On~classification of Morse--Smale flows on projective-like manifolds

V. Z. Grines ; E. Ya. Gurevich

Izvestiya. Mathematics , Tome 86 (2022) no. 5, pp. 876-902.

Résumé

In this paper, the problem of topological classification of gradient-like flows without heteroclinic intersections, given on a four-dimensional projective-like manifold, is solved. We show that a complete topological invariant for such flows is a bi-color graph that describes the mutual arrangement of closures of three-dimensional invariant manifolds of saddle equilibrium states. The problem of construction of a canonical representative in each topological equivalence class is solved.

Détail

Keywords: gradient-like flows, topological classification, projective-like manifolds, Morse function with three critical points, complex projective plane.

Bibliographie

[1] S. Smale, “On gradient dynamical systems”, Ann. of Math. (2), 74:1 (1961), 199–206 | DOI | MR | Zbl

[2] K. R. Meyer, “Energy functions for Morse Smale systems”, Amer. J. Math., 90:4 (1968), 1031–1040 | DOI | MR | Zbl

[3] V. Z. Grines, T. V. Medvedev, and O. V. Pochinka, Dynamical systems on 2- and 3-manifolds, Dev. Math., 46, Springer, Cham, 2016 | DOI | MR | Zbl

[4] V. V. Prasolov, Elements of combinatorial and differential topology, MCCME, Moscow, 2004; English transl. Grad. Stud. Math., 74, Amer. Math. Soc., Providence, RI, 2006 | DOI | MR | Zbl

[5] J. Eells, Jr., and N. H. Kuiper, “Manifolds which are like projective planes”, Inst. Hautes Études Sci. Publ. Math., 14 (1962), 5–6 | DOI | MR | Zbl

[6] D. Pixton, “Wild unstable manifolds”, Topology, 16:2 (1977), 167–172 | DOI | MR | Zbl

[7] C. Bonatti and V. Grines, “Knots as topological invariants for gradient-like diffeomorphisms of the sphere $S^3$”, J. Dynam. Control Systems, 6:4 (2000), 579–602 | DOI | MR | Zbl

[8] C. Bonatti, V. Grines, and O. Pochinka, “Topological classification of Morse–Smale diffeomorphisms on 3-manifolds”, Duke Math. J., 168:13 (2019), 2507–2558 | DOI | MR | Zbl

[9] V. S. Medvedev and E. V. Zhuzhoma, “Morse–Smale systems with few non-wandering points”, Topology Appl., 160:3 (2013), 498–507 | DOI | MR | Zbl

[10] E. V. Zhuzhoma and V. S. Medvedev, “Continuous Morse–Smale flows with three equilibrium positions”, Mat. Sb., 207:5 (2016), 69–92 ; English transl. Sb. Math., 207:5 (2016), 702–723 | DOI | MR | Zbl | DOI

[11] V. Z. Grines, E. Ya. Gurevich, and O. V. Pochinka, “A combinatorial invariant of Morse–Smale diffeomorphisms without heteroclinic intersections on the sphere $S^n$, $n\ge 4$”, Mat. Zametki, 105:1 (2019), 136–141 ; English transl. Math. Notes, 105:1 (2019), 132–136 | DOI | MR | Zbl | DOI

[12] V. A. Rokhlin and D. B. Fuks, Beginner's course in topology. Geometric chapters, Nauka, Moscow, 1977 ; English transl. D. B. Fuks and V. A. Rokhlin, Universitext, Springer-Verlag, Berlin, 1984 | MR | Zbl | MR | Zbl

[13] D. Rolfsen, Knots and links, AMS Chelsea Publishing, 346, Reprint with corr. of the 1976 original, Amer. Math. Soc., Providence, RI, 2003 | MR | Zbl

[14] C. McA. Gordon and J. Luecke, “Knots are determined by their complements”, J. Amer. Math. Soc., 2:2 (1989), 371–415 | DOI | MR | Zbl

[15] M. W. Hirsch, Differential topology, Grad. Texts in Math., 33, Springer-Verlag, New York–Heidelberg, 1976 ; Russian transl. Mir, Moscow, 1979 | DOI | MR | Zbl | MR | Zbl

[16] V. Z. Grines, E. V. Zhuzhoma, and V. S. Medvedev, “New relations for Morse–Smale systems with trivially embedded one-dimensional separatrices”, Mat. Sb., 194:7 (2003), 25–56 ; English transl. Sb. Math., 194:7 (2003), 979–1007 | DOI | MR | Zbl | DOI

[17] J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells, Ann. of Math. Stud., 51, Princeton Univ. Press, Princeton, NJ, 1963 ; Russian transl. Moscow, Mir, 1965 | DOI | MR | Zbl | MR

[18] S. Smale, “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73:6 (1967), 747–817 ; Russian transl. Uspekhi Mat. Nauk, 25:1(151) (1970), 113–185 | MR | Zbl | MR

[19] M. Brown, “A proof of the generalized Schoenflies theorem”, Bull. Amer. Math. Soc., 66:2 (1960), 74–76 | DOI | MR | Zbl

[20] M. Brown, “Locally flat imbeddings of topological manifolds”, Ann. of Math. (2), 75:2 (1962), 331–341 | DOI | MR | Zbl

[21] J. Palis, Jr., and W. de Melo, Geometric theory of dynamical systems. An introduction, Transl. from the Portuguese, Springer-Verlag, New York–Berlin, 1982 ; Russian transl. Mir, Moscow, 1986 | DOI | MR | Zbl | MR

[22] D. M. Grobman, “Homeomorphism of systems of differential equations”, Dokl. Akad. Nauk SSSR, 128:5 (1959), 880–881 (Russian) | MR | Zbl

[23] P. Hartman, “A lemma in the theory of structural stability of differential equations”, Proc. Amer. Math. Soc., 11:4 (1960), 610–620 | DOI | MR | Zbl

[24] V. Guillemin and A. Pollack, Differential topology, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1974 | MR | Zbl

[25] Y. Matsumoto, An introduction to Morse theory, Transl. from the Japan., Transl. Math. Monogr., 208, Iwanami Series in Modern Mathematics, Amer. Math. Soc., Providence, RI, 2002 | DOI | MR | Zbl

[26] V. Z. Grines, E. Ya. Gurevich, V. S. Medvedev, and E. V. Zhuzhoma, “On topology of manifolds admitting a gradient-like flow with a prescribed non-wandering set”, Matem. Tr., 21:2 (2018), 163–180 ; English transl. Siberian Adv. Math., 29:2 (2019), 116–127 | DOI | MR | Zbl | DOI

[27] V. Z. Grines, E. Ya. Gurevich, and V. S. Medvedev, “On realization of topological conjugacy classes of Morse–Smale cascades on the sphere $S^n$”, Trudy Mat. Inst. Steklova, 310 (2020), 119–134 ; English transl. Proc. Steklov Inst. Math., 310 (2020), 108–123 | DOI | MR | Zbl | DOI