Linking and the Morse complex

Michael Usher

Annales de la faculté des sciences de Toulouse Mathématiques (2014)

  • Volume: 23, Issue: 1, page 25-94
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Usher, Michael. "Linking and the Morse complex." Annales de la faculté des sciences de Toulouse Mathématiques 23.1 (2014): 25-94. <http://eudml.org/doc/275413>.

@article{Usher2014,
abstract = {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.},
author = {Usher, Michael},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Morse function; linking number; Betti numbers; Morse complex; Floer theory},
language = {eng},
number = {1},
pages = {25-94},
publisher = {Université Paul Sabatier, Toulouse},
title = {Linking and the Morse complex},
url = {http://eudml.org/doc/275413},
volume = {23},
year = {2014},
}

TY - JOUR
AU - Usher, Michael
TI - Linking and the Morse complex
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2014
PB - Université Paul Sabatier, Toulouse
VL - 23
IS - 1
SP - 25
EP - 94
AB - 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.
LA - eng
KW - Morse function; linking number; Betti numbers; Morse complex; Floer theory
UR - http://eudml.org/doc/275413
ER -

References

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