On the Seifert form at infinity associated with polynomial maps

by András Némethi The Ohio State University, Columbus, OH, USA e-mail: nemethi@math.ohio-state.edu

The Seifert forms are unimodular bilinear forms (over Z) associated with some geometrical objects: the spinnable structures.

A spinnable structure (or open book decomposition) on a closed manifold M is a triple S = {F, m, g} such that F is a compact manifold (with boundary), m : F → F is a diffeomorphism such that m|_{∂F} =identity, and g : T (F, m) → M is a diffeomorphism, where T (F, m) is a closed manifold defined as follows. It is obtained from F × [0, 1] by identifying (x, 1) with (m(x), 0) for all x ∈ F, and (x, t) with (x, t′) for all x ∈ ∂F and t, t′ ∈ [0, 1]. The spinnable structure S is called simple if F is a handle-body obtained from a ball by attaching handles of index ≤ [dim M/2].

A closed, oriented (2n − 1)–manifold is called Alexander manifold, if H_{n}(M, Z) = H_{n−1}(M, Z) = 0. If it has a simple spinnable structure, then H_{n−1}(F, Z) is torsion free.

At the homological level, the geometry of a spinnable structure is coded in its Seifert form. Let S = {F, m, g} be a simple spinnable structure on an Alexander manifold M^{2n−1}. Then the bilinear form H̃_{n−1}(F)⊗H̃_{n−1}(F) → Z, given by (α, β) =linking number(g_{#}(α × 0), g_{#}(β × 1/2)) is called the Seifert form of S.

The power of the Seifert form can be emphasized by the following result of M. Kato [4], and (independently) A. Durfee [3]: there is a one-to-one correspondence of isomorphism classes of simple spinnable structures on a 1connected Alexander manifold M^{2n−1} with congruence classes of unimodular matrices via Seifert matrices (provided that n ≥ 4).

We recall the following fundamental example. If f : (C^{n}, 0) → (C, 0) is the germ of an analytic function, which defines an isolated singularity, then by a result of J. Milnor [6], for any (2n − 1)–dimensional sphere S_{ϵ}^{2n−1} (centered at the origin, and with sufficiently small radius ϵ), the map f/|f| : S_{ϵ}^{2n−1} \ f^{−1}(0) → S^{1} is a C^{∞} locally trivial fibration with fiber F. Moreover, this fibration is equivalent to the fibration f : f^{−1}(∂D_{r}) ∩ B_{ϵ}^{n} → ∂D_{r}, where D_{r} is a disc centered at the origin with radius r (0 < r << ϵ), and B_{ϵ}^{n} is the open ball centered at the origin with radius ϵ.

The advantage of the first (Milnor) fibration is that it provides a simple spinnable structure of the (2n−1)–dimensional sphere S_{ϵ}^{2n−1}. The associated Seifert form is denoted by Γ(f), and it is (maybe) the most powerful topological invariant of f. In general, there is no algorithm to determine it, and even in special cases its computation can be very difficult. For this reason, the following result of K. Sakamoto (or equivalently, the result of P. Deligne, which solves the problem at the variation map level) is crucial:

Theorem: [16, 17] Assume that g : (C^{m}, 0) → (C, 0) and h : (C^{n}, 0) → (C, 0) define isolated singularities. Define f : (C^{n+m}, 0) → (C, 0) by f(x, y) = g(x) + h(y). Then

Γ(f) = (−1)^{mn} Γ(g) ⊗ Γ(h).

Actually, the Milnor fiber F of f can be identified with the join G ∗ H of the Milnor fibers G and H of g and h respectively, therefore H̃_{n+m−1}(F) = H̃_{m−1}(G) ⊗ H̃_{n−1}(H), hence the above formula makes sense by this identification.

The goal of this paper is to present a similar result for global polynomial maps f : C^{n} → C.

We say that the Milnor fibration at infinity of the poynomial map f : C^{n} → C exists, if for R sufficiently big

f/|f| : S_{R}^{2n−1} \ f^{−1}(0) → S^{1}

is a C^{∞} locally trivial fibration (S_{R}^{2n−1} = ∂B_{R}^{n}).

The main difficulty is that, in general, the Milnor fibration at infinity does not exist (see, for example, [13, 14, 12]). On the other hand, in the last years, a large number of families of polynomial maps were constructed with nice behaviour at infinity.

