Homotopical dynamics of gradient flows

Octavian Cornea

Banach Center Publications (1998)

  • Volume: 45, Issue: 1, page 41-46
  • ISSN: 0137-6934

Abstract

top
In this paper we will be interested in results surrounding the following basic question: what are the homotopy properties that one can extract from a gradient flow? We approach this question by decomposing it into three parts: 1. Identify what are the homotopical objects that are provided by the flow (e.g. critical points, Conley indexes). 2. Discover what are the relations that have to be satisfied by these objects (e.g. Morse inequalities, Lusternik-Schnirelmann type inequalities). 3. (The Realizability Problem.) Given some homotopy objects that satisfy the relations from 2., is there a corresponding flow? Is this flow unique up to some equivalence relation? We will consider only gradient flows induced by functions with isolated critical points, restrict our discussion to the finite dimensional, compact context and concentrate on the non-Morse case. One way to look at these questions in this case is to first treat numerical invariants, then local invariants, and finally, pairwise invariants (that concern pairs of consecutive critical points)... At each level, we can look at the points 1.,2.,3. above.

How to cite

top

Cornea, Octavian. "Homotopical dynamics of gradient flows." Banach Center Publications 45.1 (1998): 41-46. <http://eudml.org/doc/208908>.

@article{Cornea1998,
abstract = {In this paper we will be interested in results surrounding the following basic question: what are the homotopy properties that one can extract from a gradient flow? We approach this question by decomposing it into three parts: 1. Identify what are the homotopical objects that are provided by the flow (e.g. critical points, Conley indexes). 2. Discover what are the relations that have to be satisfied by these objects (e.g. Morse inequalities, Lusternik-Schnirelmann type inequalities). 3. (The Realizability Problem.) Given some homotopy objects that satisfy the relations from 2., is there a corresponding flow? Is this flow unique up to some equivalence relation? We will consider only gradient flows induced by functions with isolated critical points, restrict our discussion to the finite dimensional, compact context and concentrate on the non-Morse case. One way to look at these questions in this case is to first treat numerical invariants, then local invariants, and finally, pairwise invariants (that concern pairs of consecutive critical points)... At each level, we can look at the points 1.,2.,3. above.},
author = {Cornea, Octavian},
journal = {Banach Center Publications},
keywords = {Conley index; homotopy properties; gradient flow; isolated critical points; non-Morse case},
language = {eng},
number = {1},
pages = {41-46},
title = {Homotopical dynamics of gradient flows},
url = {http://eudml.org/doc/208908},
volume = {45},
year = {1998},
}

TY - JOUR
AU - Cornea, Octavian
TI - Homotopical dynamics of gradient flows
JO - Banach Center Publications
PY - 1998
VL - 45
IS - 1
SP - 41
EP - 46
AB - In this paper we will be interested in results surrounding the following basic question: what are the homotopy properties that one can extract from a gradient flow? We approach this question by decomposing it into three parts: 1. Identify what are the homotopical objects that are provided by the flow (e.g. critical points, Conley indexes). 2. Discover what are the relations that have to be satisfied by these objects (e.g. Morse inequalities, Lusternik-Schnirelmann type inequalities). 3. (The Realizability Problem.) Given some homotopy objects that satisfy the relations from 2., is there a corresponding flow? Is this flow unique up to some equivalence relation? We will consider only gradient flows induced by functions with isolated critical points, restrict our discussion to the finite dimensional, compact context and concentrate on the non-Morse case. One way to look at these questions in this case is to first treat numerical invariants, then local invariants, and finally, pairwise invariants (that concern pairs of consecutive critical points)... At each level, we can look at the points 1.,2.,3. above.
LA - eng
KW - Conley index; homotopy properties; gradient flow; isolated critical points; non-Morse case
UR - http://eudml.org/doc/208908
ER -

