Alternative Definitions of Three-dimensional Lens Spaces
The three dimensional lens space L(p,q) is often defined to be a solid ball with the following identification: first mark p equidistant points on the equator of the solid ball, denote them a0 to ap-1, then on the boundary of the ball, draw geodesic lines connecting the points to the north and south pole. Now identify spherical triangles by identifying the north pole to the south pole and the points ai with ai+q and ai+1 with ai+q+1. The resulting space is homeomorphic to the lens space .
Another related definition is to view the solid ball as the following solid bipyramid: construct a planar regular p sided polygon. Put two points n and s directly above and below the center of the polygon. Construct the bipyramid by joining each point of the regular p sided polygon to n and s. Fill in the bipyramid to make it solid and give the triangles on the boundary the same identification as above.
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