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Journal Articles Manuscripta mathematica Year : 2016

Open book structures on semi-algebraic manifolds


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.
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hal-01064170 , version 1 (15-09-2014)
hal-01064170 , version 2 (16-09-2014)



Nicolas Dutertre, Raimundo N. Araújo dos Santos, Ying Chen, Antonio Andrade. Open book structures on semi-algebraic manifolds. Manuscripta mathematica, 2016, 149 (1), pp.18. ⟨10.1007/s00229-015-0772-4⟩. ⟨hal-01064170v2⟩
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