In this paper, we will assume that our polynomials f : C^{n} → C satisfy the following condition:

(C) (Regularity at infinity) For any t ∈ C, the fiber f^{−1}(t) is either smooth or has only isolated singularities, and there exists a sufficiently small neighbourhood D_{t} of t, and a sufficiently large R(t) >> 0 such that for any R ≥ R(t)

f : (f^{−1}(D_{t}) \ B_{R}^{n}, f^{−1}(D_{t}) ∩ S_{R}^{2n−1}) → D_{t}

is a trivial fibration over D_{t}.

If a polynomial f satisfies the condition (C), then it has the following properties as well:

(P1) The bifurcation set of f is exactly the set Σ_{f} of critical values, i.e.

f|_{C^{n}\f^{−1}(Σ_{f})} : C^{n} \ f^{−1}(Σ_{f}) → C \ Σ_{f}

is a C^{∞} locally trivial fibration, with a (smooth) fiber F which has the homotopy type of a bouquet ∨S^{n−1} of (n − 1)–dimensional spheres. (For the proof of (P1), see the corresponding arguments in [1, 7, 8, 12, 11].)

(P2) For any disc D with the property Σ_{f} ⊂ D, there exists R_{0} >> 0 such that for any R ≥ R_{0}

(∞) f : (f^{−1}(∂D) ∩ B̄_{R}^{n}, f^{−1}(∂D) ∩ S_{R}^{2n−1}) → ∂D

is a locally trivial fibration of pair of spaces, such that its restriction on f^{−1}(∂D) ∩ S_{R}^{2n−1} is trivial, and actually it can be extended to a trivial fibration f^{−1}(D) ∩ S_{R}^{2n−1} → D. (The proof is easy.)

The fibration f^{−1}(∂D) → ∂D is called the fibration of f at infinity, and it is equivalent to the fibration f^{−1}(∂D) ∩ B̄_{R}^{n} → ∂D.

(P3) The Milnor fibration f/|f| : S_{R}^{2n−1} \ f^{−1}(0) → S^{1} exists and it is equivalent to the fibration of f at infinity. In particular, the fiber of f/|f| is diffeomorphic to the generic fiber f^{−1}(t) of f (See the corresponding proofs in [14, 13] for n = 2, and notice that the dimension is not important; see also [12].)

Polynomial maps which satisfy the condition (C) provide simple spinnable structures of S_{R}^{2n−1} via their Milnor fibration at infinity.

Examples:

1. In the case of plane curves (i.e. for n = 2) Neumann and Rudolf [14] and Neumann [13] studied the Milnor fibration at infinity of “good curves” (i.e. polynomials with condition (C)) proving among others that the link of a generic fiber at infinity is fiberable if and only if f is “good”. (In general, it is not true that the existence of the Milnor fibration at infinity implies the condition (C), see for example the case of “semi-tame” polynomials [12]; or the example f(x, y) = x(xy − 1).).

2. Broughton introduced in [1] his “tame” polynomials, and proved that they satisfy the condition (C), and he proved properties (P1-P2) for them. The author generalized Broughton’s results for the larger class of “quasitame” polynomials [7, 8]. Moreover, A. Zaharia and the author extended all these results for the larger class of “M-tame” polynomials, and actually they proved also that the property (P3) is satisfied even for a larger class of (the “semitame”) polynomials [11, 12]. In [12] the interested reader can find some more examples and even some counter-examples.

3. Let f = f_{d} + f_{d−1} + … be the decomposition of f in its homogeneous parts. Assume that f satisfies:

{∂f_{d}/∂x_{1} = ⋯ = ∂f_{d}/∂x_{n} = f_{d−1} = 0} = {0}.

Then A. Dimca [2] proved that f is “quasitame”, in particular it satisfies (C,P1,P2,P3) (via the results of [7, 8, 12]). For more properties of these polynomials (and more motivation for the present paper), see [5] (and the forthcoming joint papers of R. García López and the author).

The main result of this note is the following affine analogue of Sakamoto’s local result:

Theorem: Assume that the polynomials g : C^{m} → C, h : C^{n} → C and f : C^{m+n} → C (f(x, y) = g(x) + h(y)) satisfy the condition (C). Assume that Γ(g), Γ(h) respectively Γ(f) denote their Seifert forms associated with their Milnor fibration at infinity. Then

Γ(f) = (−1)^{mn} Γ(g) ⊗ Γ(h).

