Open book structures on semi-algebraic manifolds (1409.4316v2)
Abstract: Given a $C2$ 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.