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A manifold of odd dimension that arises as the [[Orbit|orbit]] space of the isometric free action of a cyclic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l0581701.png" /> on the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l0581702.png" /> (cf. [[Action of a group on a manifold|Action of a group on a manifold]]). It is convenient to take for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l0581703.png" /> the unit sphere in the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l0581704.png" /> in which a basis is fixed. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l0581705.png" /> acts on each coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l0581706.png" /> by multiplying it by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l0581707.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l0581708.png" /> is invertible modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l0581709.png" />, that is, there are numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817012.png" />). This specifies an isometric free (thanks to the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817013.png" /> is invertible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817014.png" />) action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817016.png" />, and any such action has this form described in a suitable coordinate system. The [[Reidemeister torsion|Reidemeister torsion]] corresponding to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817017.png" />-th root of unity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817018.png" /> is defined for a lens space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817019.png" /> constructed in this way by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817020.png" />. Any piecewise-linear lens space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817021.png" /> homeomorphic to it must have equal (up to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817022.png" />) torsion, and it turns out that the sets of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817024.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817025.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817027.png" />), and the [[Fundamental group|fundamental group]] is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817028.png" /> in view of the fact that the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817029.png" /> is the [[Universal covering|universal covering]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817030.png" />. The homology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817031.png" /> coincides up to dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817032.png" /> with the homology of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817033.png" />, that is, it is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817034.png" /> in all dimensions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817035.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817037.png" />. The direct limit of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817038.png" /> gives an [[Eilenberg–MacLane space|Eilenberg–MacLane space]] of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817039.png" />. Two lens spaces are homotopy equivalent if and only if the linking coefficients (cf. [[Linking coefficient|Linking coefficient]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817040.png" /> coincide, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817041.png" /> 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.
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In the three-dimensional case lens spaces coincide with manifolds that have a [[Heegaard diagram|Heegaard diagram]] of genus 1, and so they are Seifert manifolds (cf. [[Seifert manifold|Seifert manifold]]). It is convenient to represent the fundamental domain of the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817042.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058170/l05817043.png" /> as a  "lens" , i.e. the union of a spherical segment and its mirror image; this is how the name lens surface arose.
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A manifold of odd dimension that arises as the [[Orbit|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|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|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|fundamental group]] is equal to  $  \mathbf Z _ {h} $
 +
in view of the fact that the sphere  $  S  ^ {2n-} 1 $
 +
is the [[Universal covering|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|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|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|Heegaard diagram]] of genus 1, and so they are Seifert manifolds (cf. [[Seifert manifold|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  , ''Selected work'' , '''2''' , Moscow  (1972)  pp. 728  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. de Rham,  "Sur la théorie des intersections et les intégrales multiples"  ''Comm. Math. Helv.'' , '''4'''  (1932)  pp. 151–154</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Seifert,  W. Threlfall,  "A textbook of topology" , Acad. Press  (1980)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Poincaré,  , ''Selected work'' , '''2''' , Moscow  (1972)  pp. 728  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G. de Rham,  "Sur la théorie des intersections et les intégrales multiples"  ''Comm. Math. Helv.'' , '''4'''  (1932)  pp. 151–154</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Seifert,  W. Threlfall,  "A textbook of topology" , Acad. Press  (1980)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR></table>

Revision as of 22:16, 5 June 2020


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
How to Cite This Entry:
Lens space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lens_space&oldid=13837
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article