Genus:

The genus of a closed surface of Euler characteristic X(M) is given by (2-X(M))/2 if X(M) is even and (1-X(M))/2 if it is odd.

The genus of a surface counts how many "handles" the surface has. For orientable surfaces, this is the number of "holes" in the surface. For example, a torus has X(M) = 0, so the genus is 1; it has one hole and one handle.

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8/12/94 dpvc@geom.umn.edu -- The Geometry Center