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Representing Homology Classes on Surfaces

Published online by Cambridge University Press:  20 November 2018

James A. Schafer
James A. Schafer*
Affiliation:
University of MarylandCollege Park, Maryland
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Let T2 = S1×S1, where S1 is the unit circle, and let {α, β} be the integral basis of H1(T2) induced by the 2 S1-factors. It is well known that 0 ≠ X = pα + qβ is represented by a simple closed curve (i.e. the homotopy class αppq contains a simple closed curve) if and only if gcd(p, q) = 1. It is the purpose of this note to extend this theorem to oriented surfaces of genus g.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 3 , 01 September 1976 , pp. 373 - 374
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Magnus, W., Karrass, A., and Solitar, D., Combinatorial Group Theory, Interscience, New York, London, 1966.Google Scholar
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3. Zieschang, H., Algorithmen fur einfache Kurven auf Flâchen: Math. Scand., 17 (1965), 17–40.Google Scholar