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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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David B Massey (2019)

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Mihai Tibar, Raimundo Santos (2008)

http://arxiv.org/licenses/nonexclusive-distrib/1.0/