Open book structures on semi-algebraic manifolds

Abstract

International audienceGiven a $C^2$ semi-algebraic mapping $F: \mathbb{R}^N \rightarrow \mathbb{R}^p,$ we consider its restriction to $W\hookrightarrow \mathbb{R^{N}}$ an embedded closed semi-algebraic manifold of dimension $n-1\geq p\geq 2$ and introduce sufficient conditions for the existence of a fibration structure (generalized open book structure) induced by the projection $\frac{F}{\Vert F \Vert}:W\setminus F^{-1}(0)\to S^{p-1}$. Moreover, we show that the well known local and global Milnor fibrations, in the real and complex settings, follow as a byproduct by considering $W$ as spheres of small and big radii, respectively. Furthermore, we consider the composition mapping of $F$ with the canonical projection $\pi: \mathbb{R}^{p} \to \mathbb{R}^{p-1}$ and prove that the fibers of $\frac{F}{\Vert F \Vert}$ and $\frac{\pi\circ F}{\Vert \pi\circ F \Vert}$ are homotopy equivalent. We also show several formulae relating the Euler characteristics of the fiber of the projection $\frac{F}{\Vert F \Vert}$ and $W\cap F^{-1}(0).$ Similar formulae are proved for mappings obtained after composition of $F$ with canonical projections

Geometrical Conditions for the Existence of a Milnor Vector Field

“…Motivated by Theorem 3 in this section we give some descriptions of a(x) to ensure the existence of a M V F for analytic maps G. As already pointed out the results does not depend on the particular choice of open set {G 1 (x) = 0}. See Dutertre et al (2016) for details. From now on when needed we will be working on this open set without explicit mention to it.…”

On Milnor’s fibration theorem and its offspring after 50 years

“…This inspired [243,236] and the use of the canonical pencil described in Section 14. We refer for instance to [14,15,79,81,215,216,217,219,231] for recent work on the topology of the Milnor fibers. D. Massey in [176] improved (12.12) using a different viewpoint, via a generalized Łojasiewicz inequality (see the next section).…”

“…This emerged too from Milnor's seminal work in [188,189]. We also look at meromorphic functions, and remark that the much of the discussion below goes through for semi-algebraic and subanalytic maps (see [69,78,79]).…”

Lê's vanishing polyhedron for a family of mixed functions

“…This is also a consequence [, Corollary 4.6] of a more general join theorem for polar weighted homogeneous polynomials [, Theorem 4.1]. In the literature, there are several articles about the topology of the Milnor fibers of real analytic maps with isolated critical point , however, except for Oka's article on non‐degenerate mixed functions, as far as we know, there are not general theorems describing the topology of Milnor fibers of real analytic singularities.…”