Vol. 16, No. 2, 1966

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On the regions bounded by homotopic curves

Albert Marden, I. Richards and Burton Rodin

Vol. 16 (1966), No. 2, 337–339

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

Let 𝒜 and 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 are freely homotopic but not homotopic to zero then 𝒜∪ℬ bounds a closed cylinder in S.

Mathematical Subject Classification
Primary: 55.60
Published: 1 February 1966
Albert Marden
School of Mathematics
127 Vincent Hall, 206 Church St. S.E
Minneapolis MN 55455
United States
I. Richards
Burton Rodin