Open book structures on semi-algebraic manifolds

Given a C 2 semi-algebraic mapping F : R N → R p , we consider its restriction to W ↪ R N an embedded closed semi-algebraic manifold of dimension n - 1 ≥ p ≥ 2 and introduce sufficient conditions for the existence of a fibration structure (generalized open book structure) induced by the projection F ‖ F ‖ : W \ F - 1 ( 0 ) → 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 π : R p → R p - 1 and prove that the fibers of F ‖ F ‖ and π ∘ F ‖ π ∘ F ‖ are homotopy equivalent. We also show several formulae relating the Euler characteristics of the fiber of the projection F ‖ F ‖ and W ∩ F - 1 ( 0 ) . Similar formulae are proved for mappings obtained after composition of F with canonical projections.