Orbits of smooth functions on 2-torus and their homotopy types

S. I. Maksymenko; B. G. Feshchenko

doi:10.15330/ms.44.1.67-83

This result holds for a larger class of smooth functions $f\colon T^2\to\mathbb{R}$ having the following property: for every critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to\mathbb{R}$ without multiple factors.

1. A.V. Bolsinov, A.T. Fomenko, Introduction to the topology of integrable hamiltonian systems, Nauka, Moscow, 1997. (in Russian)

2. Yu.M. Burman, Morse theory for functions of two variables without critical points, Funct. Differ. Eq., 3 (1995), №1-2, 31-43.

3. Yu.M. Burman, Triangulations of surfaces with boundary and the homotopy principle for functions without critical points, Ann. Global Anal. Geom., 17 (1999), №3, 221-238.

4. C.J. Earle, J. Eells, The diffeomorphism group of a compact Riemann surface, Bull. Amer. Math. Soc., 73 (1967), 557-559.

5. A.T. Fomenko, A Morse theory for integrable Hamiltonian systems, Dokl. Akad. Nauk SSSR, 287 (1986), №5, 1071-1075. (in Russian)

6. A.T. Fomenko, Symplectic topology of completely integrable Hamiltonian systems, Uspekhi Mat. Nauk, 44 (1989), №1(265), 145-173. (in Russian)

7. A. Gramain, Le type d'homotopie du groupe des diffeomorphismes d'une surface compacte, Ann. Sci. Ecole Norm. Sup. (4), 6 (1973), 53-66.

8. A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.

9. K. Ikegami, O. Saeki, Cobordism of Morse maps and its applications to map germs, Math. Proc. Cambridge Philos. Soc., 147 (2009), №1, 235-254.

10. B. Kalmar, Cobordism group of Morse functions on unoriented surfaces, Kyushu J. Math., 59 (2005), №2, 351-363.

11. A.S. Kronrod, On functions of two variables, Uspehi Matem. Nauk (N.S.), 5 (1950), №1(35), 24-134.

12. E.A. Kudryavtseva, Realization of smooth functions on surfaces as height functions, Mat. Sb., 190 (1999), №3, 29-88.

13. E.A. Kudryavtseva, Connected components of spaces of Morse functions with fixed critical points, Vestnik Moskov. Univ. Ser. I Mat. Mekh., (2012), №1, 3-12. (in Russian)

14. E.A. Kudryavtseva, The topology of spaces of Morse functions on surfaces, Math. Notes, 92 (2012), №1-2, 219.236 (in Russian), Translation of Mat. Zametki, 92 (2012), №2, 241-261.

15. E.A. Kudryavtseva, On the homotopy type of spaces of Morse functions on surfaces, Mat. Sb., 204 (2013), №1, 79-118.

16. E.A. Kudryavtseva, D.A. Permyakov, Framed Morse functions on surfaces, Mat. Sb., 201 (2010), №4, 33-98.

17. E.V. Kulinich, On topologically equivalent Morse functions on surfaces, Methods Funct. Anal. Topology, 4 (1998), №1, 59-64.

18. S. Maksymenko, Path-components of Morse mappings spaces of surfaces, Comment. Math. Helv., 80 (2005), №3, 655-690.

19. S. Maksymenko, Components of spaces of Morse mappings, Some problems in contemporary mathematics, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 25 (1998), 135-153. (in Russian)

20. S. Maksymenko, Smooth shifts along trajectories of flows, Topology Appl., 130 (2003), №2, 183-204.

21. S. Maksymenko, Homotopy types of stabilizers and orbits of Morse functions on surfaces, Ann. Global Anal. Geom., 29 (2006), №3, 241-285.

22. S. Maksymenko, Connected components of partition preserving diffeomorphisms, Methods Funct. Anal. Topology, 15 (2009), №3, 264-279.

23. S. Maksymenko, Deformations of circle-valued morse functions on surfaces, Ukrainian Math. Journal, 62 (2010), №10, 1360-1366.

24. S. Maksymenko, Functions on surfaces and incompressible subsurfaces, Methods Funct. Anal. Topology, 16 (2010), №2, 167-182.

