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.

References

[1]	H. Poincaré, , Selected work , 2 , Moscow (1972) pp. 728 (In Russian)
[2]	G. de Rham, "Sur la théorie des intersections et les intégrales multiples" Comm. Math. Helv. , 4 (1932) pp. 151–154
[3]	H. Seifert, W. Threlfall, "A textbook of topology" , Acad. Press (1980) (Translated from German)
[4]	J.W. Milnor, O. Burlet, "Torsion et type simple d'homotopie" A. Haefliger (ed.) R. Narasimhan (ed.) , Essays on topology and related topics (Coll. Geneve, 1969) , Springer (1970) pp. 12–17