# Lens space

A manifold of odd dimension that arises as the orbit space of the isometric free action of a cyclic group $\mathbf Z _ {h}$ on the sphere $S ^ {2n-} 1$( cf. Action of a group on a manifold). It is convenient to take for $S ^ {2n-} 1$ the unit sphere in the complex space $\mathbf C ^ {n}$ in which a basis is fixed. Suppose that $\mathbf Z _ {h}$ acts on each coordinate $z _ {k}$ by multiplying it by $\zeta _ {k} = e ^ {2 \pi i m _ {k} / h }$, where $m _ {k}$ is invertible modulo $h$, that is, there are numbers $l _ {k}$ such that $m _ {k} l _ {k} \equiv 1$( $\mathop{\rm mod} h$). This specifies an isometric free (thanks to the condition that $m _ {k}$ is invertible $\mathop{\rm mod} h$) action of $\mathbf Z _ {h}$ on $S ^ {2n-} 1$, and any such action has this form described in a suitable coordinate system. The Reidemeister torsion corresponding to an $h$- th root of unity $\zeta$ is defined for a lens space $L = S ^ {2n-} 1 / \mathbf Z _ {h}$ constructed in this way by the formula $\pm \zeta ^ {q} \prod _ {k=} 1 ^ {n} ( \zeta ^ {l _ {k} } - 1 )$. Any piecewise-linear lens space $\overline{L}\;$ homeomorphic to it must have equal (up to $\pm \zeta ^ {q}$) torsion, and it turns out that the sets of numbers $\{ l _ {k} \}$ and $\{ \overline{l}\; _ {k} \}$ 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 $2n - 2$( that is, $\pi _ {i} L = 0$, $2 \leq i \leq 2 n - 2$), and the fundamental group is equal to $\mathbf Z _ {h}$ in view of the fact that the sphere $S ^ {2n-} 1$ is the universal covering for $\overline{L}\;$. The homology of $L$ coincides up to dimension $2 n - 2$ with the homology of the group $\mathbf Z _ {h}$, that is, it is equal to $\mathbf Z _ {h}$ in all dimensions from $2$ to $2 n - 2$ and $H _ {0} ( L) = H _ {2n-} 1 ( L) = \mathbf Z$. The direct limit of the spaces $L$ gives an Eilenberg–MacLane space of type $K ( \mathbf Z _ {h} , n )$. Two lens spaces are homotopy equivalent if and only if the linking coefficients (cf. Linking coefficient) $l ( a ^ {j} , a ^ {n-} j ) \in Q / \mathbf Z$ coincide, where $a$ 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 $\mathbf Z _ {h}$ on $S ^ {3}$ as a "lens" , i.e. the union of a spherical segment and its mirror image; this is how the name lens surface arose.