butterfly/kb
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity
2fbd85a83f
kb/data/en.wikipedia.org/wiki/Arnold_conjecture-0.md

835 lines
12 KiB
Markdown
Raw Blame History

This file contains invisible Unicode characters

This file contains invisible Unicode characters that are indistinguishable to humans but may be processed differently by a computer. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

---
title: "Arnold conjecture"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Arnold_conjecture"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:02:43.869021+00:00"
instance: "kb-cron"
---
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
,
ω
)
{\displaystyle (M,\omega )}
be a closed (compact without boundary) symplectic manifold. For any smooth function
H
:
M
→
R
{\displaystyle H:M\to {\mathbb {R} }}
, the symplectic form
ω
{\displaystyle \omega }
induces a Hamiltonian vector field
X
H
{\displaystyle X_{H}}
on
M
{\displaystyle M}
defined by the formula
ω
(
X
H
,
⋅
)
=
d
H
.
{\displaystyle \omega (X_{H},\cdot )=dH.}
The function
H
{\displaystyle H}
is called a Hamiltonian function.
Suppose there is a smooth 1-parameter family of Hamiltonian functions
H
t
∈
C
∞
(
M
)
{\displaystyle H_{t}\in C^{\infty }(M)}
,
t
∈
[
0
,
1
]
{\displaystyle t\in [0,1]}
. This family induces a 1-parameter family of Hamiltonian vector fields
X
H
t
{\displaystyle X_{H_{t}}}
on
M
{\displaystyle M}
. The family of vector fields integrates to a 1-parameter family of diffeomorphisms
φ
t
:
M
→
M
{\displaystyle \varphi _{t}:M\to M}
. Each individual
φ
t
{\displaystyle \varphi _{t}}
is a called a Hamiltonian diffeomorphism of
M
{\displaystyle M}
.
The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of
M
{\displaystyle M}
is greater than or equal to the number of critical points of a smooth function on
M
{\displaystyle M}
.
== Weak Arnold conjecture ==
Let
(
M
,
ω
)
{\displaystyle (M,\omega )}
be a closed symplectic manifold. A Hamiltonian diffeomorphism
φ
:
M
→
M
{\displaystyle \varphi :M\to M}
is called nondegenerate if its graph intersects the diagonal of
M
×
M
{\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
M
{\displaystyle M}
, called the Morse number of
M
{\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
F
{\displaystyle {\mathbb {F} }}
, namely
∑
i
=
0
2
n
dim
⁡
H
i
(
M
;
F
)
{\textstyle \sum _{i=0}^{2n}\dim H_{i}(M;{\mathbb {F} })}
. The weak Arnold conjecture says that
#
{
fixed points of
φ
}
≥
∑
i
=
0
2
n
dim
⁡
H
i
(
M
;
F
)
{\displaystyle \#\{{\text{fixed points of }}\varphi \}\geq \sum _{i=0}^{2n}\dim H_{i}(M;{\mathbb {F} })}
for
φ
:
M
→
M
{\displaystyle \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
′
{\displaystyle L'}
in terms of the Betti numbers of
L
{\displaystyle L}
, given that
L
′
{\displaystyle L'}
intersects L transversally and
L
′
{\displaystyle L'}
is Hamiltonian isotopic to L.
Let
(
M
,
ω
)
{\displaystyle (M,\omega )}
be a compact
2
n
{\displaystyle 2n}
-dimensional symplectic manifold, let
L
⊂
M
{\displaystyle L\subset M}
be a compact Lagrangian submanifold of
M
{\displaystyle M}
, and let
τ
:
M
→
M
{\displaystyle \tau :M\to M}
be an anti-symplectic involution, that is, a diffeomorphism
τ
:
M
→
M
{\displaystyle \tau :M\to M}
such that
τ
∗
ω
=
−
ω
{\displaystyle \tau ^{*}\omega =-\omega }
and
τ
2
=
id
M
{\displaystyle \tau ^{2}={\text{id}}_{M}}
, whose fixed point set is
L
{\displaystyle L}
.
Let
H
t
∈
C
∞
(
M
)
{\displaystyle H_{t}\in C^{\infty }(M)}
,
t
∈
[
0
,
1
]
{\displaystyle t\in [0,1]}
be a smooth family of Hamiltonian functions on
M
{\displaystyle M}
. This family generates a 1-parameter family of diffeomorphisms
φ
t
:
M
→
M
{\displaystyle \varphi _{t}:M\to M}
by flowing along the Hamiltonian vector field associated to
H
t
{\displaystyle H_{t}}
. The Arnold–Givental conjecture states that if
φ
1
(
L
)
{\displaystyle \varphi _{1}(L)}
intersects transversely with
L
{\displaystyle L}
, then
#
(
φ
1
(
L
)
∩
L
)
≥
∑
i
=
0
n
dim
⁡
H
i
(
L
;
Z
/
2
Z
)
{\displaystyle \#(\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.
Alexander Givental proved it for
(
M
,
L
)
=
(
C
P
n
,
R
P
n
)
{\displaystyle (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
,
ω
)
{\displaystyle (M,\omega )}
semi-positive.
Urs Frauenfelder proved it in the case when
(
M
,
ω
)
{\displaystyle (M,\omega )}
is a certain symplectic reduction, using gauged Floer theory.
== See also ==
Symplectomorphism#Arnold conjecture
Floer homology
Spectral invariants
Conley–Zehnder theorem
== References ==
=== Citations ===
=== 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{{citation}}: CS1 maint: unflagged free DOI (link).
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
Reference in New Issue View Git Blame Copy Permalink