A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions. In the 3-manifold case, a lens space can be visualized as the result of gluing two solid tori together by a homeomorphism of their boundaries. Often the 3-sphere and , both of which can be obtained as above, are not counted as they are considered trivial special cases. The three-dimensional lens spaces were introduced by Heinrich Tietze in 1908. They were the first known examples of 3-manifolds which were not determined by their homology and fundamental group alone, and the simplest examples of closed manifolds whose homeomorphism type is not determined by their homotopy type. J. W. Alexander in 1919 showed that the lens spaces and were not homeomorphic even though they have isomorphic fundamental groups and the same homology, though they do not have the same homotopy type. Other lens spaces (such as and ) have even the same homotopy type (and thus isomorphic fundamental groups and homology), but not the same homeomorphism type; they can thus be seen as the birth of geometric topology of manifolds as distinct from algebraic topology. There is a complete classification of three-dimensional lens spaces, by fundamental group and Reidemeister torsion. Definition The three-dimensional lens spaces are quotients of by -actions. More precisely, let and be coprime integers and consider as the unit sphere in . Then the -action on generated by the homeomorphism is free. The resulting quotient space is called the lens space . This can be generalized to higher dimensions as follows: Let be integers such that the are coprime to and consider as the unit sphere in . The lens space is the quotient of by the free -action generated by In three dimensions we have Properties The fundamental group of all the lens spaces is independent of the . Lens spaces are locally symmetric spaces, but not (fully) symmetric, with the exception of which is symmetric. (Locally symmetric spaces are symmetric spaces that are quotiented by an isometry that has no fixed points; lens spaces meet this definition.) Alternative definitions of three-dimensional lens spaces The three dimensional lens space is often defined to be a solid ball with the following identification: first mark p equally spaced points on the equator of the solid ball, denote them