I am trying to understand the Arnold conjecture in Symplectic Geometry, which basically tells us the following: If $M$ is a compact symplectic manifold and $H_t$ be a 1-periodic Hamiltonian function, then we can consider the Hamiltonian equation of motion which defines us a family $\psi_t$ of symplectomorphisms of $M$. We then consider the fixed points of $\psi_1$ and call a fixed point $x$ non-degenerate, if $\det(1- d\psi_1(x)) \neq 0$. In the case that all the fied points are non-degenerate the Arnold conjecture then is: If every fixed point of $\psi_1$ is non-degenerate, then the number of fixed points is at least the sum off all Betti numbers of $M$. \ I would now like to know the answer to the following questions: \ 1. Why is such a result helpful for our understanding of Symplectic Geometry? Why would somebody like to know whether such a conjecture is true or not? \ß 2. Why could this result be true? Can you maybe give me an explanation or reference, why Vladimir Arnold conjectured this result? \ Every answer to the above questions would be appreciated.
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3$\begingroup$ There is a book about Arnold's problems over the years and to the best of my memory there is a discussion there of this conjecture. $\endgroup$– Gil KalaiCommented Dec 24, 2012 at 22:24
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2$\begingroup$ I think the best answer you're gonna get is Arnold's actual paper on the conjecture. $\endgroup$– Chris GerigCommented Dec 24, 2012 at 23:38
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$\begingroup$ @Gil Kalai: Thank you very much. Do you know how the book is called. $\endgroup$– nicolasCommented Dec 25, 2012 at 0:04
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1$\begingroup$ The name of the book is Arnold's problems, the author is Arnold. $\endgroup$– user9072Commented Dec 25, 2012 at 15:36
4 Answers
Here is a trivial example that I read from a survey article written by Arnold in the late 80s.
Consider $T^*S^1$, the cotangent bundle of $S^1$ which we can identify with the product $\newcommand{\bR}{\mathbb{R}}$ $S^1\times\bR$. I will denote the obvious coordinates on this cylinder by $(\theta, t)$.
Like any cotangent bundle, $T^*S^1$ carries a symplectic structure, and in this case, any curve on this symplectic manifold is a lagrangian submanifold. However, there are curves, and there are curves.
Take for example the curves $C_\tau:=\lbrace t=\tau\rbrace$, $\tau$ a nonzero constant, which are disjoint from the zero section and are deformations of the zero section via the symplectic flow
$$ (\theta,t)\mapsto \Phi_\tau(\theta,t)=(\theta,t+\tau). $$
Consider next a smooth function
$$ S^1\ni\theta\mapsto f(\theta). $$
Its differential is a section of $T^*S^1$, and its graph $\Gamma_{df}=(\theta,f'(\theta))$ intersects the zero section along the critical points of $f$.
The lagrangian $\Gamma_{df}$ is a rather special deformation of the zero section: it is a Hamiltonian deformation, the points of intersection of $\Gamma_{df}|$ correspond to the periodic orbits of the Hamiltonian deformation.
Why is this fascinating? Certain pairs of lagrangian subspaces intersect in more points than predicted by topology alone, which is in itself an indication that symplectic topology is a bit more rigid than smooth topology alone.
How does the above trivial example fit the general picture?
A lagrangian submanifold $L$ of a symplectic manifold has a tubular neighborhood symplectomorphic to $T^* L$. Thus the case of cotangent bundles can be viewed as local situations of the more general cases. of lagrangian submanifolds and their hamiltonian perturbations.
Given a Hamiltonian flow $\Phi_t$ on a symplectic manifold $X$, the graph of the time $1$-map is a lagrangian submanifold in $X\times X$. Its fixed points correspond to the intersection of the graph with the diagonal in $X\times X$, which is another lagrangian submanifold. Thus the problem of intersection of lagrangian submanifolds contains as a special case the problem of existence of periodic solutions of hamiltonian systems.
Leaving aside the mysterious rigidity of symplectic topology alluded to above, the problem of existence of periodic orbits of hamiltonian systems has fascinated many classics, such as Poincare, because of it's obvious connection to the many body problem and the philosophical question: does the history of our planetary system repeat itself?
The Arnold conjecture is interesting to symplectic geometers because a lot of new math has resulted from people trying to prove it. I think the most notable of which is that Hamiltonian Floer homology was developed as a means to obtaining a proof of the conjecture. I would also argue (only on the grounds of ideology) that it is interesting that the topology of a symplectic manifold constrains the possible evolution of any dynamical system defined on it.
The motivation behind why the result could be true is that the Arnold conjecture closely resembles the Morse inequalities. Floer homology degenerates to Morse homology in the proper limit, and so really the Arnold conjecture can be thought of as a generalization of the Morse inequalities.
If you can read French, a great reference is Audin and Damian's book, "Theorie de Morse et homologie de Floer". Alternatively, Salamon has some notes which are very thorough
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1$\begingroup$ +1 for the link to Salamon's notes $\endgroup$ Commented Dec 25, 2012 at 10:15
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$\begingroup$ There is also an English translation of Audin and Damian's book. $\endgroup$– dvitekCommented Aug 27, 2015 at 13:12
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1$\begingroup$ Yes, the English translation exists now, but didn't when I originally wrote this answer! I highly recommend it. $\endgroup$ Commented Oct 26, 2015 at 21:09
In a certain sense, symplectic geometry (or safer to say symplectic topology) as we know it now was not existing before Arnold formulated these conjectures. So many would say that Arnold conjectures gave birth to symplectic geometry. At the time Arnold made this conjecture only one non-trivial statement in this directions was known - Poincaré's last geometric theorem http://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Birkhoff_theorem
The conjecture is listed as problem 1972-33 in the book Vladimir Arnold: Arnold's Problems, and in the Comments-section of this book you can you can find plenty of background information on it in a contribution by Mikhail B Sevryuk.