Remarks:

1. In [9, 10] the author proved the following fact. For any polynomial maps g and h (and f defined as above), the generic fiber F of f is the join space F = G ∗ H, where G and H are the generic fibers of g and h respectively. (For the quasi-homogeneous case, see [15].) Moreover, in [9, 10] a global Sebastiani-Thom type result is proved for the monodromy operators at infinity. If the condition (C) is satisfied, then H̃_{n+m−1}(F) = H̃_{m−1}(G)⊗H̃_{n−1}(H), and these are the only interesting homological groups of the fibers. The formula Γ(f) = (−1)^{mn}Γ(g) ⊗ Γ(h) must be understood via this identification.

2. In general, the set of spinnable structures provided by the Milnor fibrations at infinity of polynomial maps is different from the set of spinnable structures provided by the local Milnor fibrations of isolated singularities. In the case n = 2, this was clarified by W. Neumann [13] using splice diagrams: the only spinnable structures which can be represented by both local and global construction are exactly the spinnable structures provided by quasihomogeneous maps.

3. Sakamoto’s proof cannot be extended to the global situation. His proof is based on the existence of a continuous map [0, 1] × (B_{ϵ} \ {0}) → B_{ϵ}, (r, z) → r ∘ z, with the properties: 1 ∘ z = z, 0 ∘ z = 0, (rs) ∘ z = r ∘ (s ∘ z), f(r ∘ z) = rf(z), and |r ∘ z| is a strictly increasing function of r. Since, in the global case we can have many singular fibers with many singular points, a similar map as above does not exist. Our proof is based on a similar construction as in [9, 10].

4. The following question appears naturally: what spinnable structures are provided by Milnor fibration at infinity of polynomial maps?

Even in the local case, the corresponding question is still open, however some restrictions provided by algebraic spinnable structures already appeared in the literature. We think that the global problem is even more difficult.

The proof of the theorem

Consider the polynomial maps g : C^{m} → C and h : C^{n} → C. Fix a closed disc (centered at the origin) D_{g} (respectively D_{h}) such that Σ_{g} ⊂ int(D_{g}) (respectively Σ_{h} ⊂ int(D_{h})). Let D_{f} be another closed disc such that int(D_{f}) ⊃ D_{g} + D_{h}. Obviously: Σ_{f} = Σ_{g} + Σ_{h} ⊂ int(D_{f}).

Consider the map u : C^{m} ×C^{n} → C×C, u(x, y) = (g(x), h(y)). The line L_{e} = {(c, d) ∈ C × C : c + d = e} intersects D_{g} × C (respectively C × D_{h}) in D_{g}(e) (respectively in D_{h}(e)). If e ∈ ∂D_{f}, then D_{g}(e) ∩ D_{h}(e) = ∅. Moreover, there exists r sufficiently large such that for any e ∈ ∂D_{f} the ball B_{r}^{2} = {(c, d) ∈ C^{2} : |c|^{2} + |d|^{2} ≤ r^{2}} satisfies D_{g}(e) ∪ D_{h}(e) ⊂ L_{e} ∩ B_{r}^{2}.

Now, for e ∈ ∂D_{f}, by lemma 2.3 [9] one has:

u : u^{−1}(L_{e} \ (C × Σ_{h} ∪ Σ_{g} × C)) → L_{e} \ (C × Σ_{h} ∪ Σ_{g} × C)

is a C^{∞} locally trivial fibration. Therefore, u^{−1}(L_{e}) can be identified with u^{−1}(L_{e}∩B_{r}^{2}). Moreover, by (C), there exists R_{g} >> 0 (respectively R_{h} >> 0) such that:

(1) (u^{−1}(L_{e} ∩ B_{r}^{2}), u^{−1}(t)) (for t ∈ L_{e} ∩ B_{r}^{2}) has the homotopy type of

(u^{−1}(L_{e} ∩ B_{r}^{2}) ∩ (B_{R_{g}} × B_{R_{h}}), u^{−1}(t) ∩ (B_{R_{g}} × B_{R_{h}})).

(2) u^{−1}(L_{e}∩B_{r}^{2}\(C×Σ_{h}∪Σ_{g}×C))∩(B_{R_{g}}×B_{R_{h}}) → L_{e}∩B_{r}^{2}\(C×Σ_{h}∪Σ_{g}×C)

is a locally trivial fibration with fiber

G × H = (g^{−1}(c) ∩ B_{R_{g}}) × (h^{−1}(d) ∩ B_{R_{h}}), ((c, d) generic).

