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Manuscripta Mathematica

, Volume 149, Issue 1–2, pp 205–222 | Cite as

Open book structures on semi-algebraic manifolds

  • N. Dutertre
  • R. N. Araújo dos Santos
  • Ying Chen
  • Antonio Andrade do Espirito Santo
Article
  • 90 Downloads

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.

Mathematics Subject Classification

14P10 32S55 58K15 58K65 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • N. Dutertre
    • 1
  • R. N. Araújo dos Santos
    • 2
  • Ying Chen
    • 2
  • Antonio Andrade do Espirito Santo
    • 3
  1. 1.CNRS, Centrale Marseille, I2M, UMR 7373Aix-Marseille UniversitéMarseilleFrance
  2. 2.Departamento de Matemática, Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São PauloSão CarlosBrazil
  3. 3.Centro de Ciências Exatas e TecnológicasUniversidade Federal do Recôncavo da BahiaCruz das AlmasBrazil

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