Lens Space - Alternative Definitions of Three-dimensional Lens Spaces

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 a_p-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 a_i with a_i+q and a_i+1 with a_i+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.