(3) The restriction of the above fibration on ∂(B_{R_{g}} × B_{R_{h}}):

u^{−1}(L_{e}∩B_{r}^{2}\(C×Σ_{h}∪Σ_{g} ×C))∩∂(B_{R_{g}} ×B_{R_{h}}) → L_{e}∩B_{r}^{2}\(C×Σ_{h}∪Σ_{g} ×C)

can be extended to a trivial fibration

u^{−1}(L_{e} ∩ B_{r}^{2}) ∩ ∂(B_{R_{g}} × B_{R_{h}}) → L_{e} ∩ B_{r}^{2},

with fiber ∂(G × H).

By the above facts, for e ∈ ∂D_{f}, one can identify the fiber f^{−1}(e) = u^{−1}(L_{e}) with (u^{−1}(L_{e} ∩ B_{r}^{2})) ∩ (B_{R_{g}} × B_{R_{h}}). Actually, the (2n + 2m − 2)– dimensional manifold with boundary

(F, ∂F) = (f^{−1}(e) ∩ B_{R_{f}} , f^{−1}(e) ∩ ∂B_{R_{f}}) (for R_{f} >> 0)

can be identified with the manifold (with corners and boundary):

(u^{−1}(L_{e} ∩B_{r}^{2})∩(B_{R_{g}} ×B_{R_{h}}), u^{−1}(L_{e} ∩∂B_{r}^{2})∪[u^{−1}(L_{e} ∩B_{r}^{2})∩∂(B_{R_{g}} ×B_{R_{h}})]).

Fix e_{0} ∈ ∂D_{f}. Consider a diffeomorpfism v = v_{e_{0}} : R^{2} → L_{e_{0}} such that v^{−1}(D_{g}(e_{0})) ⊂ (−∞, 0) × R and v^{−1}(D_{h}(e_{0})) ⊂ (0, ∞) × R (cf. lemma 2.1.

[9]). Take l := v({0} × R) ∩ B_{r}^{2}, this is the “segment” which separates D_{g}(e_{0}) and D_{h}(e_{0}) in L_{e_{0}} ∩ B_{r}^{2}. Set G = v((−∞, 0] × R) ∩ B_{r}^{2} and H = v([0, ∞) × R) ∩ B_{r}^{2}; therefore G ∩ H = l and G ∪ H = L_{e_{0}} ∩ B_{r}^{2}, D_{g}(e_{0}) ⊂ G and D_{h}(e_{0}) ⊂ H.

By lemma 2.3 [9] one has (≈ denotes a diffeomorphism):

(4) u^{−1}(G) ∩ (B_{R_{g}} × B_{R_{h}}) ≈ B_{R_{g}} × H u^{−1}(H) ∩ (B_{R_{g}} × B_{R_{h}}) ≈ G × B_{R_{h}} u^{−1}(l) ∩ (B_{R_{g}} × B_{R_{h}}) ≈ G × H × l.

Let m_{g} : (G, ∂G) → (G, ∂G) be a geometric monodromy at infinity of g (i.e. a characteristic map of the fibration (∞) of g at infinity) with m_{g}|_{∂G} =identity. Similarly define m_{h}. Then the following holds:

(5) A geometric monodromy of u : u^{−1}(∂D_{g}(e_{0})) → ∂D_{g}(e_{0}) is given by m_{g} × id;

a geometric monodromy of u : u^{−1}(∂D_{h}(e_{0})) → ∂D_{h}(e_{0}) is given by id × m_{h}.

By (P3), (i.e. by the identification of the fibration of f at infinity and the Milnor fibration at infinity), and using Alexander duality, similarly as in the local case, the Seifert form of f can be identified (modulo a sign) with the variation map

V ar_{f} : H_{n+m−1}(F, ∂F) → H_{n+m−1}(F).

(Similarly for g and h.) Hence the theorem can be reformulated in terms of the variation maps: V ar_{f} = ±V ar_{g} ⊗ V ar_{h}. In the sequel, we verify this formula. For simlicity, we will assume that n ≥ 2, m ≥ 2. The case n = 1 or m = 1 can be proved similarly (if we replace H_{0} by H̃_{0}).

In the sequel, we will identify the fiber F with:

u^{−1}(L_{e} ∩ B_{r}^{2}) ∩ (B_{R_{g}} × B_{R_{h}}).