25. S. Maksymenko, Functions with isolated singularities on surfaces, Geometry and topology of functions on manifolds, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 7 (2010), №4, 7-66.

26. S. Maksymenko, Homotopy types of right stabilizers and orbits of smooth functions functions on surfaces, Ukr. Math. Journ., 64 (2012), №9, 1186-1203. (in Russian)

27. S. Maksymenko, Structure of fundamental groups of orbits of smooth functions on surfaces, 2014.

28. S. Maksymenko, B. Feshchenko, Homotopy properties of spaces of smooth functions on 2-torus, Ukr. Math. Journ., 66 (2014), №9, 1205-1212. (in Russian)

29. S. Maksymenko, B. Feshchenko, Smooth functions on 2-torus whose kronrod-reeb graph contains a cycle, Methods Funct. Anal. Topology, 21 (2015), №1, 22-40.

30. Ya. Masumoto, O. Saeki, A smooth function on a manifold with given Reeb graph, Kyushu J. Math., 65 (2011), №1, 75-84.

31. S.V. Matveev, A.T. Fomenko, V.V. Sharko, Round Morse functions and isoenergetic surfaces of integrable Hamiltonian systems, Mat. Sb. (N.S.), 135(177) (1988), №3, 325-345.

32. G. Reeb, Sur certaines proprietes topologiques des varietes feuilletees, Actualites Sci. Ind., №1183, Hermann & Cie., Paris, 1952, Publ. Inst. Math. Univ. Strasbourg 11, 5.89, 155-156.

33. V.V. Sharko, Functions on surfaces. I, Some problems in contemporary mathematics, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 25 (1998), 408-434. (in Russian)

34. V.V. Sharko, Smooth and topological equivalence of functions on surfaces, Ukr. Mat. Zh., 55 (2003), №5, 687-700.

35. V.V. Sharko, About Kronrod-Reeb graph of a function on a manifold, Methods Funct. Anal. Topology, 12 (2006), №4, 389-396.

