Manuscripta Mathematica

, Volume 149, Issue 1–2, pp 205–222 | Cite as

Open book structures on semi-algebraic manifolds

  • N. Dutertre
  • R. N. Araújo dos Santos
  • Ying Chen
  • Antonio Andrade do Espirito Santo


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.

Mathematics Subject Classification

14P10 32S55 58K15 58K65 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Araújo dos Santos R.N., Tibăr M.: Real map germs and higher open book structures. Geom. Dedicata 147, 177–185 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Araújo dos Santos R.N., Chen Y., Tibăr M.: Open book structures from real mappings. Cent. Eur. J. Math. 11(5), 817–828 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Araújo dos Santos, R.N., Chen, Y., Tibăr, M.: Real polynomial maps and singular open books at infinity. arXiv:1401.8286, to appear in Math. Scand
  4. 4.
    Bocknak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry, vol. 36. Springer, Berlin (1998)Google Scholar
  5. 5.
    Chen Y.: Milnor fibration at infinity for mixed polynomials. Cent. Eur. J. Math. 12(1), 28–38 (2014)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cisneros-Molina J.L., Seade J., Snoussi J.: Milnor fibrations and d-regularity for real analytic singularities. Int. J. Math. 21(4), 419–434 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Durfee A.: Neighborhoods of algebraic sets. Trans. Am. Math. Soc. 276(2), 517–530 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dutertre, N., Araújo dos Santos, R.N.: Topology of real Milnor fibrations for non-isolated singularities. Submitted for publication. arXiv:1211.6233
  9. 9.
    Massey, D.: Real analytic Milnor fibrations and a strong Łojasiewicz inequality. In: Manoel, M., Romero Fuster, M.C., Wall, C.T.C. (eds.) Real and Complex Singularities, pp. 268–292, London Math. Soc. Lecture Note Ser., 380. Cambridge University Press, Cambridge (2010)Google Scholar
  10. 10.
    Milnor J.: Singular Points of Complex Hypersurfaces, Vol. 61. Princeton University Press, Princeton (1968)Google Scholar
  11. 11.
    Némethi A., Zaharia A.: Milnor fibration at infinity. Indag. Math. (N.S.) 3(3), 323–335 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Oka M.: Non degenerate mixed functions. Kodai Math. J. 33(1), 1–62 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Thom R.: Ensembles et morphismes stratifiés. Bull. Am. Math. Soc. 75, 240–284 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wall C.T.C.: Topological invariance of the Milnor number mod 2. Topology 22(3), 345–350 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • N. Dutertre
    • 1
  • R. N. Araújo dos Santos
    • 2
  • Ying Chen
    • 2
  • Antonio Andrade do Espirito Santo
    • 3
  1. 1.CNRS, Centrale Marseille, I2M, UMR 7373Aix-Marseille UniversitéMarseilleFrance
  2. 2.Departamento de Matemática, Instituto de Ciências Matemáticas e de ComputaçãoUniversidade de São PauloSão CarlosBrazil
  3. 3.Centro de Ciências Exatas e TecnológicasUniversidade Federal do Recôncavo da BahiaCruz das AlmasBrazil

Personalised recommendations