Free lines for homeomorphisms of the open annulus

Abstract

Let H H be a homeomorphism of the open annulus S 1 × R S^1 \times \textbf {R} isotopic to the identity and let h h be a lift of H H to the universal cover R × R \textbf {R} \times \textbf {R} without fixed point. Then we show that h h admits a Brouwer line which is a lift of a properly imbedded line joining one end to the other in the annulus or H H admits a free essential simple closed curve.