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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.

### Most cited references3

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### Non-degenerate mixed functions

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

(2019)
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### Real map germs and higher open books

(2008)
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