Arnold conjecture

The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential 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 a Hamiltonian vector field $X_{H}$ on $M$ defined by the formula

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

The function $H$ is called a Hamiltonian 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 of diffeomorphisms $\varphi _{t}:M\to M$. Each individual $\varphi _{t}$ is a called a Hamiltonian diffeomorphism of $M$.

The strong Arnold conjecture states 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 called nondegenerate if 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 a Morse function on $M$, called the Morse number of $M$.

In view of the Morse inequality, the Morse number is greater than or equal to the sum of Betti numbers over a field ${\mathbb {F} }$, namely ${\textstyle \sum _{i=0}^{2n}\dim H_{i}(M;{\mathbb {F} })}$. The weak Arnold conjecture says 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

The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, gives a lower bound on the number of intersection points of two Lagrangian submanifolds L and $L'$ in terms of the Betti numbers of $L$, given that $L'$ intersects L transversally and $L'$ is Hamiltonian isotopic to L.

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 of Hamiltonian functions on $M$. This family generates a 1-parameter family of diffeomorphisms $\varphi _{t}:M\to M$ by flowing along the Hamiltonian vector field associated 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.

• Givental proved it for $(M,L)=(\mathbb {CP} ^{n},\mathbb {RP} ^{n})$.
• Yong-Geun Oh proved it for real forms of compact Hermitian spaces with suitable assumptions on the Maslov indices.
• Lazzarini proved it for negative monotone case under suitable assumptions on the minimal Maslov number.
• Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono proved it for $(M,\omega )$ semi-positive.
• Urs Frauenfelder proved it in the case when $(M,\omega )$ is a certain symplectic reduction, using gauged Floer 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, MR 2076142.
• Fukaya, Kenji; Oh, Yong-Geun; Ohta, Hiroshi; Ono, Kaoru (2009), Lagrangian intersection Floer theory - anomaly and obstruction, International Press, ISBN 978-0-8218-5253-8
• 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, S2CID 123546007, Zbl 0724.58031
• 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, MR 1179726.
• 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, ISBN 978-3-0348-9948-2