Just before his death, Poincar\'e stated a `theorem' which had arisen in his studies on the celestial mechanics of the solar system: an area preserving self-diffeomorphism of the annulus which rotates the boundary circles in opposite directions leaves at least two points fixed. After many years, this result was proven by G. D. Birkhoff using very special arguments in two dimensions. In the 1960's, V. Arnold saw how to generalize the Poincar\'e-Birkhoff `Twist' Theorem in the light of modern symplectic geometry. He conjectured that the number of fixed points of a Hamiltonian symplectomorphism (i.e. the analogue of the annular twist) on a symplectic manifold is at least as great as the number of critical points for any function on the manifold. In recent times, as a consequence of Lusternik-Schnirelmann theory and the difficulty of the original conjecture, the Arnold conjecture was reformulated, replacing the number of critical points by the cuplength of the manifold. In this form, Floer proved the conjecture for manifolds M with $\pi_2(M) = 0$ by creating a magnificent new Morse-type analytic homology theory, Floer homology. This talk will focus on how more classical algebraic topological invariants may be used to prove the original Arnold conjecture.
This talk is intended for a general mathematical audience.