Open book structures on semi-algebraic manifolds

TL;DR

This paper establishes conditions for fibration structures on semi-algebraic manifolds induced by semi-algebraic mappings, connecting to Milnor fibrations and analyzing fiber homotopy types and Euler characteristics.

Contribution

It introduces sufficient conditions for generalized open book structures on semi-algebraic manifolds and links these to classical Milnor fibrations, also analyzing fiber homotopy and Euler characteristic relations.

Findings

Established conditions for fibration structures on semi-algebraic manifolds.

Connected generalized open book structures to Milnor fibrations in real and complex cases.

Derived formulas relating Euler characteristics of fibers and intersections with zero sets.

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.