logo AFST
Linking and the Morse complex
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 1, pp. 25-94.

For a Morse function f on a compact oriented manifold M, we show that f has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in M whose components have nontrivial linking number, such that the minimal value of f on one of the components is larger than its maximal value on the other. Indeed we characterize the precise number of critical points of f in terms of the Betti numbers of M and the behavior of f with respect to links. This can be viewed as a refinement, in the case of compact manifolds, of the Rabinowitz Saddle Point Theorem. Our approach, inspired in part by techniques of chain-level symplectic Floer theory, involves associating to collections of chains in M algebraic operations on the Morse complex of f, which yields relationships between the linking numbers of homologically trivial (pseudo-)cycles in M and an algebraic linking pairing on the Morse complex.

Pour une fonction de Morse f sur une variété compacte orientée M, nous montrons que f a un nombre de points critiques supérieur au nombre requis par les inégalités de Morse si, et seulement si, il existe une certaine classe d’entrelacs dans M, dont les composantes ont un nombre d’enlacement non trivial, telle que la valeur minimale de f sur l’une des composantes est supérieure à sa valeur maximale sur l’autre composante. Nous définissons le nombre exact de points critiques de f en fonction des nombres de Betti de M et du comportement de f par rapport aux entrelacs. Ce résultat peut être vu comme un raffinement, dans le cas des variétés compactes, du théorème du point selle de Rabinowitz. Notre approche, partiellement inspirée des techniques de théorie symplectique de Floer au niveau des chaînes, est basée sur l’association d’opérations algébriques sur le complexe de Morse de f à certaines collections de chaînes de M, ce qui induit des relations entre les nombres d’enlacement des (pseudo-)cycles homologiquement triviaux de M d’une part, et un accouplement d’enlacement algébrique sur le complexe de Morse d’autre part.

Published online:
DOI: 10.5802/afst.1397
@article{AFST_2014_6_23_1_25_0,
     author = {Michael Usher},
     title = {Linking and the {Morse} complex},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {25--94},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 23},
     number = {1},
     year = {2014},
     doi = {10.5802/afst.1397},
     zbl = {1301.53095},
     mrnumber = {3204731},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1397/}
}
TY  - JOUR
TI  - Linking and the Morse complex
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2014
DA  - 2014///
SP  - 25
EP  - 94
VL  - Ser. 6, 23
IS  - 1
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1397/
UR  - https://zbmath.org/?q=an%3A1301.53095
UR  - https://www.ams.org/mathscinet-getitem?mr=3204731
UR  - https://doi.org/10.5802/afst.1397
DO  - 10.5802/afst.1397
LA  - en
ID  - AFST_2014_6_23_1_25_0
ER  - 
%0 Journal Article
%T Linking and the Morse complex
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2014
%P 25-94
%V Ser. 6, 23
%N 1
%I Université Paul Sabatier, Toulouse
%U https://doi.org/10.5802/afst.1397
%R 10.5802/afst.1397
%G en
%F AFST_2014_6_23_1_25_0
Michael Usher. Linking and the Morse complex. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 1, pp. 25-94. doi : 10.5802/afst.1397. https://afst.centre-mersenne.org/articles/10.5802/afst.1397/

[1] Biran (P.), Cornea (O.).— Lagrangian topology and enumerative geometry, Geom. Topol. 16, no. 2, p. 963-1052 (2012). | MR: 2928987 | Zbl: 1253.53079

[2] Burghelea (D.), Haller (S.).— On the topology and analysis of a closed one form. I (Novikov’s theory revisited), Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., vol. 38, Enseignement Math., Geneva, p. 133-175 (2001). | MR: 1929325 | Zbl: 1017.57013

[3] Chang (K-C).— Infinite-dimensional Morse theory and multiple solution problems, Progress in Nonlinear Differential Equations and their Applications, 6, Birkhäuser Boston Inc., Boston, MA (1993). | MR: 1196690 | Zbl: 0779.58005

[4] Cornea (O.).— Homotopical dynamics. IV. Hopf invariants and Hamiltonian ows, Comm. Pure Appl. Math. 55, no. 8, p. 1033-1088 (2002). | MR: 1900179 | Zbl: 1024.37040

[5] Cornea (O.), Ranicki (A.).— Rigidity and gluing for Morse and Novikov complexes, J. Eur. Math. Soc. (JEMS) 5, no. 4, p. 343-394 (2003). | MR: 2017851 | Zbl: 1052.57052

[6] Douady (A.).— Variétés à bord anguleux et voisinages tubulaires, Séminaire Henri Cartan, 1961/62, Exp. 1, Secrétariat mathématique, Paris, 1961/1962, p. 11. | Numdam | MR: 160221 | Zbl: 0116.40304

[7] Fukaya (K.), Oh (Y-G), Ohta (H.), Ono (K.).— Lagrangian intersection Floer theory: anomaly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI (2009). | MR: 2553465 | Zbl: 1181.53002

[8] Fukaya (K.).— Morse homotopy, A1-category, and Floer homologies, Proceedings of GARC Workshop on Geometry and Topology ’93 (Seoul, 1993) (Seoul), Lecture Notes Ser., vol. 18, Seoul Nat. Univ., p. 1-102 (1993). | MR: 1270931 | Zbl: 0853.57030

[9] Jänich (K.).— On the classification of O(n)-manifolds, Math. Ann. 176, p. 53-76 (1968). | MR: 226674 | Zbl: 0153.53801

[10] Latour (F.).— Existence de 1-formes fermées non singulières dans une classe de cohomologie de de Rham, Inst. Hautes Études Sci. Publ. Math. (1994), no. 80, p. 135-194 (1995). | EuDML: 104099 | Numdam | MR: 1320607 | Zbl: 0837.58002

[11] McDuff (D.) and Salamon (D.).— J-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications, vol. 52, American Mathematical Society, Providence, RI (2004). | MR: 2045629 | Zbl: 1064.53051

[12] Qin (L.).— An application of topological equivalence to Morse theory (2011).

[13] Rabinowitz (P. H.).— Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, Published for the Conference Board of the Mathematical Sciences, Washington, DC (1986). | MR: 845785 | Zbl: 0609.58002

[14] Schwarz (M.).— Morse homology, Progress in Mathematics, vol. 111, Birkhäuser Verlag, Basel (1993). | MR: 1239174 | Zbl: 0806.57020

[15] Schwarz (M.).— Equivalences for Morse homology, Geometry and topology in dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), Contemp. Math., vol. 246, Amer. Math. Soc., Providence, RI, p. 197-216 (1999). | MR: 1732382 | Zbl: 0951.55009

[16] Smale (S.).— An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87, p. 861-866 (1965). | MR: 185604 | Zbl: 0143.35301

[17] Thom (R.).— Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, p. 17-86 (1954). | EuDML: 139072 | MR: 61823 | Zbl: 0057.15502

[18] Usher (M.).— Duality in filtered Floer-Novikov complexes, J. Topol. Anal. 2, no. 2, p. 233-258 (2010). | MR: 2652908 | Zbl: 1196.53051

[19] Usher (M.).— Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds, Israel J. Math. 184, p. 1-57 (2011). | MR: 2823968 | Zbl: 1253.53085

[20] Zinger (A.).— Pseudocycles and integral homology, Trans. Amer. Math. Soc. 360, no. 5, p. 2741-2765 (2008). | MR: 2373332 | Zbl: 1213.57031

Cited by Sources: