Less formally, it is a surface in the shape of a bagel or doughnut. It can be made from a rectangular piece of plane by identifying the top and bottom edges (i.e., rolling it into a tube) and then identifying the left and right edges (that is, bending the tube around so that the ends meet).
An n-handled torus is a sphere with n handles; it has Euler characteristic equal to 2 - 2 n.
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The Geometry Center