to , 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 with and with . 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. Classification of 3-dimensional lens spaces Classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces and are: homotopy equivalent if and only if for some ; homeomorphic if and only if . In this case they are "obviously" homeomorphic, as one can easily produce a homeomorphism. It is harder to show that these are the only homeomorphic lens spaces. The invariant that gives the homotopy classification of 3-dimensional lens spaces is the torsion linking form. The homeomorphism classification is more subtle, and is given by Reidemeister torsion. This was given in as a classification up to PL homeomorphism, but it was shown in to be a homeomorphism classification. In modern terms, lens spaces are determined by simple homotopy type, and there are no normal invariants (like characteristic classes) or surgery obstruction. A knot-theoretic classification is given in : let C be a closed curve in the lens space which lifts to a knot in the universal cover of the lens space. If the lifted knot has a trivial Alexander polynomial, compute the torsion linking form on the pair (C,C) – then this gives the homeomorphism classification. Another invariant is the homotopy type of the configuration spaces – showed that homotopy equivalent but not homeomorphic lens spaces may have configuration spaces with different homotopy types, which can be detected by different Massey products. See also Spherical 3-manifold References Glen Bredon, Topology and Geometry, Springer Graduate Texts in Mathematics 139, 1993. Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002. Allen Hatcher, Notes on basic 3-manifold topology. (Explains classification of L(p,q) up to homeomorphism.) Herbert Seifert and William Threlfall, A textbook of topology, Pure and