On the regions bounded by homotopic curves

Albert Marden, I. Richards and Burton Rodin

Vol. 16 (1966), No. 2, 337–339

DOI: 10.2140/pjm.1966.16.337

Abstract

We give a short proof of the following theorem of H. I. Levine [2]:

Let 𝒜 and ℬ be nonintersecting simple closed curves on an orientable surface S (compact or not).

(i) If 𝒜 is homotopic to zero, then 𝒜 bounds a closed disk in S.

(ii) If 𝒜 and ℬ are freely homotopic but not homotopic to zero then 𝒜∪ℬ bounds a closed cylinder in S.

Mathematical Subject Classification

Primary: 55.60

Milestones

Published: 1 February 1966