Geodesic
Representing Homology Classes on Surfaces
Canadian mathematical bulletin, Tome 19 (1976) no. 3, pp. 373-374

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Let T 2 = S 1×S 1, where S1 is the unit circle, and let {α, β} be the integral basis of H 1(T 2) induced by the 2 S 1-factors. It is well known that 0 ≠ X = pα + q β is represented by a simple closed curve (i.e. the homotopy class α p p q 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. 
Schafer, James A. Representing Homology Classes on Surfaces. Canadian mathematical bulletin, Tome 19 (1976) no. 3, pp. 373-374. doi: 10.4153/CMB-1976-058-4
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     author = {Schafer, James A.},
     title = {Representing {Homology} {Classes} on {Surfaces}},
     journal = {Canadian mathematical bulletin},
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     year = {1976},
     volume = {19},
     number = {3},
     doi = {10.4153/CMB-1976-058-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1976-058-4/}
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