Lens space

A lens space is a particular type of 3-manifold obtained by gluing two solid tori along their boundaries via a homeomorphism of the torus determined by a pair of coprime integers (p,q). Introduced by Heinrich Tietze in 1908, lens spaces are the simplest class of 3-manifolds beyond the 3-sphere itself, and they provided the first examples of manifolds that are homotopy equivalent but not homeomorphic — proving that homology and fundamental group alone cannot distinguish 3-manifolds. Every lens space is a Seifert fibered space and admits spherical geometry, placing it among the non-hyperbolic cases in the geometrization program.