose integral curves contract B_{ϵ,δ} to the origin.

(b) From the long exact sequence of the pair (B_{ϵ,δ}, F), prove that it is enough to verify that (∗) H_{q}(B_{ϵ,δ}, F, Z) = Z^{μ} if q = n, and = 0 otherwise.

(c) Verify (∗) for a non–degenerate singularity.

(d) Consider a morsification f_{a} of f as in (E.1.6). Let {z_{i}}_{i=1}^{μ(f)} be the set of critical points and c_{i} = f(z_{i}) ∈ int(D_{δ}) the set of critical values. Let B_{i}(ϵ′) be a Milnor ball of f with center z_{i}, D_{i} = D_{i}(δ′) a small disc with center c_{i} (0 < δ′ << ϵ′). Take c′_{i} ∈ ∂D_{i} and γ_{i} : [0, 1] → D_{δ} (i = 1, …, μ) such that γ_{i}(0) = δ, γ_{i}(1) = c′_{i}, and Γ := ∪_{i}(imγ_{i}) ∪ (∪_{i}D_{i}) ↪ D_{δ} admits a strong deformation retract r. Use Ehresmann’s Theorem to prove that f_{a} : f_{a}^{−1}(D_{δ} \ ∪_{i}{c′_{i}}) ∩ B_{ϵ} → D_{δ} \ ∪_{i}{c′_{i}} is a fiber bundle, hence the deformation retract r can be lifted. Notice also that (f_{a}^{−1}(D_{i}) ∩ B_{ϵ}) \ B_{i} → D_{i} is a trivial fiber bundle. Therefore: H_{∗}(B_{ϵ,δ}, F) =

= H_{∗}(f_{a}^{−1}(D_{δ}) ∩ B_{ϵ}, f_{a}^{−1}(δ) ∩ B_{ϵ})  (transversality)
= H_{∗}(f_{a}^{−1}(Γ) ∩ B_{ϵ}, f_{a}^{−1}(δ) ∩ B_{ϵ})  (deformation retract)
= H_{∗}(f_{a}^{−1}(Γ) ∩ B_{ϵ}, f_{a}^{−1}(∪_{i}imγ_{i}) ∩ B_{ϵ})  (deformation retract)
= ⊕_{i} H_{∗}(f_{a}^{−1}(D_{i}) ∩ B_{ϵ}, f_{a}^{−1}(c′_{i}) ∩ B_{ϵ})  (excision)
= ⊕_{i} H_{∗}(f_{a}^{−1}(D_{i}) ∩ B_{ϵ}, f_{a}^{−1}(c′_{i}) ∪ (f_{a}^{−1}(D_{i}) \ B_{i})) (deformation retract)
= ⊕_{i} H_{∗}(f_{a}^{−1}(D_{i}) ∩ B_{i}, f_{a}^{−1}(c′_{i}) ∩ B_{i}))  (excision).

Now, apply (c). E.2.5. Assume that n = 0 and f : (C, 0) → (C, 0) is given by f(z) = z^{a}. Find F, m, h and prove that Δ(t) = (t^{a} − 1)/(t − 1). E.2.6. A’Campo’s formula. [2] Assume that n = 1 (A’Campo’s result is for arbitrary n). As above, let G_{f} be the embedded resolution graph of f. For each w ∈ W let δ_{w} = #V_{w}. Prove that

\frac{t − 1}{Δ(t)} = ∏_{w∈W}(t^{m_{w}} − 1)^{2−δ_{w}}.

(This proves the first part (2.18.a) of the Monodromy theorem.) In particular, χ(F) = 1 − μ(f) = ∑_{w} m_{w}(2 − δ_{w}). Hint: If ϕ is as in (1.14), consider F′ = ϕ^{−1}(F) ⊂ Y, which is diffeomorphic to F. Then there exists a map π : F′ → D such that for any singular point P of D: π^{−1}(P) is a disjoint union of S^{1}’s (which do not have any contribution in the Euler–characteristic or in the characteristic polynomial); and π^{−1}(E_{w}^{reg}) → E_{w}^{reg} (E_{w}^{reg}=regular part of E_{w}) is an m_{w}–covering with a Galois action corresponding to the monodromy (w ∈ W). Notice also that χ(E_{w}^{reg}) = 2 − δ_{w}. (cf. 4.3 and 4.11) E.2.7. [A’Campo][3] Assume n = 1. Using (E.2.6), prove that if f is singular (i.e. ∂f(0) = 0) then trace(h) = ±1. E.2.8. Recall that cone(Z) = [0, 1] × Z/(0, z) ∼ (0, z′), and there is a natural inclusion Z ↪ cone(Z), z ↦ [1, z].

(a) Prove that f^{−1}([0, δ]) ∩ B_{ϵ} (0 < δ << ϵ << 1) is contractible.

(b) Prove that there exists a homotopy equivalence:

φ : (cone(F), F) → (f^{−1}([0, δ]) ∩ B_{ϵ}, f^{−1}(δ) ∩ B_{ϵ})

(where F = f^{−1}(δ) ∩ B_{ϵ}), such that f(φ([t, z])) = tδ. Hint: Use the Curve Selection Lemma (cf. the hint of E.2.4.a). E.2.9. The join space.

Let X and Y be two spaces. The join space X ∗Y is defined by X ×[0, 1]×Y/ ∼, where (x, 0, y) ∼ (x, 0, y′) and (x, 1, y) ∼ (x′, 1, y). There is a natural projection p : X∗Y → [0, 1] [x, t, y] ↦ t.

(a) By the identifications p^{−1}(1/2) = X × Y, p^{−1}([0, 1/2]) = X × cone(Y), ([x, t, y] ↦ (x, [2t, y])), and p^{−1}([1/2, 1]) = cone(X) × Y, prove that

X ∗ Y = X × cone(Y) ⋃_{X×Y} cone(X) × Y.

(b) [Milnor] By the Künneth formula and the Mayer–Vietoris exact sequence of the above decomposition, prove that:

(∗)  H̃_{r}(X ∗ Y, R) = ⨁_{p+q=r−1} H̃_{p}(X, R) ⊗ H̃_{q}(Y, R).

Moreover, if either H_{∗}(X, Z) or H_{∗}(Y, Z) is free, then (∗) is valid even with Z–coefficients.

(c) If m_{X} : X → X and m_{Y} : Y → Y are homeomorphism, then m_{X} ∗ m_{Y} is defined via [x, t, y] ↦ [m_{X}(x), t, m_{Y} (y)]. Show that (∗) is compatible with the morphisms, i.e. (m_{X} ∗ m_{Y})_{∗,r} = ⊕_{p+q=r−1}(m_{X})_{∗,p} ⊗ (m_{Y})_{∗,q}. E.2.10. Prove the following homeomorphisms:

(a) S^{n} ∗ S^{m} = S^{n+m+1};

(b) S^{0} ∗ X = suspension of X;

(c) Prove that (⋁_{α} S^{n}) ∗ (⋁_{β} S^{m}) and ⋁_{αβ} S^{n+m+1} have the same homotopy types. E.2.11. Sebastiani–Thom theorem.

Let g : (C^{n+1}, 0) → (C, 0) and h : (C^{m+1}, 0) → (C, 0) be isolated singularities. Define f : (C^{n+1} × C^{m+1}, 0) → (C, 0) by f(x, y) = g(x) + h(y).

(a) Prove that f is an isolated singularity, and M(f) = M(g) ⊗ M(h). In particular, μ(f) = μ(g) · μ(h).

(b) Consider the map u : (C^{n+1} ×C^{m+1}, 0) → (C×C, 0), u(x, y) = (g(x), h(y)). Then (in a good neighborhood system, whose description is left to the reader) u is a C^{∞} locally trivial fibration over C × C \ {cd = 0}, with fiber F_{g} × F_{h}.

(c) For any θ ∈ [0, 2π], consider L_{θ} = {(c, d) ∈ C^{2}; c+d = δe^{2πiθ}}∩B_{r} (δ << r << 1). Then u^{−1}(L_{θ}) can be identified with f^{−1}(δe^{2πiθ}).

(d) Set P_{θ} = L_{θ} ∩ {c = 0} and Q_{θ} = L_{θ} ∩ {d = 0}, and let S_{θ} be the segment [P_{θ}, Q_{θ}] ⊂ L_{θ}. Then the inclusion S_{θ} ⊂ L_{θ} admits a strong deformation retract, which (by (b)) can be lifted to u. Therefore, F_{f} has the same homotopy type as u^{−1}(S_{θ}).

(e) Let M_{θ} be the midpoint of S_{θ}. Verify that u^{−1}(M_{θ}) = F_{g} × F_{h}, u^{−1}([P_{θ}, M_{θ}]) = (cone(F_{g})) × F_{h}, u^{−1}([M_{θ}, Q_{θ}]) = F_{g} × cone(F_{h}) (cf E.2.8). Therefore, u^{−1}(S_{θ}) = F_{g} ∗ F_{h}.

(f) u^{−1}(M_{θ}) = g^{−1}(δe^{2πiθ}/2) × h^{−1}(δe^{2πiθ}/2), θ moving in [0, 1] induces on the right hand side of this identity the monodromy m_{g}×m_{h}, hence (cf. E.2.9.c) m_{f} can be identified with m_{g} ∗ m_{h}.

(g) Using (E.2.9), prove that one has the following identifications (this is called Sebastiani-Thom formula [53]): H_{n+m+1}(F_{f}, Z) = H̃_{n}(F_{g}, Z)⊗H̃_{m}(F_{h}, Z) and h_{f} = h_{g}⊗h_{h}.

(h) If Δ_{f}(t) = ∏_{i=1}^{μ(f)}(t − ζ_{i}) (ζ_{i} ∈ S^{1}), is the characteristic polynomial of the germ f, then we denote Div(f) := ∑_{i}(ζ_{i}) ∈ Z[S^{1}]. Note that the free abelian group Z[S^{1}] generated by the elements of S^{1} is a ring by (ζ)(ζ′) = (ζζ′).

With the notation of this exercise, prove that: Div(f) = Div(g) · Div(h). E.2.12. Brieskorn polynomials. Let Λ_{a} = ∑_{j=1}^{a}(e^{2πij/a}) ∈ Z[S^{1}]. (For the definition of Div(f), see 2.11.h.)

(a) If f : (C, 0) → (C, 0), f(z) = z^{a}, then Div(f) = Λ_{a} − (1).

(b) If f = ∑_{i} z_{i}^{a_{i}}, then Div(f) = ∏_{i}(Λ_{a_{i}} − (1)).

(c) Λ_{a}Λ_{b} = (a, b)Λ_{[a,b]}.

Remark.* [Milnor–Orlik] [28] If f is quasi–homogeneous isolated singularity of weights {w_{i}}_{i}, then the following formula of Milnor–Orlik gives Div(f) (hence Δ(t) too) in terms of weights.:

Div(f) = ∏_{i}(v_{i}^{−1}Λ_{u_{i}} − (1)),

where w_{i} = u_{i}/v_{i} and u_{i}, v_{i} are positive relatively prime integers. E.2.13. Consider the following two germs in two variables: f = xy^{3} + x^{8}y and g = xy^{4} + x^{6}y. Prove that their Div is the same (Λ_{23} + (1)), their Milnor number is the same (=24), the number of irreducible components is the same (=3), but their topological type is different. E.2.14.* Let P_{M(f)}(t) be the Poincaré polynomial of a quasi–homogeneous singularity (cf. E.1.8). P_{M(f)}(t) = ∑_{i=1}^{μ(f)} t^{r_{i}} can be codified (because r_{i} ∈ Q, cf. 2.18) in Z[Q] in the form ∑_{i=1}^{μ(f)}(r_{i}). On the other hand, there is a natural map q : Z[Q] → Z[Q/Z] → Z[S^{1}] induced by (r) ↦ (e^{2πir}). Prove that q(∑_{i}(r_{i})) = Div(f), i.e. by (E.1.8) and by the remark at the end of (E.2.12), one has:

q(∏_{i} \frac{t^{1/w_{i}} − t}{1 − t^{1/w_{i}}} ) = ∏_{i}(v_{i}^{−1}Λ_{u_{i}} − (1)).

