Editing Symplectomorphism (section)

==Arnold conjecture==
{{main|Arnold conjecture}}
A celebrated conjecture of [[Vladimir Arnold]] relates the ''minimum'' number of [[Fixed point (mathematics)|fixed points]] for a Hamiltonian symplectomorphism <math>\varphi: M \to M</math>, in case <math>M</math> is a compact [[symplectic manifold]], to [[Morse theory]] (see <ref>{{cite book |last1=Arnolʹd |first1=Vladimir |title=Mathematical methods of classical mechanics |series=Graduate Texts in Mathematics |date=1978 |volume=60 |publisher=Springer-Verlag |location=New York |doi=10.1007/978-1-4757-1693-1 |isbn=978-1-4757-1693-1 |url=https://link.springer.com/book/10.1007/978-1-4757-1693-1}}</ref>). More precisely, the conjecture states that <math>\varphi</math> has at least as many fixed points as the number of [[critical point (mathematics)|critical point]]s that a smooth function on <math>M</math> must have. Certain weaker version of this conjecture has been proved: when <math>\varphi</math> is "nondegenerate", the number of fixed points is bounded from below by the sum of [[Betti number]]s of <math>M</math> (see,<ref>{{cite journal |last1=Fukaya |first1=Kenji |last2=Ono |first2=Kaoru |title=Arnold conjecture and Gromov-Witten invariants |journal=Topology |date=September 1999 |volume=38 |issue=5 |pages=933–1048 |doi=10.1016/S0040-9383(98)00042-1 |doi-access=free }}</ref><ref>{{cite journal |last1=Liu |first1=Gang |last2=Tian |first2=Gang |title=Floer homology and Arnold conjecture |journal=Journal of Differential Geometry |date=1998 |volume=49 |issue=1 |pages=1–74 |doi=10.4310/jdg/1214460936 |doi-access=free }}</ref>). The most important development in symplectic geometry triggered by this famous conjecture is the birth of [[Floer homology]] (see <ref>{{cite journal |last1=Floer |first1=Andreas |title=Symplectic fixed points and holomorphic spheres |journal=Communications in Mathematical Physics |date=1989 |volume=120 |issue=4 |pages=575–611 |doi=10.1007/BF01260388 |s2cid=123345003 |url=https://link.springer.com/article/10.1007/BF01260388}}</ref>), named after [[Andreas Floer]].