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.^[1]

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

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

The function ${\displaystyle H}$ is called a Hamiltonian function.

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

The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of ${\displaystyle M}$ is greater than or equal to the number of critical points of a smooth function on ${\displaystyle M}$.^[2][3]

Let ${\displaystyle (M,\omega )}$ be a closed symplectic manifold. A Hamiltonian diffeomorphism ${\displaystyle \varphi :M\to M}$ is called nondegenerate if its graph intersects the diagonal of ${\displaystyle M\times M}$ 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 ${\displaystyle M}$, called the Morse number of ${\displaystyle M}$.

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

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

for ${\displaystyle \varphi :M\to M}$ a nondegenerate Hamiltonian diffeomorphism.^[2][3]

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 ${\displaystyle L'}$ in terms of the Betti numbers of ${\displaystyle L}$, given that ${\displaystyle L'}$ intersects L transversally and ${\displaystyle L'}$ is Hamiltonian isotopic to L.

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

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

${\displaystyle \#(\varphi _{1}(L)\cap L)\geq \sum _{i=0}^{n}\dim H_{i}(L;\mathbb {Z} /2\mathbb {Z} )}$.^[4]

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

• Givental proved it for ${\displaystyle (M,L)=(\mathbb {CP} ^{n},\mathbb {RP} ^{n})}$.^[5]
• Yong-Geun Oh proved it for real forms of compact Hermitian spaces with suitable assumptions on the Maslov indices.^[6]
• 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 ${\displaystyle (M,\omega )}$ semi-positive.^[7]
• Urs Frauenfelder proved it in the case when ${\displaystyle (M,\omega )}$ is a certain symplectic reduction, using gauged Floer theory.^[4]

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

• 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