Consider its decomposition in u^{−1}(G) ∩ F and u^{−1}(H) ∩ F. Then, by (4), for q = n+m−2 and q = n+m−1 one has: H_{q}(u^{−1}(G)∩F) = H_{q}(u^{−1}(H)∩F) = 0. Therefore, by Mayer–Vietoris argument, the boundary map:

∂ : H_{n+m−1}(F) → H_{n+m−2}(u^{−1}(l) ∩ F)

is an isomorphism. Geometrically, ∂ can be described as follows: if [γ] ∈ H_{n+m−1}(F) and the cycle γ is in generic position with respect to u^{−1}(l), then ∂[γ] = [γ ∩ u^{−1}(l)].

Let P ∈ l be an arbitrary point on l. Then by the local triviality of u over l, the natural strong deformation retract l ↦ P induces an isomorphism:

r : H_{n+m−2}(u^{−1}(l) ∩ F) → H_{n+m−2}(u^{−1}(P) ∩ (B_{R_{g}} × B_{R_{h}})) =
= H_{n+m−2}(G × H) = H_{m−1}(G) ⊗ H_{n−1}(H).

Notice that u^{−1}(l)∩F is a manifold (with corners) of dimension (2n+2m−3), and with boundary:

[u^{−1}(∂l) ∩ (B_{R_{g}} × B_{R_{h}})] ∪ [u^{−1}(l) ∩ ∂(B_{R_{g}} × B_{R_{h}})],

and by duality:

H_{n+m−1}(u^{−1}(l) ∩ F, ∂(u^{−1}(l) ∩ F)) = H^{n+m−2}(u^{−1}(l) ∩ F).

Then, by a duality argument, or by similar Mayer–Vietoris argument as above, one can prove that the natural inclusion induces an isomorphism:

i : H_{n+m−1}(u^{−1}(l) ∩ F, ∂(u^{−1}(l) ∩ F)) → H_{n+m−1}(F, ∂F).

We want to investigate the composition V = r ∘ ∂ ∘ V ar ∘ i:

[figure: H_{m−1}(G, ∂G) ⊗ H_{n−1}(H, ∂H) ⊗ H_{1}(l, ∂l) = H_{n+m−1}(u^{−1}(l) ∩ F, ∂(u^{−1}(l) ∩ F)) →^{i} H_{n+m−1}(F, ∂F); H_{m−1}(G) ⊗ H_{n−1}(H) ⟵^{r} H_{n+m−2}(u^{−1}(l) ∩ F) ⟵^{∂} H_{n+m−1}(F); V; V ar_{f}]

If m_{f} : (F, ∂F) → (F, ∂F) is the geometric monodromy of f at infinity (with m_{f}|_{∂F} =identity) then the variation map, by its very definition, is V ar[γ] = [m_{f}(γ) − γ]. Let [α] ∈ H_{m−1}(G, ∂G) and [β] ∈ H_{n−1}(H, ∂H) be the homology classes of the relative cycles α, respectively β. In order to find the map V, we wish to describe the intersection of the geometric cycle m_{f}(α × β × l) − (α × β × l) with u^{−1}(l).

Moving e along ∂D_{f}, we can construct a continuous family v_{e} : R^{2} → L_{e} with similar properties as v_{e_{0}} (cf. [9]). Then the monodromy action on the line l = l_{e_{0}} is the following:

Then the cycle γ = m_{f}(α × β × l) − (α × β × l) can be identified with the total space of a fibering p with fiber α × β and base space the loop C in L_{e_{0}}:

The intersection γ ∩ u^{−1}(l) is the collection of the fibers of p over the (oriented) intersection C ∩ l = P − A + B − C. Now, (5) and the retract r′ : u^{−1}(l) → u^{−1}(P) give the identifications:

r′(u^{−1}(P)) = α × β, r′(u^{−1}(A)) = α × m_{h}(β),
r′(u^{−1}(B)) = m_{g}(α) × m_{h}(β), r′(u^{−1}(C)) = m_{g}(α) × β.

Therefore:

V [α × β] = ±[α × β − α × m_{h}(β) + m_{g}(α) × m_{h}(β) − m_{g}(α) × β] =

= ±[(m_{g}(α) − α) × (m_{h}(β) − β)] = ±V ar_{g}(α) ⊗ V ar_{h}(β).

This gives Γ(f) = ±Γ(g) ⊗ Γ(h).

The sign ± is a universal sign which depends only on n and m, and can be determined also by a careful study of the orientation of the cycles which are intersected. But, this sign does not depend on the polynomials g and h, therefore it is the same as in the case of generic homogeneous polynomials. Since for homogeneous polynomials the local and global theory agree, this sign is the same as in the local theory (i.e. as in Sakamoto’s theorem) (provided that we use the same orientation conventions).

This ends the proof of the theorem.

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