§3. The Milnor fibration and the Seifert form.

3.1. Consider the local fibration (2.5):

(E, ∂E) := (f^{−1}(∂D_{δ}) ∩ B_{ϵ}, f^{−1}(∂D_{δ}) ∩ S_{ϵ}) →^{f} ∂D_{δ} = S^{1}

with fiber (F, ∂F). The Wang exact sequence provides:

(3.2) 0 → H_{n+1}(E) → H_{n}(F) ⟶^{h−id} H_{n}(F) → H_{n}(E) → 0.

But this exact sequence gives very little supplementary information about the local structure. More precisely: (3.2) gives the homology groups of E, but provides no new relation, restriction for (F, ∂F) or h. The next theorem identifies E with S_{ϵ}^{2n+1} \ ∂F, then via this identification (3.2) becomes a very strong exact sequence.

We start with a definition. 3.3. Definition. A “fibered knot K ⊂ S^{2n+1}” (or open book decomposition of S^{2n+1}) is an embedding of an (n − 2)–connected (2n − 1)–manifold K in S^{2n+1} (where K is empty if n = 0), together with a smooth fiber bundle ϕ : S^{2n+1} \ K → S^{1} that has the following properties:

(i) There is a tubular (disk) neighborhood T of K and a bundle equivalence α of T to the trivial bundle K × D with

[figure: T \ K; K × (D \ {0}); S^{1}; α|T \ K; ϕ|T \ K; (x, c) ↦ c/|c|]

(ii) for all θ, the fiber F_{θ}^{o} = ϕ^{−1}(e^{2πiθ}) is (n − 1)–connected. (Its closure F_{θ} is a (2n)– manifold with boundary K by (i).) (Above S^{1} and S^{2n+1} are considered with their natural orientations.) 3.4. Example.* In the classical knot–theory, by a result of Stallings and Neuwirth, the following facts are equivalent:

(a) K ⊂ S^{3} is a fibered knot;

(b) the commutator subgroup [G, G] of G := π_{1}(S^{3} \ K) is a free group;

(c) the commutator group [G, G] is a finitely generated group. (cf. Exercise 3.2)

3.5. Theorem. [Milnor] [26] Fix an isolated singularity f. (Recall K_{f} = f^{−1}(0) ∩ S_{ϵ}.) Then:

(a) For ϵ sufficiently small ϕ = f/|f| : S_{ϵ} \ K_{f} → S^{1} is a C^{∞} locally trivial fibration (called the Milnor fibration of f), which defines an open book decomposition of S_{ϵ}.

(b) The local fibration f : f^{−1}(∂D_{δ})∩B_{ϵ} → ∂D_{δ} and the Milnor fibration are bundle isomorphic. Actually, if we consider F_{θ} := f^{−1}(e^{2πiθ}[δ, ∞)) ∩ B_{ϵ}, then f^{−1}(δe^{2πiθ}) ∩ B_{ϵ} ⊂ F_{θ} can be “pushed out” by a flow (along F_{θ}), isotopically to F_{θ} ∩ S_{ϵ} = ϕ^{−1}(e^{2πiθ}). (In the sequel, we will denote both fibers f^{−1}(δe^{2πiθ}) and ϕ^{−1}(e^{2πiθ}) simply by F_{θ}.)

3.6. The Wang exact sequence revisited. By (3.5), the space E in (3.2) is S_{ϵ} \ K_{f}. Notice that:

H_{q}(S_{ϵ} \ K_{f}, Z) = H^{2n−q}(K_{f}, Z) (by Alexander duality)
= H_{q−1}(K_{f}, Z) (by Poincaré duality).

Therefore, (3.2) reads as:

0 → H_{n}(K_{f}, Z) → H_{n}(F, Z) ⟶^{h−id} H_{n}(F, Z) → H_{n−1}(K_{f}, Z) → 0.

3.7. Corollary.

(a) rank H_{n}(K_{f}) = rank H_{n−1}(K_{f}) = dim ker(h − id), in particular, K_{f} is a rational homology sphere if and only if Δ(1) = det(id − h) ≠ 0.

(b) K_{f} is an integer homology sphere if and only if Δ(1) = ±1. 3.8. Corollary (3.7) is very useful for the study of the nilpotent part of h as well. Notice that the dimension of the generalized 1–eigenspace dim H_{n}(F)_{1} can be determined from Δ(t) (i.e. it is the exponent of (t − 1) in Δ(t)). On the other hand, rank H_{n}(K_{f}) = dim ker(h − 1) ≤ dim H_{n}(F)_{1}, and the inequality is strict if and only if there exists at least one Jordan block of h with eigenvalue 1 and size ≥ 2. (If we want to test the Jordan blocks corresponding to λ = e^{2πik/l}, with k/l ∉ Z, then we replace f by f + z^{l}, where z is a new variable. Then by Sebastiani–Thom theorem, it is enough to test the λ = 1 eigenvalue for this new germ. Cf. E.3.12.) 3.9. The exact sequences (2.12) and (3.2) (or 3.6) have big similarities. We will show that they can be identified via the variation map (and some duality isomorphisms).

The (Gysin) isomorphism α : H_{q}(S_{ϵ} \ K_{f}, Z) → H_{q−1}(K_{f}, Z) and its inverse β : H_{q−1}(K_{f}, Z) → H_{q}(S_{ϵ} \ K_{f}, Z) are defined as follows. If c is a closed (oriented) q–cycle in S_{ϵ} \ K_{f}, consider a (q + 1)–cycle d in S_{ϵ} with ∂d = c and d ∩ | K_{f}. Then α([c]) = [d ∩ K_{f}]. Conversely, if e is a closed (q − 1)–cycle in K_{f}, let π : T → K_{f} be the projection of a tubular neighbourhood of K_{f} onto K_{f}, then β([e]) = [∂π^{−1}e] = [π^{−1}(e) ∩ ∂T]. Now, α = β^{−1}, and the following diagram is commutative (please, verify!):

[figure: 0 → H_{n+1}(S_{ϵ} \ K_{f}); 0 → H_{n}(K_{f}); H_{n}(F); H_{n}(F); H_{n}(F); H_{n}(F); H_{n}(F, K_{f}); H_{n}(F)^{∗}; H_{n}(S_{ϵ} \ K_{f}) → 0; H_{n−1}(K_{f}) → 0; ∂; h − 1; j; b; β; ≃; ≃; β; V]

Now, the Five lemma implies:

3.10. Theorem. V : H_{n}(F, K_{f}, Z) → H_{n}(F, Z) is an isomorphism.

3.11. Corollary of (2.23) and (3.10). Let θ : H_{n}(F, Z) → H_{n}(F, Z)^{∗∗}, θ(u)(φ) = φ(u) be the natural isomorphism. Then h and (,) can be computed from V as follows:

h = −(−1)^{n}V (θ^{−1} ∘ V^{∗})^{−1},
b = −V^{−1} − (−1)^{n}(θ^{−1} ∘ V^{∗})^{−1}.

3.12. Corollary.

(a) ker b = ker(h − 1) = H_{n}(K, Z), (recall 2.22: V ∘ b = h − 1),

(b) if b is non-degenerate (equivalently, if Δ(1) ≠ 0), then | det(,)| = | det(h − 1)| = |Δ(1)|. 3.13. Example. Let f = z_{1}^{2} + z_{2}^{3} + z_{3}^{5}. Then (cf. E.2.12) Div(f) = (Λ_{2} − 1)(Λ_{3} − 1)(Λ_{5} − 1) = Λ_{30} − Λ_{6} − Λ_{10} − Λ_{15} + Λ_{2} + Λ_{3} + Λ_{5} − 1, hence

Δ(t) = \frac{(t^{30} − 1)(t^{2} − 1)(t^{3} − 1)(t^{5} − 1)}{(t^{6} − 1)(t^{10} − 1)(t^{15} − 1)(t − 1)} = \frac{t^{10} − t^{5} + 1}{t^{2} − t + 1},

and Δ(1) = 1, hence (,) is unimodular (see also E.3.10). 3.14. The Seifert form.

Recall the notations: F_{θ} = f^{−1}(e^{2πiθ}[0, ∞)) ∩ S_{ϵ}, ∂F_{θ} = K_{f} = f^{−1}(0) ∩ S_{ϵ}, F_{θ}^{o} = F_{θ} \ K_{f} = ϕ^{−1}(e^{2πiθ}). The inclusion F_{θ}^{o} ↪ F_{θ} identifies their homology. Let g be a continuous map [0, 1] × F_{θ}^{o} → S_{ϵ} \ K_{f} such that g_{θ} = g(θ, ·) maps F_{0}^{o} homeomorphically onto F_{θ}^{o} and g_{0} = id_{F_{0}^{o}}. Via g_{θ} we can identify F_{θ}^{o} and F_{0}^{o} provided that 0 ≤ θ < 1, and g_{1} is a candidate for the geometric monodromy m.

The Seifert form S : H_{n}(F_{0}, Z) ⊗ H_{n}(F_{0}, Z) → Z is defined by

S(β, β′) = linking number(β, (g_{1/2})_{∗}β′),

where the linking number is considered in S_{ϵ}: linking number(β, β′)=algebraic intersection number (α, β′), where α is an (n + 1)–cycle in S_{ϵ} with ∂α = β. 3.15.* S can be identified with the following sequence of isomorphisms (all the groups are considered with Z–coefficients):

H_{n}(F_{0}) →^{∂^{−1}} H_{n+1}(S_{ϵ}, F_{0}) =^{(1)} H_{n+1}(S_{ϵ}, ∪_{0≤θ≤1/2}F_{θ})

=^{(2)} H_{n+1}(∪_{1/2≤θ≤1}F_{θ}, F_{1/2} ∪ F_{1}) =^{(3)} H_{n}(∪_{1/2≤θ≤1}F_{θ})^{∗} =^{(1)} H_{n}(F_{0})^{∗},

where (1) is induced by the inclusion F_{0} ↪ ∪_{0≤θ≤1/2}F_{θ} (which admits a strong deformation retract), (2) is given by excision, and (3) by Poincaré duality. In particular, S is unimodular. This fact will follow also from (3.16).

In fact, the Seifert form and the variation map satisfy the following relation: 3.16. Theorem. If α ∈ H_{n}(F, ∂F, Z), β ∈ H_{n}(F, Z), (with F = F_{0}) then S(V α, β) = < α, β >. Proof.

S(V α, β) = linking number(mα − α, (g_{1/2})_{∗}β)
= linking number(∂ ∪_{0≤θ≤1} (g_{θ})_{∗}α, (g_{1/2})_{∗}β)
= intersection number(∪_{0≤θ≤1}(g_{θ})_{∗}α, (g_{1/2})_{∗}β)
=< (g_{1/2})_{∗}α, (g_{1/2})_{∗}β >_{F_{1/2}}=< α, β >_{F_{0}}=< α, β > .