	Orbits of smooth functions on 2-torus and their homotopy types
Author	S. I. Maksymenko, B. G. Feshchenko maks@imath.kiev.ua, fb@imath.kiev.ua Topology department Institute of Mathematics of NAS of Ukraine
Abstract	This result holds for a larger class of smooth functions $f\colon T^2\to\mathbb{R}$ having the following property: for every critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to\mathbb{R}$ without multiple factors.
Keywords	diffeomorphism; Morse function; homotopy type
DOI	doi:10.15330/ms.44.1.67-83
Reference	1. A.V. Bolsinov, A.T. Fomenko, Introduction to the topology of integrable hamiltonian systems, Nauka, Moscow, 1997. (in Russian) 2. Yu.M. Burman, Morse theory for functions of two variables without critical points, Funct. Differ. Eq., 3 (1995), №1-2, 31-43. 3. Yu.M. Burman, Triangulations of surfaces with boundary and the homotopy principle for functions without critical points, Ann. Global Anal. Geom., 17 (1999), №3, 221-238. 4. C.J. Earle, J. Eells, The diffeomorphism group of a compact Riemann surface, Bull. Amer. Math. Soc., 73 (1967), 557-559. 5. A.T. Fomenko, A Morse theory for integrable Hamiltonian systems, Dokl. Akad. Nauk SSSR, 287 (1986), №5, 1071-1075. (in Russian) 6. A.T. Fomenko, Symplectic topology of completely integrable Hamiltonian systems, Uspekhi Mat. Nauk, 44 (1989), №1(265), 145-173. (in Russian) 7. A. Gramain, Le type d'homotopie du groupe des diffeomorphismes d'une surface compacte, Ann. Sci. Ecole Norm. Sup. (4), 6 (1973), 53-66. 8. A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. 9. K. Ikegami, O. Saeki, Cobordism of Morse maps and its applications to map germs, Math. Proc. Cambridge Philos. Soc., 147 (2009), №1, 235-254. 10. B. Kalmar, Cobordism group of Morse functions on unoriented surfaces, Kyushu J. Math., 59 (2005), №2, 351-363. 11. A.S. Kronrod, On functions of two variables, Uspehi Matem. Nauk (N.S.), 5 (1950), №1(35), 24-134. 12. E.A. Kudryavtseva, Realization of smooth functions on surfaces as height functions, Mat. Sb., 190 (1999), №3, 29-88. 13. E.A. Kudryavtseva, Connected components of spaces of Morse functions with fixed critical points, Vestnik Moskov. Univ. Ser. I Mat. Mekh., (2012), №1, 3-12. (in Russian) 14. E.A. Kudryavtseva, The topology of spaces of Morse functions on surfaces, Math. Notes, 92 (2012), №1-2, 219.236 (in Russian), Translation of Mat. Zametki, 92 (2012), №2, 241-261. 15. E.A. Kudryavtseva, On the homotopy type of spaces of Morse functions on surfaces, Mat. Sb., 204 (2013), №1, 79-118. 16. E.A. Kudryavtseva, D.A. Permyakov, Framed Morse functions on surfaces, Mat. Sb., 201 (2010), №4, 33-98. 17. E.V. Kulinich, On topologically equivalent Morse functions on surfaces, Methods Funct. Anal. Topology, 4 (1998), №1, 59-64. 18. S. Maksymenko, Path-components of Morse mappings spaces of surfaces, Comment. Math. Helv., 80 (2005), №3, 655-690. 19. S. Maksymenko, Components of spaces of Morse mappings, Some problems in contemporary mathematics, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 25 (1998), 135-153. (in Russian) 20. S. Maksymenko, Smooth shifts along trajectories of flows, Topology Appl., 130 (2003), №2, 183-204. 21. S. Maksymenko, Homotopy types of stabilizers and orbits of Morse functions on surfaces, Ann. Global Anal. Geom., 29 (2006), №3, 241-285. 22. S. Maksymenko, Connected components of partition preserving diffeomorphisms, Methods Funct. Anal. Topology, 15 (2009), №3, 264-279. 23. S. Maksymenko, Deformations of circle-valued morse functions on surfaces, Ukrainian Math. Journal, 62 (2010), №10, 1360-1366. 24. S. Maksymenko, Functions on surfaces and incompressible subsurfaces, Methods Funct. Anal. Topology, 16 (2010), №2, 167-182. 25. S. Maksymenko, Functions with isolated singularities on surfaces, Geometry and topology of functions on manifolds, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 7 (2010), №4, 7-66. 26. S. Maksymenko, Homotopy types of right stabilizers and orbits of smooth functions functions on surfaces, Ukr. Math. Journ., 64 (2012), №9, 1186-1203. (in Russian) 27. S. Maksymenko, Structure of fundamental groups of orbits of smooth functions on surfaces, 2014. 28. S. Maksymenko, B. Feshchenko, Homotopy properties of spaces of smooth functions on 2-torus, Ukr. Math. Journ., 66 (2014), №9, 1205-1212. (in Russian) 29. S. Maksymenko, B. Feshchenko, Smooth functions on 2-torus whose kronrod-reeb graph contains a cycle, Methods Funct. Anal. Topology, 21 (2015), №1, 22-40. 30. Ya. Masumoto, O. Saeki, A smooth function on a manifold with given Reeb graph, Kyushu J. Math., 65 (2011), №1, 75-84. 31. S.V. Matveev, A.T. Fomenko, V.V. Sharko, Round Morse functions and isoenergetic surfaces of integrable Hamiltonian systems, Mat. Sb. (N.S.), 135(177) (1988), №3, 325-345. 32. G. Reeb, Sur certaines proprietes topologiques des varietes feuilletees, Actualites Sci. Ind., №1183, Hermann & Cie., Paris, 1952, Publ. Inst. Math. Univ. Strasbourg 11, 5.89, 155-156. 33. V.V. Sharko, Functions on surfaces. I, Some problems in contemporary mathematics, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 25 (1998), 408-434. (in Russian) 34. V.V. Sharko, Smooth and topological equivalence of functions on surfaces, Ukr. Mat. Zh., 55 (2003), №5, 687-700. 35. V.V. Sharko, About Kronrod-Reeb graph of a function on a manifold, Methods Funct. Anal. Topology, 12 (2006), №4, 389-396.
Pages	67-83
Volume	44
Issue	1
Year	2015
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html