Arnold conjecture

Short description: Mathematical 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 $H_t\in C^{\mathrm{\infty }}(M)$, $t\in [0,1]$. 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 ${\phi }_t:M\to M$. Each individual ${\phi }_t$ 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 $\phi :M\to M$ 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 ${\sum }_{i=0}^{2n}\dim H_i(M;\mathbb{𝔽})$. The weak Arnold conjecture says that

$\mathrm{\#}\{\text{fixed points of }\phi \}\ge \sum _{i=0}^{2n}\dim H_i(M;\mathbb{𝔽})$

for $\phi :M\to M$ 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 $L^{\prime }$ in terms of the Betti numbers of $L$, given that $L^{\prime }$ intersects L transversally and $L^{\prime }$ 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 $H_t\in C^{\mathrm{\infty }}(M)$, $t\in [0,1]$ be a smooth family of Hamiltonian functions on $M$. This family generates a 1-parameter family of diffeomorphisms ${\phi }_t:M\to M$ by flowing along the Hamiltonian vector field associated to $H_t$. The Arnold–Givental conjecture states that if ${\phi }_1(L)$ intersects transversely with $L$, then

$\mathrm{\#}({\phi }_1(L)\cap L)\ge \sum _{i=0}^n\dim H_i(L;\mathbb{\mathbb{Z}}/2\mathbb{\mathbb{Z}})$.

Status

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

• Alexander Givental proved it for $(M,L)=({\mathbb{ℂ}\mathbb{\mathbb{P}}}^n,{\mathbb{ℝ}\mathbb{\mathbb{P}}}^n)$.
• 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
• 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, doi:10.1155/S1073792804133941 .
• 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, https://www.mathnet.ru/eng/faa/v23/i4/p37 
  • 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 
• 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, https://gallica.bnf.fr/ark:/12148/bpt6k5470708m/f315.item?lang=FR# .
• 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