3.17. Corollary. S(a, b) =< V^{−1}a, b >. In particular, via the identification in (2.10), S can be identified with (V^{−1})^{t}. In matrix notations, the objects S, (,), h and h^{r} satisfy:

(,) = −S + (−1)^{n+1}S^{t}
h = (−1)^{n+1}(S^{t})^{−1}S
h^{r} = (−1)^{n+1}S(S^{t})^{−1}.

(Notice that in matrix notations (,) = b^{t}.) 3.18. Some (difficult) results about the Seifert form.*

(a) By a result of J. Levine [23], the Seifert form of a fibered knot K ⊂ S^{2n+1} is an invariant of the embedding K_{f} ↪ S^{2n+1}.

(b) Moreover, for n ≥ 3, A. Durfee [12] and M. Kato [18] independently proved (based on some results of Levine) that the set of isomorphism classes of fibered knots in S^{2n+1} is equivalent (via their Seifert form) to the set of conjugacy classes of unimodular bilinear forms over Z.

In particular, the topological type of f is completely determined by the Seifert form S of f, (and it determines S modulo a conjugacy), provided that n ≥ 3.

(c) It was conjectured that in the case n = 1, the real Seifert form S of f determines the topological type of f. (This is true for irreducible germs, since by a result of Lê, already the characteristic polynomial Δ(t) determines the topological type of irreducible plane curve singularities. cf. E.3.3.b). The conjecture was disproved in the Schrauwen– Steenbrink–Stevens paper [52]. Actually, P. Du Bois and F. Michel [13] constructed two germs with different topological type, but with the same integer Seifert form. Artal– Bartolo consider the duble suspension of these curve singularities in order to construct two germs in the case n = 2 with the same integer Seifert forms but different links (in particular, with different topological types) [5].

(d) S. S.-T. Yau conjectured that the topological type of an isolated singularity f : (C^{3}, 0) → (C, 0) is determined by the characteristic polynomial Δ(t) of its monodromy operator, and by the fundamental group π_{1}(K_{f}) of its link. (Notice that both Δ(t) and π(K_{f}) depend only on the topological type of f, cf. E.3.3). The conjecture was verified in the case of quasi–homogeneous singularities by Xu and S. S.-T. Yau. This conjecture was also disproved by a counterexample constructed by Artal–Bartolo in his thesis [6].

Open problems and conjectures.

3.19. Given an isolated singularity f, find a method, or algorithm for the computation of its Seifert form S. 3.20. In the correspondence Durfee–Kato (3.18.b), characterize those Seifert forms (i.e. unimodular matrices) which correspond to the algebraic knots (i.e. (S^{2n+1}, K) = (S_{ϵ}, K_{f}) for some germ f). 3.21. Given a quasi–homogeneous isolated singularity, find its integer Seifert form (in terms of its weights, cf. E.1.9).

Exercises for the third section.

E.3.1. Prove J. Reeve’s theorem (cf. 1.13.b), namely that the intersection multiplicity of two plane curve singularities (without common component) is the linking number of their links in S_{ϵ}. E.3.2. Assume that f is an irreducible plane curve singularity. If G := π_{1}(S_{ϵ} \ K_{f}), prove that the commutator subgroup [G, G] is free. (cf. 3.4) Hint: use the homotopy long exact sequence of the Milnor fibration. E.3.3. (a) [Teissier] μ(f) depends only on the topological type of f.

(b) [Lê] Δ(t) depends only on the topological type of f.

(c) Let f = x^{2} +y^{7} +z^{14} and g = x^{3} +y^{4} +z^{12}. Show that their links are diffeomorphic, but their topological type is different (cf. E.1.15). Hint for (a-b): we assume that n ≥ 2 (the case n = 1 can be proved either by similar argument, or by the classification result 1.13). Then by the homotopy exact sequence of the Milnor fibration one has: ϕ_{∗} : π_{1}(S_{ϵ} \ K_{f}) →^{∼} π_{1}(S^{1}) = Z. The universal covering (S_{ϵ} \ K_{f})˜ of S_{ϵ} \ K_{f} can be (homotopicaly) identified with F, and μ(f) is the rank of its n^{th}–homology. Moreover, Δ(t) can be described using the Galois action of π_{1}(S_{ϵ} \ K_{f}) on H_{n}(S_{ϵ} \ K_{f})˜ . Actually, it is its Reidemeister torsion associated with the representation π_{1}(S_{ϵ} \ K_{f}) = Z → C((t)), 1 ↦ t.

We mention here that in the case n = 1, Δ(t) can be identified with the Alexander polynomial of the link K_{f} ⊂ S_{ϵ} (modulo ±t^{±i}). E.3.4. If f = ∑ z_{i}^{a_{i}}, with (a_{i}, a_{j}) = 1 for i ≠ j, then Δ(1) = 1. In particular, K_{f} is an integer homology sphere (cf. 3.7.b). E.3.5. Deligne–Sakamoto theorem* (independently). [50] (Continuation of Sebastiani–Thom theorem (E.2.11).) With the notations of (E.2.11), prove that the Seifert forms satisfy:

S_{f} = (−1)^{(n+1)(m+1)}S_{g} ⊗ S_{h}.

(For a proof in the spirit of (2.11), rewrite [40] for the local case.) E.3.6. Consider f : (C, 0) → (C, 0) given by f(z) = z^{a} (cf. E.2.5 and E.2.12.a). Show that its Seifert form (in a special base) is:

S(z ↦ z^{a}) = ( \begin{matrix}−1 & 0 & 0 & … & 0 \\ 1 & −1 & 0 & … & 0 \\ 0 & 1 & −1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & … & −1 \end{matrix} )

Hint: The goemetric Milnor fiber can be identified with a points: the a^{th}–roots of unity {P_{0}, …, P_{a−1}} in the complex plane. The geometric monodromy is P_{0} ↦ P_{1} ↦ ⋯ ↦ P_{a−1} ↦ P_{0}. Moreover, H̃_{0}(F, Z) = {∑_{i} n_{i}P_{i} : ∑_{i} n_{i} = 0}. A base is given by e_{i} := P_{i} − P_{i−1}, i = 1, 2, …, a − 1. In this base S has the wanted form. E.3.7. The non–degenerate case. Using (E.3.5-E.3.6), prove that the Seifert form of non–degenerate germ f : (C^{n+1}, 0) → (C, 0) is S = (−1)^{(n+1)(n+2)/2} (compare with the formula of the variation map (E.2.3.d)). E.3.8. Prove that if f is an isolated singularity, z_{1} and z_{2} are new varibles, and f′ = f + z_{1}^{2} + z_{2}^{2}, then S_{f′} = −S_{f}; (,)_{f′} = −(,)_{f} and h_{f′} = h_{f}. E.3.9.

(a) Using (E.3.5-E.3.6) and (3.17), compute the intersection form (,) for the following germs: f = x^{2} + y^{2} + z^{a} and g = x^{2} + y^{3} + z^{5}.

Prove that (,) is negative definite (for f one can use also (3.8)). Notice that (,)_{f} is the negative of the A_{a−1}–Cartan matrix. Compute det(,)_{f} and deduce that |T orsH_{1}(K_{f}, Z)| = a (cf. E.1.16.b). Verify this by computing Δ(1) too. (In fact H_{1}(K_{f}, Z) = π_{1}(K_{f}) = Z_{a}, cf. E.1.17.a).

(b) Prove that for any f the intersection form (,) is even, i.e. (v, v) ∈ 2Z for any v. (c)* Identify (,)_{g} with the negative of the E_{8}–Cartan matrix (the unique unimodular

even form over Z with rank 8 and signature −8). E.3.10. The Poincaré sphere. Consider the link K_{f} of f = x^{2} + y^{3} + z^{5}. Show that K_{f} is an integer homology sphere (cf. 3.13 and 3.7), but it is not a homotopy sphere (cf. E.1.17.b*, or Mumford’s theorem in 1.17). E.3.11. Exotic spheres on Brieskorn manifolds. [Brieskorn, Hirzebruch, Mayer] (The abstract link K_{f} of a Brieskorn singularity f is called Brieskorn manifold.)

Consider the Brieskorn singularity f = z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{2} + z_{5}^{2} + z_{6}^{3} + z_{7}^{5}.

(a) Using (E.3.8-E.3.9), show that (,) is negative definite, in particular σ(f) = −μ(f) = −8.

(b) Using (3.13) and (E.3.8) deduce that Δ(1) = 1, hence K_{f} is an integer homology sphere. Therefore, by (1.18), ∂F = K_{f} is homeomorphic to S^{11}.

(c) By (b), (and using that F is 5–connected), deduce that the quotient space F/∂F is a 12–dimensional topological manifold, without boundary, it is 5-connected and its signature is −8.

(d) If X is a differentiable 12–dimensional manifold, then by the Hirzebruch signature theorem its signature can be computed from its L–class, namely

signature(X) = \frac{1}{3^{3} · 5 · 7}(62p_{3} − 13p_{2}p_{1} + 2p_{1}^{3})[X],

where p_{i} ∈ H^{4i}(X, Z) are the Pontrjagin classes. Deduce that if X is 5–connected (hence p_{1} = 0), then the signature of X is divizible by 62.

(e) Deduce that the topological manifold F/∂F does not carry any differentiable structure.

(f) Deduce that ∂F is an exotic sphere (i.e. it is a differentiable manifold, homeomorphic to the standard sphere but not diffeomorphic to it.) (If ∂F were the standard sphere S^{11}, which is the boundary of the ball B^{12}, then F ∪_{S^{11}} B^{12} = F/∂F would have a differentiable structure.)

(g)* We can find exotic spheres among Brieskorn manifolds even in smaller dimension. Consider f_{k} = z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + z_{4}^{3} + z_{5}^{6k−1}. Then, for k = 1, 2, …, 28, one obtains all 28 possible differentiable structures on the 7–sphere (Milnor’s 7–spheres). (But the proof of this fact is more complicated.) E.3.12. A’Campo’s example of non–finite monodromy. [1] (cf. 3.8.)

Consider the plane curve singularity g = (x^{2} + y^{3})(x^{3} + y^{2}).

(a) Computing the resolution graph G_{g} of g (cf. 4.5), and using A’Campo’s theorem (E.2.6), deduce that Δ_{g}(t) = (t − 1)(t^{5} + 1)^{2}.

(b) Consider f = g + z^{2}, where z is a new variable. Using Sebastiani–Thom property, prove that Δ_{f}(t) = (t + 1)(t^{5} − 1)^{2}. In particular, the generalized 1–eigenspace H_{2}(F_{f})_{1} has dimension 2.

(c) Using the Appendix, find the resolution graph of (V_{f}, 0). (The minimal resolution looks as follows: there are only two irreducible exceptional divisors, both rational with self intersection e = −3, without auto intersection points, and they intersect each other in two points. Cf. Appendix.)

(d) Using (E.1.16.a), show that H_{1}(K_{f}) has rank one, therefore (by 3.7) dim ker(h_{f} − 1) = 1. Deduce that h_{f} has a (2 × 2)–Jordan block with eigenvalue 1 (cf. 3.8).

(e) Deduce that the monodromy h_{g} of g has a (2 × 2)–Jordan block with eigenvalue −1.

§4. The equivariant signature of plane curve singularities. ϵ–hermitian variation structures.

4.1. In this section we discuss the topological invariants of isolated plane curve singularities f : (C^{2}, 0) → (C, 0). Recall (cf. 1.13’) that the topological type of f is completely determined by the embedded resolution graph G_{f} of f. For G_{f}, we will use the notations of (1.14). Actually, in this section we will learn how one can read (some of) the topological invariants from G_{f}. 4.2. The number r of irreducible components is the same as the number of link components of K_{f} ⊂ S_{ϵ}^{3} (cf. E.1.11). On the graph, it is the number of arrowhead vertices #A. 4.3. The characteristic polynomial Δ(t) is given by A’Campo’s formula (cf. E.2.6):

Δ(t) = (t − 1) ∏_{w∈W}(t^{m_{w}} − 1)^{δ_{w}−2},

where δ_{w} = #V_{w}.

