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Open book structures on semi-algebraic manifolds

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      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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      Non-degenerate mixed functions

       Mutsuo Oka (2010)
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        Real analytic Milnor fibrations and a strong Łojasiewicz inequality

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          Real map germs and higher open books

          We present a general criterion for the existence of open book structures defined by real map germs \((\bR^m, 0) \to (\bR^p, 0)\), where \(m> p \ge 2\), with isolated critical point. We show that this is satisfied by weighted-homogeneous maps. We also derive sufficient conditions in case of map germs with isolated critical value.
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            Author and article information

            Journal
            2014-09-15
            2014-09-16
            1409.4316

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

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            math.AG math.GN
            ccsd

            Geometry & Topology

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