Arnold conjecture

TheArnold conjecture,named after mathematicianVladimir Arnold,is a mathematical conjecture in the field ofsymplectic geometry,a branch ofdifferential geometry.

Strong Arnold conjecture

Let$(M,\omega )$be a closed (compact without boundary)symplectic manifold.For any smooth function$H:M\to {\mathbb {R} }$,the symplectic form$\omega$induces aHamiltonian vector field$X_{H}$on$M$defined by the formula

$\omega (X_{H},\cdot )=dH.$

The function$H$is called aHamiltonian function.

Suppose there is a smooth 1-parameter family of Hamiltonian functions$H_{t}\in C^{\infty }(M)$,$t\in [0,1]$.This family induces a 1-parameter family of Hamiltonian vector fields$X_{H_{t}}$on$M$.The family of vector fields integrates to a 1-parameter family ofdiffeomorphisms$\varphi _{t}:M\to M$.Each individual$\varphi _{t}$is a called aHamiltonian diffeomorphismof$M$.

Thestrong Arnold conjecturestates that the number of fixed points of a Hamiltonian diffeomorphism of$M$is greater than or equal to the number of critical points of a smooth function on$M$.

Weak Arnold conjecture

Let$(M,\omega )$be a closed symplectic manifold. A Hamiltonian diffeomorphism$\varphi:M\to M$is callednondegenerateif its graph intersects the diagonal of$M\times M$transversely. For nondegenerate Hamiltonian diffeomorphisms, one variant of the Arnold conjecture says that the number of fixed points is at least equal to the minimal number of critical points of aMorse functionon$M$,called theMorse numberof$M$.

In view of theMorse inequality,the Morse number is greater than or equal to the sum ofBetti numbersover afield${\mathbb {F} }$,namely${\textstyle \sum _{i=0}^{2n}\dim H_{i}(M;{\mathbb {F} })}$.Theweak Arnold conjecturesays that

$\#\{{\text{fixed points of }}\varphi \}\geq \sum _{i=0}^{2n}\dim H_{i}(M;{\mathbb {F} })$

for$\varphi:M\to M$a nondegenerate Hamiltonian diffeomorphism.

Arnold–Givental conjecture

TheArnold–Givental conjecture,named after Vladimir Arnold andAlexander Givental,gives a lower bound on the number of intersection points of twoLagrangian submanifoldsLand$L'$in terms of the Betti numbers of$L$,given that$L'$intersectsLtransversally and$L'$is Hamiltonian isotopic toL.

Let$(M,\omega )$be a compact$2n$-dimensional symplectic manifold, let$L\subset M$be a compact Lagrangian submanifold of$M$,and let$\tau:M\to M$be an anti-symplectic involution, that is, a diffeomorphism$\tau:M\to M$such that$\tau ^{*}\omega =-\omega$and$\tau ^{2}={\text{id}}_{M}$,whose fixed point set is$L$.

Let$H_{t}\in C^{\infty }(M)$,$t\in [0,1]$be a smooth family ofHamiltonian functionson$M$.This family generates a 1-parameter family of diffeomorphisms$\varphi _{t}:M\to M$by flowing along theHamiltonian vector fieldassociated to$H_{t}$.The Arnold–Givental conjecture states that if$\varphi _{1}(L)$intersects transversely with$L$,then

$\#(\varphi _{1}(L)\cap L)\geq \sum _{i=0}^{n}\dim H_{i}(L;\mathbb {Z} /2\mathbb {Z} )$.

Status

The Arnold–Givental conjecture has been proved for several special cases.

• Giventalproved it for$(M,L)=(\mathbb {CP} ^{n},\mathbb {RP} ^{n})$.
• Yong-Geun Ohproved it for real forms of compact Hermitian spaces with suitable assumptions on theMaslov indices.
• Lazzarini proved it for negative monotone case under suitable assumptions on the minimal Maslov number.
• Kenji Fukaya,Yong-Geun Oh, Hiroshi Ohta, andKaoru Onoproved it for$(M,\omega )$semi-positive.
• Urs Frauenfelderproved it in the case when$(M,\omega )$is a certain symplectic reduction, usinggaugedFloer theory.

See also

• Symplectomorphism#Arnold conjecture
• Floer homology
• Spectral invariants
• Conley–Zehnder theorem

References

Bibliography

• Frauenfelder, Urs (2004), "The Arnold–Givental conjecture and moment Floer homology",International Mathematics Research Notices,2004(42): 2179–2269,arXiv:math/0309373,doi:10.1155/S1073792804133941,MR2076142.
• Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru (2009),Lagrangian intersection Floer theory - anomaly and obstruction,International Press,ISBN978-0-8218-5253-8
• Givental, A. B. (1989a),"Periodic maps in symplectic topology",Funktsional. Anal. I Prilozhen,23(4): 37–52
  • Givental, A. B. (1989b), "Periodic maps in symplectic topology (translation from Funkts. Anal. Prilozh. 23, No. 4, 37-52 (1989))",Functional Analysis and Its Applications,23(4): 287–300,doi:10.1007/BF01078943,S2CID123546007,Zbl0724.58031
• Oh, Yong-Geun (1992),"Floer cohomology and Arnol'd-Givental's conjecture of [on] Lagrangian intersections",Comptes Rendus de l'Académie des Sciences,315(3): 309–314,MR1179726.
• Oh, Yong-Geun (1995), "Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks, III: Arnold-Givental Conjecture",The Floer Memorial Volume,pp. 555–573,doi:10.1007/978-3-0348-9217-9_23,ISBN978-3-0348-9948-2