Lens Space

A lens space is a topological space studied in mathematics; the term most often refers to a particular class of three dimensional manifolds, though it can be defined in higher dimensions as well. In the three dimensional case a lens space can be pictured as the result of gluing two solid tori together along their boundaries by a homeomorphism, with the three sphere and the product of a two sphere and a circle excluded as trivial special cases even though they can be built the same way. The three dimensional lens spaces, written L(p;q), were introduced by Heinrich Tietze in 1908 and were the first known examples of three manifolds not determined by their homology and fundamental group alone; in 1919 J. W. Alexander showed that two particular lens spaces, L(5;1) and L(5;2), are not homeomorphic even though they share the same fundamental group and homology, marking what can be seen as the birth of geometric topology as distinct from algebraic topology. Three dimensional lens spaces have since been completely classified by their fundamental group and by an invariant called Reidemeister torsion.