On the topology of complex polynomials.
(Sur la topologie des polynômes complexes.)

*(French)*Zbl 0964.32028
Arnold, V. I. (ed.) et al., Singularities. The Brieskorn anniversary volume. Proceedings of the conference dedicated to Egbert Brieskorn on his 60th birthday, Oberwolfach, Germany, July 1996. Basel: Birkhäuser. Prog. Math. 162, 317-343 (1998).

It is known [see, e.g., the appendix in F. Pham, Proc. Symp. Pure Math. 40, Part 2, 319-333 (1983; Zbl 0519.49026)] that given a polynomial \(f\in\mathbb{C}[x_1,\dots,x_n]\) there exists a finite set \(\Lambda \subset\mathbb{C}\) such that \(f\) defines a locally trivial fibration over \(\mathbb{C}\setminus\Lambda\). Assuming that \(\Lambda\) is the smallest subset in \(\mathbb{C}\) satisfying such a condition, its elements are called irregular values of \(f\) and the fibers of \(f\) over points of \(\Lambda\) are called irregular fibers. In this article the authors study some topological properties of the fibration determined by \(f\) over a punctured neighborhood of an irregular value and over the complement of \(\Lambda\) in the complex plane. It is assumed that the polynomial \(f\) is such that its generic fiber has the homology type of a wedge of \((n-1)\)-dimensional spheres (called condition (CP) by the authors).

If \(t_0\in \mathbb{C}\) and \(F_0=f ^{-1}(t_0)\), there is a monodromy automorphism \(T_0:H^{n-1}(F_\delta)\to H^{n-1}(F_\delta)\), for \(F_\delta\) a fiber of \(f\) close enough to \(F_0\). Assuming that \((F_0)_{\text{red}}\) has only isolated singularities (so-called condition (CF) in the paper), in Theorem 1 the authors give a formula for the codimension of the space of invariant cocycles, i.e. \(\ker(T_0 -\text{Id})\). In Theorem 2 it is assumed that \(f\) satisfies condition (CP) and all its fibers condition (CP), and under these assumptions they give a formula for the dimension of the fixed part of \(H^{n-1} (F_\delta)\) under the action of the fundamental group of \(\mathbb{C}\setminus\Lambda\), where \(F_\delta\) denotes a generic fiber of \(f\). In Theorem 3 they consider the case of polynomials in two variables with irreducible generic fiber. If \(f\) is such a polynomial, \(F_0=F^{-1} (t_0)\) is a reduced fiber and \(D_{\varepsilon}\) denotes a small disk centered at \(t_0\), they characterize when the inclusion \(F_0\hookrightarrow f^{-1}(D_{\varepsilon})\) is a homotopy equivalence, in homological terms and also in terms of the notion of regularity at infinity.

The paper also contains a new proof of an inequality due to S. Kaliman [Pac. J. Math. 154, No. 2, 285-296 (1992; Zbl 0756.32003)] as well as several corollaries of the three main theorems mentioned above. A number of examples illustrating several phenomena which may occur in the settings considered by the authors are also included.

For the entire collection see [Zbl 0890.00033].

If \(t_0\in \mathbb{C}\) and \(F_0=f ^{-1}(t_0)\), there is a monodromy automorphism \(T_0:H^{n-1}(F_\delta)\to H^{n-1}(F_\delta)\), for \(F_\delta\) a fiber of \(f\) close enough to \(F_0\). Assuming that \((F_0)_{\text{red}}\) has only isolated singularities (so-called condition (CF) in the paper), in Theorem 1 the authors give a formula for the codimension of the space of invariant cocycles, i.e. \(\ker(T_0 -\text{Id})\). In Theorem 2 it is assumed that \(f\) satisfies condition (CP) and all its fibers condition (CP), and under these assumptions they give a formula for the dimension of the fixed part of \(H^{n-1} (F_\delta)\) under the action of the fundamental group of \(\mathbb{C}\setminus\Lambda\), where \(F_\delta\) denotes a generic fiber of \(f\). In Theorem 3 they consider the case of polynomials in two variables with irreducible generic fiber. If \(f\) is such a polynomial, \(F_0=F^{-1} (t_0)\) is a reduced fiber and \(D_{\varepsilon}\) denotes a small disk centered at \(t_0\), they characterize when the inclusion \(F_0\hookrightarrow f^{-1}(D_{\varepsilon})\) is a homotopy equivalence, in homological terms and also in terms of the notion of regularity at infinity.

The paper also contains a new proof of an inequality due to S. Kaliman [Pac. J. Math. 154, No. 2, 285-296 (1992; Zbl 0756.32003)] as well as several corollaries of the three main theorems mentioned above. A number of examples illustrating several phenomena which may occur in the settings considered by the authors are also included.

For the entire collection see [Zbl 0890.00033].

Reviewer: Ricardo Garcia Lopez (MR 99k:32069)

##### MSC:

32S50 | Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants |

32S55 | Milnor fibration; relations with knot theory |