148 D. SALAMON, FLOER HOMOLOGYwhere Hi(M, Q) denotes the singular homology of M with rational coefficients.
In contrast, the Lefschetz fixed point theorem only gives the alternating sum of
the Betti numbers as a lower bound. The Lefschetz fixed point theorem is related
to the Arnold conjecture in the same way as the Poincare-Hopf theorem (which
asserts that if a vector field has only nondegenerate zeros then the number of zeros
is bounded below by the Euler characteristic) is related to Morse theory (which
gives the sum of the Betti numbers as a lower bound for number of critical points
of a Morse function). In the special case where Ht = H is independent oft, the
Arnold conjecture is obvious. In this case all the critical points of H are constant
solutions of (1) and, in particular, are 1-periodic. Nondegeneracy of the 1-periodic
solutions implies that H is a Morse function, and hence the result follows from
Morse theory.
Exercise 1.2. Let x be a critical point of H =Ht and suppose that xis nondeg-
nerate as a 1-periodic solution of (1). Prove that x is nondegenerate as a critical
point of H. D
The Arnold conjecture (in the above form) has now been proved in full gen-erality. It was first confirmed by Eliashberg [6] for Riemann surfaces and then by
Conley and Zehnder [2] for the 2n-torus. In [18] Gromov proved the existence
of at least one fixed point under the assumption n2(M) = 0. The breakthrough
came when Floer established the Arnold conjecture for Lagrangian intersections,
and hence symplectic fixed points, again under the assumption n 2 (M) = 0. In a
series of papers [7, 8, 9, 10] Floer combined the variational approach of Conley
and Zehnder with the elliptic techniques of Gromov and the Morse-Smale-Witten
complex to develop his infinite dimensional approach to Morse theory which is now
called Floer homology. This work culminated in the paper [11], where Floer proved
the Arnold conjecture for monotone symplectic manifolds. Floer's proof was ex-
tended by Hofer-Salamon [19] and Ono [36] to the weakly monotone case, and
recently by Fukaya-Ono [14], Liu-Tian [28], and Hofer-Salamon [20, 21, 22, 23]
to the general case. Another proof was announced by Ruan [42].
Remark 1.3. There are many different forms of the Arnold conjecture. For exam-
ple, it can be formulated with any other coefficient ring (as long as it is a principal
ideal domain), or in the form that a lower bound for the number of periodic solu-
tions should (in the nondegenerate case) be the minimal number of critical points
of a Morse function. There are many examples of manifolds for which this number
is strictly larger than the sum of the Betti numbers (for any coefficient ring).
Another version of the Arnold conjecture gives a lower bound, without the
nondegeneracy condition, in terms of the Ljusternik-Schnirelman category, or again
in terms of the minimal number of critical points of any function (Morse or not) on
the manifold.
Yet another version of the Arnold conjecture refers to intersection points of
two Lagrangian submanifolds (which are related by a Hamiltonian isotopy) both
in the degenerate and nondegenerate case. There is quite a large literature on
this subject, with many partial solutions. Many question are still open, especially
concerning lower bounds which go beyond the sum of the Betti numbers, more
general coefficient rings, and Ljusternik-Schnirelman estimates for the degenerate
case. D