This gives the semisimple part of the complexified monodromy h. The above formula shows also that if N is multiple of all the multiplicities m_{w} (w ∈ W), then λ^{N} = 1 for all the eigenvalues λ of h (cf. 2.18.a).

This, together with the Monodromy Theorem (2.18) gives that (h^{N} − 1)^{2} = 0. (Actually, the monodromy restricted on the generalized 1–eigenspace is the identity.)

For the completeness of the discussion, we present here a result of W. Neumann (without proof). Neumann used the notation Δ^{1}(t) for the characteristic polynomial of the restriction h| ker(h^{N} − 1). Notice that Δ(t) and Δ^{1}(t) determine the Jordan normal form of h.

4.4. Theorem.* [Neumann] [43]

Δ^{1}(t) = (t − 1) ∏_{e∈E}(t^{d_{e}} − 1)/ ∏_{w∈W}(t^{d_{w}} − 1),

where for any e = (w_{1}, w_{2}) ∈ E we let d_{e} := g.c.d.(m_{w_{1}}, m_{w_{2}}); and for any w ∈ W we let d_{w} := g.c.d.(m_{v}| v ∈ V_{w} ∪ {w}).

4.5. Example. The minimal resolution graph of A’Campo’s plane curve singularity f = (x^{2} + y^{3})(x^{3} + y^{2}) (cf. E.3.12) is:

[figure: −2; −1; −5; −1; −2; (5); (10); (4); (10); (5); (1); (1)]

Then, by (4.4), Δ^{1}(t) = t + 1, hence there is exactly one Jordan block of size 2, and its eigenvalue is −1 (cf. E.3.12).

4.6. The Milnor fiber, together with its intersection form, can be easily determined. Indeed, as an abstract 2–dimensional real, oriented surface with r boundary components, it is completely determined by r and and the rank μ(f) of H_{1}(F). μ(f) reads from the graph as follows (cf. E.2.6 or 4.3):

1 − μ(f) = ∑_{w∈W} m_{w}(2 − δ_{w}).

(,) is skew–symmetric, dim ker(,) = r−1, and H_{1}(F, Z) = ker(,)⊕H, where (,) restricted

on H is unimodular, i.e. represented in a “good base” has the form ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ).

The abstract Milnor fiber says very little about the topological type of f. For example, f = x^{3} + y^{5} and g = x^{2} + h^{9} have diffeomorphic Milnor fibers, but obviously their open book decomposition (or link) is different (compute e.g. their characteristic polynomial.). 4.7. On the other hand, the isometric structure I = (H_{1}(F, Z); h, (,)) (i.e. h considered as an element of the orthogonal group of (,)) is a rather strong invariant of f. In E.4.3, we will give an example of two germs for which the invariants Δ(t) and Δ^{1}(t) (hence the Jordan normal forms) are the same, r and μ (hence the Milnor fibers and the intersection forms) are the same, but their isometric structure is not the same, even over C, (i.e. the monodromy operators are not conjugate in the orthogonal group of (,)). The complexified isometric structure I_{C} (where (,)_{C} is a (−1)–hermitian with (a, b) = −\overline{(b, a)}) has a direct sum decomposition ⊕(H_{n}(F, C)_{λ}, (,)_{λ}) corresponding to the generalized eigenspaces of the monodromy operator. As we will see later, if h is diagonalizable, then (H_{n}(F, C)_{λ}, (,)_{λ}) is completely determined by μ_{λ} := dim H_{n}(F, C)_{λ} (i.e. by the exponent of t − λ in Δ(t)) and σ_{λ}. In general, I_{C} is characterized by Δ(t), Δ^{1}(t) and the collection of equivariant signatures {σ_{λ}}_{λ}. Actually, even the real Seifert form can be deduced from these index type invariants. If (,) is nondegenerate (i.e. r = 1) then V = (h − 1) ∘ b^{−1} (cf. 2.22), but even in general, by a result of Neumann, Δ(t), Δ^{1}(t) and {σ_{λ}}_{λ} determine the real Seifert form. This shows the importance of these invariants.

In the following theorem, we compute the equivariant signature in terms of the resolution graph. A vertex w ∈ W is called rupture point if δ_{w} ≥ 3. The set of rupture points is denoted by R. We introduce also the following number theoretical function: ((x)) = 1/2 − {x} in x ∉ Z, and = 0 if x ∈ Z, where {x} denotes the fractional part of x. (In some papers, ((·)) is introduced with the opposite sign convention.)

4.8. Theorem. [W. Neumann] [43] Fix λ = e^{2πip/q}, where g.c.d(p, q) = 1. Then σ_{λ} = ∑_{w∈R} σ_{λ}^{w}, where for w ∈ R :

σ_{λ}^{w} = \{ \begin{matrix}0 & if q |̸ m_{w} \\ 2 ∑_{v∈V_{w}}((\frac{m_{v}·p}{q})) & if q|m_{w}. \end{matrix}

4.9. Remark. The original proof of W. Neumann is based on a difficult index theoretical result of Atiyah–Patodi–Singer, namely on the theory of eta–invariants. The theorem can be reproved using Hodge theory as follows: σ_{λ} can be described from the equivariant Hodge numbers associated with the mixed Hodge structure on H^{1}(F, C) by a result of J. Steenbrink, and these numbers are computed from G_{f} by Schrauwen–Steenbrink–Stevens. We present here a new, completely elementary proof.

Before we start the proof of (4.8), we recall the following additivity property of the signature with respect to pasting along boundary components. 4.10. Novikov’s additivity property: Notice that the Euler–characteristic, or characteristic polynomial is additive in exact sequences (e.g. with respect to the decomposition of spaces, by a Mayer–Vietoris argument). The behavior of the signature is more complicated. Nevertheless, we have the following additivity theorem for manifolds. We recall that if X is a (2k)–manifold (maybe with boundary), then H_{k}(X, R) carries a (−1)^{k}–symmetric intersection form (,), and the signature of X is defined as the signature of this form. If m : X → X is a diffeomorphism then it induces at homology level an automorphism h of H_{k}(X), which is in the orthogonal group of (,), and the equivariant signatures are the signatures of the hermitian form obtained by restricting the intersection form to the generalized eigenspaces of h.

Let X_{i} be two oriented manifolds with boundary ∂X_{i} (i = 1, 2). Assume that the disjoint union B_{1} of some of the connected components of ∂X_{1} is diffeomorphic to the disjoint union B_{2} of some of the connected components of ∂X_{2} with opposite orientation. We construct the manifold X by gluing X_{1} and X_{2} along boundary components B_{1} and −B_{2} (via that diffeomorphism). Novikov’s theorem asserts that: σ(X) = σ(X_{1}) + σ(X_{2}). If m : X → X is a diffeomorphism such that m(X_{i}) = X_{i} (i = 1, 2), then:

σ_{λ}(X) = σ_{λ}(X_{1}) + σ_{λ}(X_{2}).

4.11. Proof of 4.8. Consider an embedded resolution ϕ : (Y, D) → (C^{2}, f^{−1}(0)) as in (1.14). Replace it by one of its representatives ϕ : Y → B_{ϵ}, where B_{ϵ} is also a Milnor ball of f. Let F = f^{−1}(δ) ∩ B_{ϵ} and take F′ := ϕ^{−1}(F). Obviously, the restriction ϕ : F′ → F is an isomorphism. F′ is the nearby fiber in Y of the central fiber D = (f ∘ ϕ)^{−1}(0) ∩ B_{ϵ}. In particular, there exists a continuous surjection π : F′ → D, such that, for any singular point P of D, π^{−1}(P) is a union of S^{1}’s, and if E_{w}^{reg} denotes the regular part of E_{w}, then F′_{w} := π^{−1}(E_{w}^{reg}) → E_{w}^{reg} is an m_{w}–fold regular covering whose Galois action corresponds to the restriction of the geometric monodromy m to F′_{w}. Similarly, π : F′_{a} := π^{−1}(S_{a}^{reg}) → S_{a}^{reg} (for any a ∈ A) is an m_{a}–covering. (Since m_{a} = 1, actually this is an isomorphism.) (Fore some of the arguments, see below.)

Now, consider the decomposition F′ = (∪_{w}F′_{w}) ∪ (∪_{a}F′_{a}), which is compatible with the geometric monodromy action m. Therefore, by Novikov’s theorem, σ_{λ}(f) = ∑_{w} σ_{λ}(F′_{w}) + ∑_{a} σ_{λ}(F′_{a}). Since F′_{a} is a punctured disc with a trivial action, its equivariant signature is trivial. Moreover, if δ_{w} ≤ 2, then F′_{w} is either a disk (if δ_{w} = 1) or a punctured disk (if δ_{w} = 2), in both cases they carry no equivariant signature (because (,) is trivial). Therefore σ_{λ}(f) = ∑_{w∈R} σ_{λ}(F′_{w}).

Now, we will study more carefully the covering F′_{w} → E_{w}^{reg} (w ∈ R). If P is a regular point of D on E_{w} then F′_{w} in a neighborhood U_{P} of P has local equation x^{m_{w}} = δ, hence π is an m_{w}–fold covering. The monodromy action is the cyclic permutation of the connected components of F′_{w} ∩ U_{P}, which is exactly the Galois action of the covering group Z_{m_{w}}. We denote the corresponding representation by ρ : π_{1}(E_{w}^{reg}) → Z_{m_{w}}, where ˆ1 corresponds to the Galois action. If P is a singular point of D on E_{w}, then F′_{w} locally is given by x^{m_{w}}y^{m_{v}} = δ, where {x = 0} is the local equation of E_{w}, and the projection π is (x, y) ↦ y. If (x, y) = (0, ηe^{2πit}), 0 ≤ t ≤ 1, is a small oriented loop γ around P in E_{w}, then π^{−1}(γ) is given by x^{m_{w}} = δη^{−m_{w}}e^{−2πitm_{w}}. Therefore, ρ([γ]) = ˆ−m_{w}.

For simplicity, we will assume that δ_{w} = 3 for any w ∈ R. (Actually, this is the case, for example, for the minimal resolution graph of any irreducible plane curve singularity.) The reader can try to extend the proof to the general case.

In the sequel, we will fix a vertex w ∈ R and we will use the notations m := m_{w}, and {m_{1}, m_{2}, m_{3}} = {m_{v}}_{v∈V_{w}}. E_{w} can be diffeomorphically identified with C \ {±2}. Then π is an m–fold covering π : X → B, given by the representation ρ : L(∂^{−}, ∂^{+}) → Z_{m}, ρ(∂^{−}) = ˆ−m_{1}, ρ(∂^{+}) = ˆ−m_{2} and Galois action corresponding to ˆ1. Here L(∂^{−}, ∂^{+}) = π_{1}(B) is the free group generated by two generators ∂^{∓} = [t ↦ ∓2 + e^{2πit}].

We will assume also that g.c.d.(m, m_{1}, m_{2}, m_{3}) = 1, the general case follows easily from this one.

Set B′ = {z ∈ B : Re(z) ≤ 0}, B′′ = {z ∈ B : Re(z) ≥ 0}, C = B′ ∩ B′′, X′ = π^{−1}(B′), X′′ = π^{−1}(B′′), Y = π^{−1}(C), P = (0, i), Q = (0, −i). Since Y → C is a trivial fibration, we can fix a trivialization T : {1, 2, …, m} × C → Y, and let P_{i} = T ({i} × P), Q_{i} = T ({i} × Q).

