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# Open book structures on semi-algebraic manifolds

4 Instituto de Ciencias Matematicas e de Computaçao
ICMC-USP - Instituto de Ciências Mathemàticas e de Computação [São Carlos]
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.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-01064170
Contributor : Nicolas Dutertre Connect in order to contact the contributor
Submitted on : Tuesday, September 16, 2014 - 10:10:45 AM
Last modification on : Wednesday, November 3, 2021 - 6:46:49 AM
Long-term archiving on: : Wednesday, December 17, 2014 - 10:31:07 AM

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### Citation

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

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