# Lens space

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A manifold of odd dimension that arises as the orbit space of the isometric free action of a cyclic group on the sphere (cf. Action of a group on a manifold). It is convenient to take for the unit sphere in the complex space in which a basis is fixed. Suppose that acts on each coordinate by multiplying it by , where is invertible modulo , that is, there are numbers such that ( ). This specifies an isometric free (thanks to the condition that is invertible ) action of on , and any such action has this form described in a suitable coordinate system. The Reidemeister torsion corresponding to an -th root of unity is defined for a lens space constructed in this way by the formula . Any piecewise-linear lens space homeomorphic to it must have equal (up to ) torsion, and it turns out that the sets of numbers and must coincide. Thus, these sets characterize lens spaces uniquely up to a piecewise-linear homeomorphism and even up to an isometry; on the other hand, by the topological invariance of the torsion, they also characterize lens spaces uniquely up to a homeomorphism. A lens space is aspherical up to dimension (that is, , ), and the fundamental group is equal to in view of the fact that the sphere is the universal covering for . The homology of coincides up to dimension with the homology of the group , that is, it is equal to in all dimensions from to and . The direct limit of the spaces gives an Eilenberg–MacLane space of type . Two lens spaces are homotopy equivalent if and only if the linking coefficients (cf. Linking coefficient) coincide, where is a generator of the two-dimensional cohomology group. By means of these invariants one can establish the existence of asymmetric manifolds among lens spaces.
In the three-dimensional case lens spaces coincide with manifolds that have a Heegaard diagram of genus 1, and so they are Seifert manifolds (cf. Seifert manifold). It is convenient to represent the fundamental domain of the action of on as a "lens" , i.e. the union of a spherical segment and its mirror image; this is how the name lens surface arose.