Let γ′_{P} (resp. γ′_{Q}) be an oriented closed path in B′ with endpoints at P (resp. Q) homotop to ∂^{−}. Similarly, let γ′′_{P} (resp. γ′′_{Q}) be a closed path in B′′ with endpoints at P (resp. Q) homotop to ∂^{+}. Let e_{i}^{P} (resp. e_{i}^{Q}) an oriented path in X′, lifting of γ′_{P} (resp γ′_{Q}) with starting point P_{i} (resp Q_{i}). Similarly, let f_{i}^{P} (resp. f_{i}^{Q}) be the oriented path in X′′ with starting point P_{i} (resp Q_{i}) which is the lifting of γ′′_{P} (resp. of γ′′_{Q}).

Consider the homology exact sequence (over Z) of the pair (X, Y):

(∗) 0 → H_{1}(X) → H(X, Y) ⟶^{∂} H_{0}(Y) → H_{0}(X) → 0.

Now, H_{0}(X) = Z, because g.c.d.(m_{1}, m_{2}, m) = 1. H_{0}(Y) = Z^{m}, where a base is given by the points {P_{i}}_{i=1}^{m}. The monodromy action is P_{i} ↦ P_{i+1}. (In our notation an index i is always identified with i + km, k ∈ Z, i.e. the index, in fact, lives in Z_{m}.) Therefore, h(∑_{i} a_{i}P_{i}) = ∑_{i} a_{i}P_{i+1} = ∑_{i} a_{i−1}P_{i}, hence the monodromy action is:

h : Z^{m} → Z^{m}, h(a_{1}, …, a_{m}) = (a_{m}, a_{1}, …, a_{m−1}).

Notice that H_{1}(X, Y) = H_{1}(X′, Y) ⊕ H_{1}(X′′, Y). H_{1}(X′, Y) = Z^{m}, where a base is given by the class of paths {e_{i}^{P} }_{i} (notice the identity of the relative homology classes [e_{i}^{P}] = [e_{i}^{Q}]). Since ∂[e_{i}^{P}] = P_{i−m_{1}} −P_{i}, one has: ∂(∑ a_{i}e_{i}^{P}) = ∑ a_{i}(P_{i−m_{1}} −P_{i}) = ∑(a_{i+m_{1}} −a_{i})P_{i}, hence ∂(a) = (h^{−m_{1}} − 1)a.

Similarly, H_{1}(X′′, Y) = Z^{m} is generated by [f_{i}^{P}]_{i}, and ∂[f_{i}^{P}] = P_{i−m_{2}} − P_{i}, hence ∂(a) = (h^{−m_{2}} − 1)a. Therefore (∗) reads as:

(∗∗) 0 → H_{1}(X) → Z^{m} ⊕ Z^{m} ⟶^{∂} Z^{m} → Z → 0,

∂(a, b) = (h^{−m_{1}} − 1)a + (h^{−m_{2}} − 1)b,

where h is the permutation h(a_{1}, …, a_{m}) = (a_{m}, a_{1}, …, a_{m−1}).

If (a, b) ∈ ker ∂, then ∑ a_{i}e_{i}^{P} +∑ b_{i}f_{i}^{P} is a closed 1–cycle in X, representing an element in H_{1}(X). We want to describe the intersection form (,) on H_{1}(X) (i.e. on ker ∂) in terms of (a, b). Let (a, b) and (a′, b′) be elements of ker ∂. It is convenient to represent (a, b) as ∑ a_{i}e_{i}^{P} + ∑ b_{i}f_{i}^{P} and (a′, b′) as ∑ a′_{i}e_{i}^{Q} + ∑ b′_{i}f_{i}^{Q}. Now, notice that one has the following intersection of cycles:

(e_{i}^{P} , e_{j}^{Q}) = \{ \begin{matrix}−1 & if ĵ = î \\ +1 & if ĵ = î − m_{1} \\ 0 & otherwise. \end{matrix}

(f_{i}^{P} , f_{j}^{Q}) = \{ \begin{matrix}+1 & if ĵ = î \\ −1 & if ĵ = î + m_{2} \\ 0 & otherwise. \end{matrix}

Therefore, (∑ a_{i}e_{i}^{P} , ∑ a′_{j}e_{j}^{Q}) = ∑_{i} a_{i}(−a′_{i} +a′_{i−m_{1}}) =< a, (h^{m_{1}} −1)a′ >_{s}, where < x, y >_{s} is the standard inner product ∑ x_{i}y_{i} (or the standard hermitian product ∑ x_{i}\overline{y_{i}}). Similarly, (b_{i}f_{i}^{P} , ∑ b′_{j}f_{j}^{Q}) = ∑ b_{i}(b′_{i} − b′_{i+m_{2}}) = − < b, (h^{−m_{2}} − 1)b′ >_{s}. Therefore, the pairing:

(∗ ∗ ∗) ((a, b), (a′, b′)) =< a, (h^{m_{1}} − 1)a′ >_{s} − < b, (h^{−m_{2}} − 1)b′ >_{s}

restricted on ker ∂ provides the intersection form (,) of H_{1}(X).

Now, we can consider the eigenspace decomposition of the monodromy action h (∂_{λ} can be obtained from ∂ by replacing h by λ).

If λ = 1, then (∗∗) becomes:

0 → H_{1}(X, C)_{1} → C ⊕ C ⟶^{∂_{1}} C → C → 0,

where ∂_{1} = 0 and (,)_{1} = 0. In particular, σ_{1}(F′_{w}) = 0 (cf. E.4.2).

If λ ≠ 1, λ = e^{2πil/m}, then (∗∗) gives:

0 → H_{1}(X, C)_{λ} → C ⊕ C ⟶^{∂_{λ}} C → 0,

where ∂_{λ}(a, b) = (λ^{−m_{1}} −1)a+(λ^{−m_{2}} −1)b. Then ker ∂_{λ} is generated by (u, v) = (−(λ^{−m_{2}} − 1), λ^{−m_{1}} − 1). By a computation: ((u, v), (u, v))_{λ} =< u, (λ^{m_{1}} − 1)u >_{s} − < v, (λ^{−m_{2}} − 1)v >_{s}= (λ^{−m_{1}} − 1)(λ^{−m_{2}} − 1)(λ^{m_{1}+m_{2}} − 1) = −8i sin(πm_{1}l/m) sin(πm_{2}l/m) sin(π(m_{1} + m_{2})l/m). Therefore:

σ_{λ}(F′_{w}) = sign(sin(πm_{1}l/m) sin(πm_{2}l/m) sin(π(m_{1} + m_{2})l/m)).

Now use ((−x)) = −((x)), the fact that m_{1} + m_{2} + m_{3} ≡ 0(mod m), and (cf. E.4.8):

2((s)) + 2((t)) − 2((s + t)) = sign(sin(πs) sin(πt) sin(π(s + t))).

This ends the proof. □

4.12. ϵ-hermitian variation structures (ϵ = ±1). [34, 31]

In this subsection U is a complex vector space with a complex conjugation, denoted by ¯·. U^{∗} denotes its dual Hom_{C}(U, C). There is a natural isomorphism θ : U → U^{∗∗}, given by θ(u)(φ) = φ(u). If φ ∈ Hom_{C}(U, U′), then φ̄ ∈ Hom_{C}(U, U′) is defined by φ̄(x) := \overline{φ(x̄)}. The dual φ^{∗} : U^{′∗} → U^{∗} of φ is defined by φ^{∗}(ψ) = ψ ∘ φ. 4.13. Definition. An ϵ–hermitian variation structure is a system (U; b, h, V), where U is as above, and:

(a) b : U → U^{∗} is a C–linear endomorphism with \overline{b^{∗} ∘ θ} = ϵb;

(b) h is a b–orthogonal automorphism of U, i.e. \overline{h^{∗}} ∘ b ∘ h = b;

(c) V : U^{∗} → U is a C–linear endomorphism, with \overline{θ^{−1} ∘ V^{∗}} = −ϵV ∘ h̄^{∗}, and V ∘ b = h − id.

If b is an isomorphism, then V = (h − id)b^{−1}, in particular, the structure (U; b, h, V) is completely determined by the isometric structure (U; b, h).

If V is an isomorphism, then h = −ϵV \overline{(θ^{−1} ∘ V^{∗})}^{−1}, and b = −V^{−1} − ϵ\overline{(θ^{−1} ∘ V^{∗})}^{−1}. The structures with this property are called “simple variation structures”. For the classification of (non-degenerate) isometric structures, see [Milnor] [27], and for the classification of simple variation structures, see [Némethi] [34].

Any base {e_{i}}_{i} of U defines a dual base {e_{i}^{∗}}_{i} of U^{∗} by e_{j}^{∗}(ē_{i}) = 1 if j = i and = 0 otherwise. It is convenient in the description of the variation structures to use the matrix representation in a convenient base of U and its dual base. Then θ corresponds to the identity matrix. If an endomorphism φ : U → U′, in a given base, has matrix representation A, then φ̄^{∗} in the dual base is represented by the transposed matrix Ā^{t}. 4.14. Example/Definition of V(f). If f : (C^{n+1}, 0) → (C, 0) is an isolated singularity, then by the results of section 2, (U = H_{n}(F, C), b =the complex (−1)^{n}-symmetric hermitian intersection form, h=the complexified monodromy operator, and V =the complexified variation map) form a (−1)^{n}-hermitian variation structure, which is simple, by (3.10), (cf. (3.11) as well). This variation structure is denoted by V(f).

For simplicity of our discussion, in all our examples and result in the sequel, we will assume that h_{nil} = 0, i.e. the complex monodromy operator is diagonalizable. 4.15. Examples of simple ϵ–hermitian variation structures: [34] (see also [44, 43].) We will use the notation ϵ = (−1)^{n} (cf. 4.14). For |λ| = 1, we define:

W_{λ}(±1) = (C; ±i^{−n^{2}}, λ, ±(λ − 1)i^{n^{2}}), if λ ≠ 1, and
W_{1}(±1) = (C; 0, 1_{C}, ±i^{n^{2}+1}) if λ = 1.

It is easy to verify that they are (−1)^{n}–hermitian variation structures.

4.16. Theorem. [34] Any simple ϵ–hermitian variation structure, with h_{nil} = 0 and eigenvalues of h on the unit circle S^{1}, is uniquely expressible as a direct sum of indecomposable ones up to order of summands and isomorphism. The indecomposable structures are: W_{λ}(±1) (λ ∈ S^{1}).

Proof. First put h in the Jordan normal form. Then, via the relations (4.13), the classification is just the classification of hermitian forms. □

Now, we will consider the (ϵ = −1)–variation structure V(f) associated with an isolated plane curve singularity with h_{nil} = 0 (cf. 4.14). By (4.16) one has:

(4.17) V(f) = ⨁_{λ} p_{λ}^{+} · W_{λ}(+1) ⊕ ⨁_{λ} p_{λ}^{−} · W_{λ}(−1).

The question is: can any combination, as represented in the right hand side of (4.17), be realized as the variation structure V(f) of some f? The answer is no. The first restriction appears immediately. Since V(f) is the complexification of a real structure, it is stable with respect to the complex conjugation. Therefore (cf. E.4.2):

(4.18) p_{λ}^{+} = p− ; λ̄ provided that λ ≠ 1.

The next restriction is not so evident. By (4.13.c), the restriction of V on ker(h^{∗} − 1) satisfies \overline{V}^{∗} = V, hence it is a symmetric form. It can be diagonalized with p_{1}^{+} entries of −1 and p_{1}^{−} entries of +1 on the diagonal.

