Open book structures on semi-algebraic manifolds

Abstract

Given a $C^2$ semi-algebraic mapping $F:{\mathbb{R}}^N\to {\mathbb{R}}^p,$ we consider its restriction to $W\hookrightarrow {\mathbb{R}}^{\mathbb{N}}$ an embedded closed semi-algebraic manifold of dimension $n-1\ge p\ge 2$ 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.

Cited by in corpus (1)

• Fibrations structure and degree formulae for Milnor fibers