A collection of one ore more closed curves on a surface is called a cycle or more generally a 1-cycle since it is 1-dimensional.

A cycle is trivial if it is the boundary curve of some region of the surface, and is non-trivial if it does not bound a region. For example, on a sphere, every curve bounds a region, so all cycles are trivial, but on a torus, a curve that goes around the hole does not bound a region, and so is non-trivial.

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8/12/94 -- The Geometry Center