4.19. Proposition. [Neumann] [43] p_{1}^{−} = 0, or equivalently, the Seifert form S restricted on ker(h − 1) = ker(,) is a symmetric negative definite form. In particular, p_{1}^{+} = dim ker(h − 1) = r − 1. Proof. The Milnor fiber has r oriented boundary components corresponding to the irreducible components {f_{i}}_{i=1}^{r} of f. Their homology classes in H_{1}(F, Z) are denoted by {d_{i}}_{i}. They generate ker(h − 1) = ker(,) with the relation ∑_{i} d_{i} = 0. Therefore, V (∑_{i} a_{i}d_{i}, ∑_{j} a_{j}d_{j}) = ∑_{i} a_{i}V (d_{i}, ∑_{j≠i}(a_{j} − a_{i})d_{j}) = ∑_{i≠j} a_{i}(a_{j} − a_{i})V (d_{i}, d_{j}). Now, V (d_{i}, d_{j}) = V (d_{j}, d_{i}) = i_{0}(f_{i}, f_{j}) > 0 (cf. Reeve’s theorem 1.13 or E.3.1). Hence: V (∑ a_{i}d_{i}, ∑ a_{i}d_{i}) = − ∑_{i<j}(a_{i} − a_{j})^{2} · i_{0}(f_{i}, f_{j}). □ 4.20. Let μ_{λ} = dim H_{1}(F, Z)_{λ} and σ_{λ} the corresponding equivariant signature. Then, for λ ≠ 1 and ϵ = (−1)^{n} = −1, one has: σ_{λ}(W_{λ}(±1)) = σ_{λ}(±i^{−1}) = ±1. Therefore:

(4.21) μ_{λ} = p_{λ}^{+} + p_{λ}^{−}, σ_{λ} = p_{λ}^{+} − p_{λ}^{−}.

4.22. Corollary. [Neumann] Assume that f : (C^{2}, 0) → (C, 0) satisfies h_{nil} = 0. Then:

V(f) = (r − 1) · W_{1}(+1) ⊕ ⨁_{λ≠1} \frac{μ_{λ} + σ_{λ}}{2} · W_{λ}(+1) ⊕ ⨁_{λ≠1} \frac{μ_{λ} − σ_{λ}}{2} · W_{λ}(−1).

(Notice that for λ ≠ 1, the relation p_{λ}^{+} = p− ; λ̄ corresponds to μ_{λ} = μλ̄ and σ_{λ} = −σλ̄.)

Open problems and conjectures.

4.23. Classify the ϵ–hermitian variation structures. (The problem is important for the following reason. Let f : (X, x) → (C, 0) be an analytic germ such that both (X, x) and (f^{−1}(0), 0) have only isolated singularities (and the dimension is arbitrary). Then, similarly as in the hypersurface case, f defines a variation structure, which is simple if and only if the link of (X, x) is a rational homology sphere, which is not the case in general.) In order to understand the difficulties of the classification, see E.4.9. 4.24. Generalize (4.8) for arbitrary dimensions. 4.25. Find algebraicity restrictions, as in (4.19), for higher dimensional variation structures (cf. [34]).

Exercises for the fourth section.

E.4.1. (a) Compute {σ_{λ}(f)}_{λ} for f = x^{2} + y^{3}. Prove that V(x^{2} + y^{3}) = W_{exp(πi/3)}(+1) ⊕ W_{exp(−πi/3)}(−1).

(b) Prove that V(x^{2} + y^{2}) = W_{1}(+1). E.4.2. (a) Prove that (,)| ker(h − 1) is trivial.

(b) Prove that σ_{λ} = (−1)^{n}σλ̄ (for any λ), and p_{λ}^{−} = p+ ; λ̄ (for λ ≠ 1, cf. 4.18).

(c) Using either (a) or (b), prove that σ_{1} = 0 provided that n = 1.

(d) If n = 2, then in general σ_{1} ≠ 0. Isn’t this in contradiction with (a)?

(e) Prove that σ_{1} ≠ 0 if f = (x^{2} + y^{3})(x^{3} + y^{2}) + z^{2} (cf. E.3.12 and 4.5). Hint for (e): Notice that the different generalized eigenspaces are orthogonal. In our case H_{1}(F)_{1} is two-dimensional, and ker(,) is its one–dimensional subspace. Therefore σ_{1} = 0 would imply that (,)_{1} is trivial, hence H_{1}(F)_{1} = ker(,), which is false. Remark.* Actually, from Hodge theory, (see, e.g. [34]), for any isolated hypersurface singularity with n = 2, the equivariant signature σ_{1}(f) is non-negative. Therefore, in our case (E.4.2.e): σ_{1}(f) = 1. E.4.3.* Marie-Claire Grima’s family. (Cf. [44].)

Assume that p + r = q + s and ps < qr, and g.c.d.(pe, qf) = g.c.d.(re, sf) = g.c.d.(pf, qe) = g.c.d.(rf, se) = 1. Set f = (x^{pe} + y^{qf})(x^{re} + y^{sf}) and g = (x^{pf} + y^{qe})(x^{rf} + y^{se}). Then f and g have the same Δ(t), Δ^{1}(t), μ, r, h⊗C, F, (,). But, their equivariant signatures are different.

Compute the case (p, q, r, s, e, f) = (1, 3, 5, 3, 2, 1). E.4.4.* The quasi–homogeneous case. The complex (real) Seifert form. [Némethi] [34]

Assume that f : (C^{n+1}, 0) → (C, 0) is quasi–homogeneous of weights {w_{i}}_{i=1}^{n+1}. Fix a set of monomials {z^{α} : α ∈ Λ ⊂ N^{n+1}}, such that {[z^{α}]}_{α} is a base of M(f) (cf. E.1.8.a). We define (cf. E.1.8.b) l(α) = ∑_{i=1}^{n+1}(α_{i} + 1)/w_{i}. Then

V(f) = ⨁_{α∈Λ} W_{exp(2πil(α))}((−1)^{[l(α)]+n}),

where [·] is the integer part function. Remark. The codification of {z^{α}}_{α∈Λ} in Z[Q] via z^{α} ↦ (l(α)) gives P_{M(f)}(t); its codification in Z[Q/2Z] via z^{α} ↦ W_{exp(2πil(α))}((−1)^{[l(α)]+n}) gives V(f), and finally the codification in Z[Q/Z] via z^{α} ↦ (exp(2πil(α))) provides Δ(t). The natural projections Z[Q] → Z[Q/2Z] → Z[Q/Z] give the corresponding relations between the Poincaré polynomial, (real) variation map (= real Seifert form) and characteristic polynomial. E.4.5. (a) Prove that V(x^{2} + y^{2} + z^{a}) = ⊕_{k=1}^{a−1}W_{exp(2πik/a)}(−1).

(b) Using (E.4.4), compute V(x^{2} + y^{3} + z^{5}).

(c) Notice that in both cases l(α) ∈ (1, 2) for any α ∈ Λ, hence (,) is negative definite. Remark. If n = 2, f quasi–homogeneous, then l(α) ∈ (1, 2) for any α ∈ Λ if and only if (f^{−1}(0), 0) is a rational normal surface singularity. Using Hodge theory (namely the spectrum of f), this can be generalized for any hypersurface singularity. E.4.6. Non–additivity of the signature.

Consider the space X from (4.11) together with the monodromy action X → X and its decomposition X = X′ ∪ X′′, which is compatible with the monodromy action. Then σ_{λ}(X′) = σ_{λ}(X′′) = 0 for any λ, but σ_{λ}(X) ≠ 0 in general. Actually, the important invariant σ_{λ}(f) appears exactly as the measure of this non–additivity.

Is this in contradiction with Novikov’s additivity theorem?

(In general, the non–additivity defect was computed by C.T.C. Wall.) E.4.7. (a) The function ((x)) is odd, and its Fourier sine expansion is:

((x)) = \frac{1}{π} ∑_{k≥1} \frac{1}{k} sin(2πkx).

(b) For any x ∈ R, l, k ∈ N (with g.c.d.(k, l) = d) one has:

∑_{j=0}^{l−1}((k · \frac{x + j}{l})) = d · ((kx/d)).

E.4.8. Prove that

2((s)) + 2((t)) − 2((s + t)) = sign( sin(πs) sin(πt) sin(π(s + t))).

E.4.9. The following matrices define an indecomposable (+1)-variation structure, but h has two Jordan blocks:

b = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 1 \end{matrix} ) , h = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} ) , V = ( \begin{matrix}0 & −1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) .

§5. The signature and Dedekind sums.

5.1. In this section we will present Brieskorn’s formula for the signature of the Milnor fiber of Brieskorn singularities (in arbitrary dimension), and we will relate it with a famous, classical lattice point problem. In the second part, we will compute the signature of singularities of type f(x, y, z) = g(x, y) + z^{a} in terms of generalized Dedekind sums, showing the deep arithmetical nature of the signature.

We hope also, that this section will convince the reader about the elegance of the variation structures.

Since the definition of a variation structure W_{λ}(±1) depends on ϵ = (−1)^{n}, sometimes we write W_{λ}(±1)_{n}. 5.2. Consider a Brieskorn singularity f : (C^{n+1}, 0) → (C, 0) given by f = ∑_{i=1}^{n+1} z_{i}^{a_{i}}, and assume that n is even. Then the intersection form (,) of the Milnor fiber is symmetric. We will compute its signature σ(f) in terms of {a_{i}}_{i}.

Start with the singularity z ↦ z^{a} (i.e. n = 0). Then:

(5.3) V(z^{a}) = ⨁_{k=1}^{a−1} W_{exp(2πik/a)}(+1)_{n=0}.

Since W_{ξ}(+1)_{n=0} = (C; 1, ξ, ξ − 1), this is equivalent to:

(5.4) V(z^{a}) = ⨁_{k=1}^{a−1}(C; 1, e^{2πik/a}, e^{2πik/a} − 1).

By Sebastiani–Thom and Deligne–Sakamoto theorem (cf. E.2.11 and E.3.5), the variation map V (f) and the monodromy h(f) of f = ∑_{i} z_{i}^{a_{i}} satisfy:

(V (f), h(f)) = ⊕′(V_{k}, h_{k}),

where ⊕′ = ⊕_{k_{1}=1}^{a_{1}−1} ⋯ ⊕_{k_{n+1}=1}^{a_{n+1}−1}, k = (k_{1}, …, k_{n+1}), and:

V_{k} = (−1)^{n(n+1)/2}(e^{2πik_{1}/a_{1}} − 1) ⋯ (e^{2πik_{n+1}/a_{n+1}} − 1), and h_{k} = e^{2πi ∑_{j=1}^{n+1} k_{j}/a_{j}}.

Now, since b = (h − 1)V^{−1}, one has b(f) = ⊕′b_{k}, where b_{k} = (h_{k} − 1)/V_{k}. By a computation:

b_{k} = \frac{sin(π ∑_{j} k_{j}/a_{j})}{2^{n} ∏_{j} sin(πk_{j}/a_{j})},

which has signature sign sin(π ∑_{j} k_{j}/a_{j}). Therefore:

5.5. Theorem. [Brieskorn] [9] If n is even, then:

σ(∑_{i=1}^{n+1} z_{i}^{a_{i}}) = ∑_{k_{1}=1}^{a_{1}−1} ⋯ ∑_{k_{n+1}=1}^{a_{n+1}−1} sign sin(π ∑_{j=1}^{n+1} k_{j}/a_{j}).