References

top
  1. [1] T. Bartsch, Critical point theory and group actions, Springer 1993. 
  2. [2] M. Clapp and D. Puppe, The generalized Lusternik-Schnirelmann category of a product space, Trans. of the A.M.S. 321 (1990), 525-532. Zbl0709.55001
  3. [3] R. Cohen, J. D. S. Jones and G. Segal, Homotopical Morse Theory, preprint 1992. 
  4. [4] R. Cohen, J. D. S. Jones and G. Segal, Floer's infinite dimensional Morse theory and homotopy theory, The Floer memorial Volume, Birkhäuser (1995). Zbl0843.58019
  5. [5] Ch. Conley, Isolated invariant sets and the Morse index, CBMS Regional Conf. Ser. in Math. 38, Amer. Math. Soc., Providence, R. I., 1976. 
  6. [6] O. Cornea, Cone-length and Lusternik Schnirelmann category, Topology 33 (1994), 95-111. Zbl0811.55004
  7. [7] O. Cornea, Strong L.S.-category equals cone-length, Topology 34 (1995), 377-381. Zbl0833.55004
  8. [8] O. Cornea, Cone-decompositions and degenerate critical points, Proc. of the London Math. Soc. 77 (1998), 437-461. 
  9. [9] O. Cornea, Spanier-Whitehead duality and critical points, Homotopy Theory via Algebraic Geometry and Group Representations, AMS Contemp. Math. 220 (1998). 
  10. [10] O. Cornea, Some properties of relative L.S.-category, Fields Inst. Commun. 19, AMS (1998), 67-72. Zbl0898.55002
  11. [11] O. Cornea, Y. Felix and J. M. Lemaire, Rational Category and Cone-length of Poincaré Complexes, Topology 37 (1998), 743-748. Zbl0899.55003
  12. [12] O. Cornea, Homotopical Dynamics: Suspension and Duality, to appear in Erg. Theory & Dyn. Syst. Zbl0984.37017
  13. [13] O. Cornea, Homotopical Dynamics II: Hopf invariants smoothings and the Morse complex, preprint August 1998. 
  14. [14] E. N. Dancer, Degenerate critical points, homotopy indices and Morse inequalities, J. Reine Ang. Math. 350 (1984), 1-22. Zbl0525.58012
  15. [15] N. Dupont, A counterexample to the Lemaire-Sigrist conjecture, to appear in Topology. 
  16. [16] Y. Eliashberg and M. Gromov, Lagrangian Intersection Theory, preprint December 1996. 
  17. [17] E. Fadell and S. Y. Husseini, Relative category, products and coproducts, preprint 1994. Zbl0860.55011
  18. [18] Y. Félix and S. Halperin, Rational L.S. category and its applications, Trans. of the A.M.S. 273 (1982), 1-37 . 
  19. [19] Y. Félix and J. C. Thomas, Sur la structure des espaces de L.S. catégorie deux, Ill. J. of Math. 30 (1986), 574-593 . Zbl0585.55010
  20. [20] J. Franks, Morse-Smale flows and homotopy theory, Topology 18 (1979), 199-215. Zbl0426.58013
  21. [21] R. Franzosa, Connection matrices for the Conley index, Trans. AMS. Zbl0896.58053
  22. [22] T. Ganea, Lusternik-Schnirelmann category and strong category, Ill. J. of Math. 11 (1967), 417-427. Zbl0149.40703
  23. [23] I. M. James, On category, in the sense of Lusternik and Schnirelmann, Topology 17 (1978), 331-348. Zbl0408.55008
  24. [24] J.-M. Lemaire and F. Sigrist, Sur les invariants d'homotopie rationnelles liés à la L.S. catégorie, Comm. Math. Helv. 56 (1981), 103-122. 
  25. [25] L. Ljusternik and L. Schnirelmann, Méthodes topologiques dans les problèmes variationnels, Hermann, Paris, (1934). 
  26. [26] C. McCord, Poincaré-Lefschetz duality for the homology Conley index, Trans. of the A.M.S. 329 (1992), 233-252. Zbl0755.55011
  27. [27] P. M. Moyaux, Two lower bounds for the relative Lusternik Schnirelmann category, preprint Spring 1998. 
  28. [28] J. Pears, Degenerate Critical Points and the Conley Index, Thesis, University of Edinburgh, 1995. 
  29. [29] Y. Rudyak, On category weight and its applications, To appear in Topology. 
  30. [30] Y. Rudyak and J. Oprea, The Lusternik Schnirelmann category of Symplectic Manifolds, preprint, Summer 1997. Zbl0931.53039
  31. [31] D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans of the A.M.S. 291 (1985) 1-41. Zbl0573.58020
  32. [32] J. Strom, Two special cases of the Ganea conjecture, To appear in Trans. of the A.M.S. 
  33. [33] F. Takens, The Lusternik-Schnirelmann categories of a product space. Comp. Math. 22 (1970), 175-180. Zbl0198.28302
  34. [34] L. Vanderbrouck, Adjunction spaces satisfying the Ganea conjecture, preprint 1997. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.