Open book structures on semi-algebraic manifolds

Abstract

Given 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.