# Algebraic Topology and the Arnold Conjecture

### John Oprea

### Department of Mathematics, Cleveland State University

### 4:00 PM, October 30, 1997

### Room 102, Bradley Hall

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*.