On the minimal number of critical points of smooth maps between closed manifolds

Russian Journal of Mathematical Physics - Tập 16 - Trang 363-370 - 2009
D. Andrica1, L. Funar2, E. Kudryavtseva3
1Faculty of Mathematics and Computer Science, “Babes-Bolyai” University of Cluj, Cluj-Napoca, Romania
2Institut Fourier BP 74, UMR 5582, Université de Grenoble I, Saint-Martin-d’Hères cedex, France
3Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

Tóm tắt

New information concerning the minimal number of critical points of smooth proper mappings between closed connected surfaces (possibly with boundary) without critical points on the boundary is presented.

Tài liệu tham khảo

D. Andrica and L. Funar, “On Smooth Maps with Finitely Many Critical Points,” J. London Math. Soc. 69, 783–800 (2004). D. Andrica and L. Funar, “On SmoothMaps with Finitely Many Critical Points. Addendum,” J. London Math. Soc. 73, 231–236 (2006). I. Berstein and A. L. Edmonds, “On the Construction of Branched Coverings of Low-Dimensional Manifolds,” Trans. Amer. Math. Soc. 247, 87–124 (1979). S. A. Bogatyi, D. L. Gonçalves, E.A. Kudryavtseva, and H. Zieschang, “Realization of Primitive Branched Coverings over Closed Surfaces,” in Advances in Topological Quantum Field Theory, NATO Sci. Ser. II Math. Phys. Chem. 179 (Kluwer Acad. Publ., Dordrecht, 2004), pp. 297–316. S. A. Bogatyi, D. L. Gonçalves, E.A. Kudryavtseva, and H. Zieschang, “Realization of Primitive Branched Coverings over Closed Surfaces Following the Hurwitz Approach,” Cent. Eur. J. Math. 1, 184–197 (2003). V. Braungardt and D. Kotschick, “Clustering of Critical Points in Lefschetz Fibrations and the Symplectic Szpiro Inequality,” Trans. Amer. Math. Soc. 355, 3217–3226 (2003). P. T. Church and J.G. Timourian, “Differentiable Maps with 0-Dimensional Critical Set I,” Pacific J. Math. 41, 615–630 (1972). P. T. Church and J.G. Timourian, “Continuous Maps with 0-Dimensional Branch Set,” Indiana Univ. Math. J. 23, 949–958 (1974). P. T. Church and J.G. Timourian, “Differentiable Maps with 0-Dimensional Critical Set II,” Indiana Univ. Math. J. 24, 17–28 (1974). S.K. Donaldson, “Lefschetz Pencils on Symplectic Manifolds,” J. Differential Geom. 53, 205–236 (1999). S.K. Donaldson, “Lefschetz Pencils and Mapping Class Groups,” in Problems on Mapping Class Groups and Related Topics, ed. by B. Farb,Proc. Sympos. Pure Math. 74 (Amer. Math. Soc., Providence, 2006), pp. 151–163. A. Edmonds, R. Kulkarni, and R. Stong, “Realizability of Branched Coverings of Surfaces,” Trans. Amer. Math. Soc. 282, 773–790 (1984). J.B. Etnyre and T. Fuller, “Realizing 4-Manifolds as Achiral Lefschetz Fibrations,” Int. Math. Res. Not., Art. ID 70272 (2006). C. L. Ezell, “Branch Point Structure of Covering Maps onto Nonorientable Surfaces,” Trans. Amer. Math. Soc. 243, 123–133 (1978). L. Funar, C. Pintea, and P. Zhang, “Examples of Smooth Maps with Finitely Many Critical Points in Dimensions (4, 3), (8, 5), and (16, 9),” math.GT/0803.0665. L. Funar, “Smooth Maps with Finitely Many Critical Points in Dimensions (4, 3) and (8, 5),” in prepar. R. E. Gompf and A. I. Stipsicz, 4-Manifolds and Kirby Calculus (Amer. Math. Soc., Providence, 1999). J. L. Harer, “Pencils of Curves of 4-Manifolds,” PhD Thesis (Univ. California, Berkeley, 1979). D. H. Husemoller, “Ramified Coverings of Riemann Surfaces,” Duke Math. J. 29, 167–174 (1962). H. C. King, “Topological Type of Isolated Singularities,” Ann. of Math. 107, 385–397 (1978). M. Korkmaz and B. Ozbagci, “Minimal Number of Singular Fibers in a Lefschetz Fibration,” Proc. Amer. Math. Soc. 129(5), 1545–1549 (2001). S. Łojasiewicz, “Triangulation of Semi-Analytic Sets,” Ann. Sc. Norm. Super. Pisa (3) 18, 449–474 (1964). Y. Matsumoto, “Handlebody Decompositions of 4-Manifolds and Torus Fibrations,” Osaka J. Math. 33, 805–822 (1996). C. Pintea, “Continuous Mappings with an Infinite Number of Topologically Critical Points,” Ann. Polon. Math. 67, 87–93 (1997). V. V. Prasolov and A.B. Sossinsky, Knots, Links, Braids and 3-Manifolds. An Introduction to the New Invariants in Low-Dimensional Topology, Transl. Math. Monogr. 154 (Amer. Math. Soc., 1997). G. B. Shabat and V. A. Voevodsky, “Drawing Curves over Number Fields,” in Grothendieck Festschrift, ed. by P. Cartier, Progress in Math. 88, Vol. 3 (Birkhäuser, 1990), pp. 199–227. A. I. Stipsicz, “On the Number of Vanishing Cycles in Lefschetz Fibrations,” Math. Res. Lett. 6(3–4), 449–456 (1999). A. I. Stipsicz, “Singular Fibers in Lefschetz Fibrations on Manifolds with b +2 = 1,” Topology Appl. 117 (1), 9–21 (2002). F. Takens, “Isolated Critical Points of C ∞ and C ω Functions,” Indag. Math. 29, 238–243 (1967).