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Linking and the Morse complex
Michael Usher
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 1, p. 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.

@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},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 23},
     number = {1},
     year = {2014},
     pages = {25-94},
     doi = {10.5802/afst.1397},
     mrnumber = {3204731},
     zbl = {1301.53095},
     language = {en},
     url = {https://afst.centre-mersenne.org/item/AFST_2014_6_23_1_25_0}
}
Usher, Michael. 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. afst.centre-mersenne.org/item/AFST_2014_6_23_1_25_0/

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