5.6. Corollary. Consider the following lattice point counting:

S_{t} := #{k ∈ Z^{n+1} : 1 ≤ k_{j} ≤ a_{j} − 1, (1 ≤ j ≤ n + 1), t < ∑_{j=1}^{n+1} k_{j}/a_{j} < t + 1}.

Then: σ(∑_{j} z_{j}^{a_{j}}) = ∑_{t}(−1)^{t}S_{t}. 5.7. If n = 2 then this formula is much simpler. First notice that, in general:

#{(k_{1}, …, k_{n+1}) ∈ Z^{n+1}; ∑_{j} k_{j}/a_{j} ∈ Z} = μ_{1},

where μ_{1} is the dimension of the 1–eigenspace H_{n}(F)_{1}. This follows from (E.2.12). Moreover, that exercise also shows that in our case (n = 2) one has:

μ_{1} = \frac{(a_{1}, a_{2})(a_{2}, a_{3})(a_{3}, a_{1})}{(a_{1}, a_{2}, a_{3})} − (a_{1}, a_{2}) − (a_{2}, a_{3}) − (a_{3}, a_{1}) + 2.

Now, if n = 2 and t ∉ {0, 1, 2} then S_{t} = 0. Moreover:

S_{0} = S_{2} by the symmetry of the lattice points,

S_{0} + S_{1} + S_{2} = μ_{≠1}(f) = (a_{1} − 1)(a_{2} − 1)(a_{3} − 1) − μ_{1},
S_{0} − S_{1} + S_{2} = σ(f).

Hence, (if n = 2):

(5.8) 4 · S_{0} = σ(f) + (a_{1} − 1)(a_{2} − 1)(a_{3} − 1) − μ_{1}.

Therefore, the computation of the signature is equivalent to the counting of the lattice points in the open tetrahedron T (a_{1}, a_{2}, a_{3}) with vertices (0, 0, 0), (a_{1}, 0, 0), (0, a_{2}, 0), and (0, 0, a_{3}). In particular cases (e.g. when a_{1}, a_{2}, a_{3} are small numbers) it is easy to count these lattice points, but in general it is not. There was a famous classical problem to give a precise formula of the number of these lattice points (i.e. of S_{0}) in terms of the numbers a_{1}, a_{2} and a_{3}. The problem was solved by L. J. Mordell [29] when the numbers {a_{i}} are pairwise relative prime; the general case was solved recently by Pommersheim. The formula provides S_{0} in terms of some Dedekind sums involving the a_{j}’s. In (5.18), we will see that our theorem (5.15) applied in the particular case of Brieskorn singularities will provide Mordell’s theorem.

5.9. Now, we will start to compute the signature of f(x, y, z) = g(x, y) + z^{a}, where g defines an isolated plane curve singularity (with h_{nil}(g) = 0) and a is an arbitrary integer.

Again, we will use Deligne–Sakamoto result: V (f) = V (g) ⊗ V (z^{a}). By theorem (4.22), we have to compute only tensor products of type: W_{λ}(±1)_{n=1} ⊗ V(z^{a})_{n=0} (and V(z^{a}) has the decomposition (5.4), hence we need to compute only one–dimensional tensor products). We proceed as follows: we make the product of the variation maps and eigenvalues, and compute b from the relation (λ − 1)/V. 5.10. Lemma. (a)

σ(W_{1}(+1)_{n=1} ⊗ V(z^{a})) = −a.

(b) Let λ = e^{2πiα} ≠ 1, then:

b(W_{λ}(±1)_{n=1} ⊗ W_{exp(2πik/a)}(+1)_{n=0})_{n=2} = ∓\frac{1}{2} · \frac{sin π(α + k/a)}{sin(πk/a) · sin(πα)}.

Proof. (a) Notice that V (W_{1}(+1)) = −1, hence tensorized with (e^{2πik/a} − 1) gives V = −(e^{2πik/a} − 1) and h = e^{2πik/a}, hence b = (h − 1)/V = −1 for any k = 1, …, a − 1. In the case of (b), the variation map of the tensor product is ±i(λ − 1)(e^{2πik/a} − 1), and its monodromy is λe^{2πik/a}. □ The expression of the right hand side of (b) can be transformed as follows: 5.11. Lemma.

∑_{k=1}^{a−1} sign \frac{sin π(α + \frac{k}{a})}{sin πα} = 2a((α)) − 2((aα)).

Proof. Consider the two cases l/a < α < (l + 1)/a and α = l/a. □ Now, (4.22), (5.10) and (5.11) give:

5.12. Theorem. [Némethi] [35, 36, 37] Assume that h_{nil}(g) = 0. Then, with the notation λ = e^{2πiα}, one has:

σ(g(x, y) + z^{a}) = −(r_{g} − 1)(a − 1) + 2 ∑_{λ≠1} σ_{λ}(g) · (((aα)) − a((α))).

5.13. Example.

(a) Let g = x^{2} + y^{2}. Then V(g) = W_{1}(+1), hence σ(g + z^{a}) = −(a − 1) (cf. E.3.9 and E.4.5).

(b) Let g = x^{2} + y^{3}. Then r_{g} = 1, and σ_{λ}(g) = ±1 if λ = exp(±2πi/6) (cf. E.4.1). Therefore:

σ(x^{2} + y^{3} + z^{a}) = 4 · (((a/6)) − a((1/6))).

Therefore, if 1 ≤ a < 6 then σ(f) = −μ(f) = 2(1 − a), hence (,)_{f} is negative definite. If a = 6, then σ(f) = −8, μ(f) = 10 and μ_{1}(f) = 2, hence (,)_{f} is negative semi–definite. If a = 7, then (,)_{f} is non–degenerate with μ = 12 and σ = −8. Actually, if a = 6t + 1, then σ(f) = −8t.

Now, using theorem (4.8), the signature σ(f) of g + z^{a} can be computed from the embedded resolution graph of g as follows:

σ(g + z^{a}) = −(r_{g} − 1)(a − 1) + 4 ∑_{w∈R} ∑_{v∈V_{w}} ∑_{l=1}^{m_{w}−1}((\frac{lm_{v}}{m_{w}})) · ( ((\frac{la}{m_{w}})) − a((\frac{l}{m_{w}})) ).

5.14. Definition. [Dedekind, Rademacher] [47, 46] For integers a, b, c (a ≠ 0) the generalized Dedekind sum is defined as

s(b, c; a) = ∑_{k=1}^{a−1}((\frac{kb}{a})) · ((\frac{kc}{a})).

The classical Dedekind sum of the pair (a, b) is s(b, 1; a).

5.15. Theorem. [Némethi] [36, 37] Assume that h_{nil}(g) = 0. From the embedded resolution graph G_{g} of g, the signature of g + z^{a} reads as follows:

σ(g + z^{a}) = −(r_{g} − 1)(a − 1) + 4 ∑_{w∈R} ∑_{v∈V_{w}}(s(m_{v}, a; m_{w}) − a · s(m_{v}, 1; m_{w})).

5.16. Corollary. [Conjectured by Brieskorn, Durfee and Zagier, first proved by Neumann]. The correspondence (a ↦ σ(g +z^{a})) is a sum of a periodic function and a linear function. 5.17. Example. If g = x^{2} + y^{3} then R has only one element {w} with m_{w} = 6 and {m_{v}}_{v∈V_{w}} = {1, 2, 3}. 5.18. Example.* [Némethi] [37]

Consider g : (C^{2}, 0) → (C, 0), g(z_{1}, z_{2}) = z_{1}^{a_{1}} + z_{2}^{a_{2}}, and set a = a_{3}. For simplicity, we will assume that g.c.d.(a_{1}, a_{2}) = 1. Fix integers b_{1} and b_{2} such that b_{1}a_{1} + b_{2}a_{2} = 1. Then W has only one element {w} with m_{w} = a_{1}a_{2} and the multiplicities {m_{v}}_{v∈V_{w}} modulo m_{w} are given by {−a_{1}b_{1}, −a_{2}b_{2}, 1}. Now, using the following properties of the Dedekind sums:

(5.19) s(b, c; a) = (a, b, c) · s(\frac{b}{(a, b)}, \frac{c}{(a, b, c)}; \frac{a}{(a, b)}), and

(5.20) s(b, c; a) = s(kb, kc; a) if (k, a) = 1;

and the famous Dedekind Reciprocity Law:

s(b, 1; a) + s(a, 1; b) = −\frac{1}{4} + \frac{a^{2} + b^{2} + (a, b)^{2}}{12ab} ,

one can deduce that:

σ(z_{1}^{a_{1}} + z_{2}^{a_{2}} + z_{3}^{a_{3}}) = −a_{3} \frac{(a_{1}^{2} − 1)(a_{2}^{2} − 1)}{3a_{1}a_{2}} + 4 · ∑_{v∈V_{w}} s(m_{v}, a_{3}; a_{1}a_{2}).

(Recall that the set {m_{v}} is given above.)

This formula is equivalent (via 5.7) with Mordell’s formula which gives the number of lattice points in the open tetrahedron T (a_{1}, a_{2}, a_{3}) (in Mordell’s case the a_{i}’s are pairwise relatively prime numbers). If we drop the condition g.c.d.(a_{1}, a_{2}) = 1, then by a similar computation, we reobtain a recent result of Pommersheim, which counts the number of lattice points in the open tetrahedron (with no restriction about the a_{i}’s). 5.21. Remark. Notice that the formula (5.12) or (5.15) is not valid if h_{nil}(g) ≠ 0. Take, for example, A’Campo’s germ g = (x^{2} + y^{3})(x^{2} + y^{3}) (cf. E.3.12 and 4.5). Set f = g + z^{2} i.e. a = 2. Then, even without computation, using the symmetry of the graph G_{g}, the right hand side of (5.15) gives an odd number. Notice that μ(f) = 11, and one has, in general, the relation μ(f)+σ(f) ≡ dim ker(,)_{f} (mod 2). Since in our case dim ker(,)_{f} = 1 (cf E.3.2 or E.4.2), σ(f) is even. (Actually, σ(f) = −8, see E.5.5.).

The signature σ(g + z^{a}), without any restriction about g, is given in:

5.22. Theorem.* [Némethi] [36] For any isolated plane curve singularity g and positive integer a, one has:

σ(g + z^{a}) = −(r_{g} − 1)(a − 1) + ∑_{e∈E}(g.c.d.(a, d_{e}) − 1) − ∑_{w∈W}(g.c.d.(a, d_{w}) − 1)+

+4 ∑_{w∈R} ∑_{v∈V_{w}}(s(m_{v}, a; m_{w}) − a · s(m_{v}, 1; m_{w})).

Above, similarly as in (4.4): for any e = (w_{1}, w_{2}) ∈ E we let d_{e} := g.c.d.(m_{w_{1}}, m_{w_{2}}), and for any w ∈ W, we let d_{w} := g.c.d.(m_{v}|v ∈ V_{w} ∪ {w}). 5.23. Remark.

(a) In [Némethi] [32, 33] the (equivariant) signature of f = g + z^{a} is clarified for arbitrary n.

(b) (Cf. 2.26-2.27) In [37] there is a very short proof of the negativity of the signature of f = g+z^{a}. Stronger inequalities are proved in [35, 36]. For a review about the signature of f = g + z^{a}, see [38].

(c) The semi–ring structure of “simple” variation structures (cf. 4.13) (i.e. the tensor product of indecomposable blocks, without the restriction h_{nil} = 0) is given in [Némethi] [31].

Some more results about the signature*. Case n = 2.

5.24. Assume that f : (C^{3}, 0) → (C, 0) defines an isolated singularity. We recall that in general the diffeomorphism type of its link K_{f} says very little about the smoothing invariants μ(f), or σ(f) (cf. E.1.15.b). Nevertheless, there are some connections between these two (i.e. link and smoothing) type of invariants. Moreover, there is a big interest in particular cases, when K_{f} determines the smoothing invariants μ(f) resp. σ(f).

First, we recall an important invariant of K_{f}. Let ϕ : (Y, E) → (f^{−1}(0), 0) be a resolution with E := ϕ^{−1}(0). Then the canonical class (associated with ϕ) is an element K ∈ H_{2}(Y, Q). We use the same notation {E_{w}}_{w∈W} for the 2–homology classes of the irreducible exceptional divisors {E_{w}}_{w} (which is a base in H_{2}(Y, Q)). Then K = ∑ r_{w}E_{w}. K satisfies the relations (adjunction formula):

−K · E_{w} = E_{w}^{2} + χ(E_{w}) for any w ∈ W.

Since the intersection matrix (A_{αβ}) (cf. 1.15) is negative definite, the above relations determine K. In fact, in our case, K associated with the graph of (V_{f}, 0) provided by an isolated hypersurface singularity, has integer coefficients {r_{w}}_{w}. The invariant K^{2} is the auto–intersection of this class, i.e. K^{2} = K · K = ∑_{i,j} r_{i}r_{j}E_{i} · E_{j}. Notice that K depends on the choice of the resolution ϕ, but the numerical invariant K^{2} + #W is independent of ϕ, and depends only on the link K_{f}. 5.25. Theorem.* [Durfee] [14] Let f : (C^{3}, 0) → (C, 0) be an isolated singularity and define the numerical invariant K^{2} + #W from one of the resolutions of (V_{f}, 0). As usual, K_{f} denotes the link of f. Then:

3σ(f) + 2μ(f) + 2 · dim H_{1}(K_{f}) + K^{2} + #W = 0.

In particular, modulo the link K_{f}, the smoothing invariants μ(f) and σ(f) are equivalent. Now, it is an interesting problem to find particular cases, when the three–dimensional manifold K_{f} determines both μ(f) and σ(f). A very interesting result is the following.

Casson defined an invariant λ(Σ) for any integer homology 3–manifold Σ (a fact generalized for rational homology spheres by Walker). Hence, if the link K_{f} is an integer homology sphere, then it has its well–defined Casson invariant λ(K_{f}). 5.26. Theorem.* Let f be as before, and assume that K_{f} is an integer homology sphere. Then 8 · λ(K_{f}) = σ(f) in the following cases:

(a) if f is a Brieskorn singularity; [Fintushel–Stern] [15]

(b) if f(x, y, z) = g(x, y) + z^{a}, [Neumann–Wahl] [45].

In particular, for these singularities, the smoothing invariants μ(f) and σ(f) are determined completely by the link only !

Open problems and conjectures.

5.27. [Némethi] [36] For any isolated plane curve singularity g and positive integer a, show that

σ(g + z^{a}) ≤ −\frac{a + 1}{3a} μ(g + z^{a}).

5.28. (cf. 2.28) For any isolated plane curve singularity g and positive integer a, show that:

σ(g + z^{a}) > σ(g + z^{a+1}).

5.29. (cf. 5.26) [Neumann–Wahl] [45] If the link of the isolated singularity f : (C^{3}, 0) → (C, 0) is an integer homology sphere then 8 · λ(K_{f}) = σ(f).

Exercises for the fifth section.

E.5.1. Prove (5.19) (Use E.4.7.b) and (5.20).

E.5.2. Prove that

−s(b, c; a) = ∑_{k=1}^{a−1}((\frac{kb}{a})) · \{\frac{kc}{a}\}.

(Use: ∑_{k=1}^{a−1}((\frac{kb}{a})) = 0, cf. E.4.7.b) E.5.3. If a > 0, prove that:

|s(b, c; a)| ≤ \frac{(a − 1)(a − 2)}{12a}.

E.5.4. Compute the signature of the Milnor fiber of f = x^{3} + y^{5} + z^{a} for any a ≥ 1. E.5.5. Consider the singularity f = (x^{2} + y^{3})(x^{3} + y^{2}) + z^{2} (cf. E.3.12).

(a) Using (5.22), prove that σ(f) = −8.

(b) Consider the minimal resolution of (V_{f}, 0) (as it is described in (E.3.12.c). Prove that K = −(E_{1} + E_{2}), and K^{2} = −2. Verify the relation (5.25) for f. E.5.6. Prove that:

(a) σ(x^{d} + y^{d} + z^{d}) = −(d − 1)(d^{2} + d − 3)/3.

(b) σ(x^{a} + y^{a} + z^{c}) = a − 1 + 4a · s(c, 1; a) − c(a^{2} − 1)/3.

(c) Consider again f = x^{d} + y^{d} + z^{d}. Then its minimal resolution graph has only one irreducible exceptional divisor E, with genus (d − 1)(d − 2)/2 and E^{2} = −d. Moreover, K = −(d − 2)E. Recompute σ(f) using (5.25). E.5.7. Let g^{−1}(0) be the union of three cusps of the form x^{2} + y^{3} with distinct tangent lines, and take f = g + z^{2}. Then μ(f) = 28 and σ(f) = −18. E.5.8. The signature of quasi–homogeneous singularities.* [Steenbrink] [55]

Let f : (C^{n+1}, 0) → (C, 0) be a quasi–homogeneous isolated singularity with weights {w_{i}}_{i}. Define the function l(α) for α ∈ Λ as in (E.4.4). Then:

σ(f) = ∑_{{α∈Λ \atop l(α)∉Z}}(−1)^{[l(α]}.

Show, that in the case of Brieskorn singularities, this gives exactly the Brieskorn formula (5.5). Remark 1. The above formula of Steenbrink follows easily from (E.4.4), but originally, the proof of (the author’s real Seifert form formula) (E.4.4) is based on Steenbrink’s result [55] on the Hodge numbers of a quasi-homogeneous singularity, which implies both relations. Remark 2. Take f = ∑_{j=1}^{3} z_{j}^{a_{j}} (i.e. n = 2), such that the numbers a_{j} are pairwise relatively prime. Then S_{0} = 0 is equivalent to l(α) ∈ (1, 2) for any α ∈ Λ. Therefore, S_{0} = (μ + σ)/4 is the measure of the non–rationality of the singularity (f^{−1}(0), 0) (cf. E.4.5). (S_{0} is called the “geometric genus of the singularity”, which can be defined for any isolated singularity f.)

Appendix.

I guess that the result of this appendix is known for specialists, but I never have seen it written down in this form. The idea and the method is already in the book of Laufer [19].

We start with an isolated plane curve singularity g. We give a recipe how one can obtain the resolution graph G(V_{f}) of the normal surface singularity (V_{f}, 0) (where f = g + z^{a}) from the embedded resolution graph G_{g} of g and from the integer a. Actually, we will consider the germ z : (V_{f}, 0) → (C, 0) (induced by the projection (x, y, z) ↦ z), and we will determine the resolution graph G_{z} of this germ z. Obviously, G(V_{f}) is the graph G_{z} without arrowheads, so it is simpler, but computing G_{z} has an advantage. Sometimes, it is easier to compute multiplicities than Euler–numbers (characteristic classes). So, computing G_{z}, first we compute all the multiplicities of the germ z, then some of the Euler–numbers can be more easily computed using the relation (1.16). For the convenience of the reader, we recall (1.16):

(A0) e_{w}m_{w} + ∑_{v∈V_{w}} m_{v} = 0 for any w ∈ W.

1. For any three positive integers u, v and a, with (u, v, a) = 1, we consider the congruence:

v + x · \frac{u}{(u, a)} ≡ 0 (mod \frac{a}{(u, a)}).

Let 0 ≤ x_{1} < a/(u, a) be its solution, and take:

(A1) v + x_{1} · \frac{u}{(u, a)} = m_{1} · \frac{a}{(u, a)}.

If x_{1} ≠ 0, then consider the continuous fraction:

\frac{a/(u, a)}{x_{1}} = k_{1} − \frac{1}{k_{2} − \frac{1}{... − \frac{1}{k_{s}}}} , k_{1}, …, k_{s} ≥ 2.

Consider the “string”:

[figure: G(u, v, a) :; (\frac{u}{(u,a)}); (\frac{v}{(v,a)}); −k_{1}; −k_{2}; −k_{s}; (m_{1}); (m_{2}); (m_{s}); ⋯]

where all vertices have genus g = 0 (i.e. they represent rational irreducible exceptional divisors), their self intersection (Euler) numbers are −k_{1}, …, −k_{s} respectively. The arrowheads have multiplicities u/(u, a) and v/(v, a), and the first vertex has multiplicity m_{1} given by (A1). Therefore, m_{2}, …, m_{s} can be easily computed using (A0), namely

−k_{1}m_{1} + \frac{u}{(u, a)} + m_{2} = 0, and − k_{i}m_{i} + m_{i−1} + m_{i+1} = 0 for i ≥ 2.

If x_{1} = 0, then the string G(u, v, a) has no vertices, it is only an edge.

2. Now, consider the embedded resolution graph G_{g} of g. It is a tree. For the definition of d_{e} (e ∈ E) and d_{w} (w ∈ W) see (4.4) or (5.22). Recall also that δ_{w} = #V_{w}.

The graph G_{z} can be considered as a (branched) “covering” q : G_{z} → G_{g}.

(a) Above w ∈ W(G_{g}) there are (d_{w}, a) vertices of G_{z}, each with multiplicity m_{w}/(m_{w}, a) and genus g̃, where:

2 − 2g̃ = \frac{(2 − δ_{w})(m_{w}, a) + ∑_{v∈V_{w}} (m_{w}, m_{v}, a)}{(d_{w}, a)}.

The vertices in q^{−1}(w) can be indexed by the group Z_{(d_{w},a)}.

(b) An edge e = (w_{1}, w_{2}) of G_{g}

[figure: (m_{w_{1}}); (m_{w_{2}})]

is covered by (d_{e}, a) copies of strings of type G(m_{w_{1}}/(d_{e}, a), m_{w_{2}}/(d_{e}, a), a/(d_{e}, a)). These strings can be indexed by the group Z_{(d_{e},a)}. The arrowheads of the strings are identified with the vertices q^{−1}(w_{1}) respectively q^{−1}(w_{2}) via the natural morphisms Z_{(d_{e},a)} → Z_{(d_{w_{1}},a)}, respectively Z_{(d_{e},a)} → Z_{(d_{w_{2}},a)}.

(c) An arrowhead of G_{g}

[figure: (m_{w}); (1)]

is covered by one string of type G(m_{w}, 1, a), whose right arrowhead will remain an arrowhead of G_{z} with multiplicity 1, and its left arrowhead is identified with the unique vertex above w.

(d) In this way, we obtain all the vertices and edges of G_{z}, and all the multiplicities (which are the multiplicities of the germ z). Moreover, by the description of the strings (cf. part 1), one has all the auto–intersection numbers of the vertices which are situated on the new strings. The auto–intersection numbers of the vertices q^{−1}(w) (w ∈ W(G_{g})) can be computed using (A0).

3. If we drop the arrowheads and multiplicities of G_{z}, we obtain G(V_{f}). The graphs G_{z} and G(V_{f}), in general, are not minimal. They can be simplified by blowing down the (−1)–rational curves.

Exercise. (a) If f = x^{2} + y^{3} + z^{6}, then the minimal resolution graph G_{z} is:

[figure: g = 1, e = −1; (1); (1)]

(b) If f = (x^{2} + y^{3})(x^{3} + y^{2}) + z^{2}, then the minimal resolution graph G(V_{f}) is:

[figure: −3; −3; and g_{w} = 0 for both vertices.]

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