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| {{no footnotes|date=February 2012}}
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| A '''lens space''' is an example of a [[topological space]], considered in [[mathematics]]. The term often refers to a specific class of [[3-manifold]]s, but in general can be defined for higher dimensions.
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| In the 3-manifold case, a lens space can be visualized as the result of gluing two [[solid torus|solid tori]] together by a homeomorphism of their boundaries. Often the [[3-sphere]] and <math>S^2 \times S^1</math>, both of which can be obtained as above, are not counted as they are considered trivial special cases.
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| The three-dimensional lens spaces <math>L(p,q)</math> were introduced by [[Heinrich Tietze|Tietze]] in 1908. They were the first known examples of 3-manifolds which were not determined by their [[homology (mathematics)|homology]] and [[fundamental group]] alone, and the simplest examples of closed manifolds whose homeomorphism type is not determined by their homotopy type. [[James_Waddell_Alexander_II|J.W. Alexander]] in 1919 showed that the lens spaces <math>L(5;1)</math> and <math>L(5;2)</math> 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 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]]. | | The entire game happens around an artifact of three triangles called 'Triforce'. A sleeper hit by many definitions, Metro 2033 charts the tumultuous existence of humans living underground, after a nuclear holocaust wiped away the topside earth. Battle Cards is a fun card game and the animations add to the game. To start your series of running stick figure action games run click here. At this time there exists only a short pitch reel comprised of clips from Hannibal and other TV shows and films, which serves to illustrate the proposed show's general tone. |
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| There is a complete classification of three-dimensional lens spaces, by [[fundamental group]] and [[Reidemeister torsion]].
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| == Definition ==
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| The three-dimensional lens spaces <math>L(p;q)</math> are quotients of <math>S^3</math> by <math>\mathbb{Z}/p</math>-actions. More precisely, let <math>p</math> and <math>q</math> be [[coprime]] integers and consider <math>S^3 </math> as the unit sphere in <math> \mathbb C^2</math>. Then the <math> \mathbb{Z}/p</math>-action on <math>S^3</math> generated by
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| :<math>[1].(z_1,z_2):=(e^{2\pi i/p} \cdot z_1, e^{2\pi i q/p}\cdot z_2)</math>
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| is free as p and q were coprime. The resulting [[Group action#Orbits and stabilizers|quotient space]] is called the '''lens space''' <math>L(p;q)</math>.
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| This can be generalized to higher dimensions as follows: Let <math>p,q_1,\ldots,q_n</math> be integers such that the <math>q_i</math> are coprime to <math>p</math> and consider <math>S^{2n-1}</math> as the unit sphere in <math> \mathbb C^n</math>. The lens space <math>L(p;q_1,\ldots q_n)</math> is the quotient of <math>S^{2n-1}</math> by the free <math>\mathbb Z/p</math>-action generated by
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| :<math>[1].(z_1,\ldots,z_n):=(e^{2\pi iq_1/p} \cdot z_1,\ldots, e^{2\pi i q_n/p}\cdot z_n).</math>
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| In three dimensions we have <math>L(p;q)=L(p;1,q).</math>
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| The fundamental group of all the lens spaces <math>L(p;q_1,\ldots, q_n)</math> is <math>\mathbb Z/p</math> independent of the <math>q_i</math>.
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| == Alternative definitions of three-dimensional lens spaces ==
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| The three dimensional lens space L(p,q) is often defined to be a solid ball with the following identification: first mark ''p'' equidistant points on the equator of the solid ball, denote them ''a<sub>0</sub>'' to ''a<sub>p-1</sub>'', then on the boundary of the ball, draw geodesic lines connecting the points to the north and south pole. Now identify spherical triangles by identifying the north pole to the south pole and the points ''a<sub>i</sub>'' with ''a<sub>i+q</sub>'' and ''a<sub>i+1</sub>'' with ''a<sub>i+q+1</sub>''. The resulting space is homeomorphic to the lens space <math>L(p,q)</math>.
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| Another related definition is to view the solid ball as the following solid [[bipyramid]]: construct a planar regular ''p'' sided [[regular polygon|polygon]]. Put two points ''n'' and ''s'' directly above and below the center of the polygon. Construct the bipyramid by joining each point of the regular ''p'' sided polygon to ''n'' and ''s''. Fill in the bipyramid to make it solid and give the triangles on the boundary the same identification as above.
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| == Classification of 3-dimensional lens spaces ==
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| Classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces <math>L(p;q_1)</math> and
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| <math>L(p;q_2)</math> are:
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| #homotopy equivalent if and only if <math>q_1 q_2 \equiv \pm n^2 \pmod{p}</math> for some <math>n \in \mathbb{N}</math>;
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| #homeomorphic if and only if <math>q_1 \equiv \pm q_2^{\pm 1} \pmod{p}</math>.
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| #:In this case they are "obviously" homeomorphic, as one can easily produce a homeomorphism. It is harder to show that these are the only homeomorphic lens spaces.
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| The invariant that gives the homotopy classification of 3-dimensional lens spaces is the [[torsion linking form]].
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| The homeomorphism classification is more subtle, and is given by [[Reidemeister torsion]]. This was given in {{harv|Reidemeister|1935}} as a classification up to [[PL homeomorphism]], but it was shown in {{harv|Brody|1960}} to be a homeomorphism classification. In modern terms, lens spaces are determined by [[simple homotopy|''simple'' homotopy]] type, and there are no normal invariants (like [[characteristic classes]]) or [[surgery obstruction]].
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| A [[knot theory|knot-theoretic]] classification is given in {{harv|Przytycki|Yasuhara|2003}}:
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| let ''C'' be a closed curve in the lens space which lifts to a knot in the universal cover of the lens space. If the lifted knot has a trivial [[Alexander polynomial]], compute the torsion linking form on the pair (C,C) – then this gives the homeomorphism classification.
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| Another invariant is the homotopy type of the [[configuration space]]s – {{harv|Salvatore|Longoni|2004}} showed that homotopy equivalent but not homeomorphic lens spaces may have configuration spaces with different homotopy types, which can be detected by different [[Massey product]]s.
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| == See also ==
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| * [[Spherical 3-manifold]]
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| == References ==
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| * G. Bredon, ''Topology and Geometry'', Springer Graduate Texts in Mathematics 139, 1993.
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| * {{citation| doi=10.2307/1969884| first=E. J. |last=Brody |title=The topological classification of the lens spaces|journal=[[Annals of Mathematics]] | series=2 | volume=71| issue=1 | year=1960 | pages=163–184| jstor=1969884}}
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| * A. Hatcher, ''[http://www.math.cornell.edu/~hatcher/AT/ATpage.html Algebraic Topology],'' Cambridge University Press, 2002.
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| * A. Hatcher, ''[http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html Notes on basic 3-manifold topology],''
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| *:(explains classification of ''L(p,q)'' up to homeomorphism)
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| * {{citation|last1=Przytycki|authorlink=Józef H. Przytycki|last2=Yasuhara|title=Symmetry of Links and Classification of Lens Spaces|journal=Geom. Ded.|volume=98|issue=1|year=2003}}
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| *{{citation|first=Kurt |last=Reidemeister|authorlink=Kurt Reidemeister|title=Homotopieringe und Linsenräume|journal= Abh. Math. Sem. Univ. Hamburg |volume= 11 |year=1935|issue=1|pages= 102–109|doi=10.1007/BF02940717}}
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| * {{citation|last1= Salvatore| first1=Paolo| last2=Longoni| first2=Riccardo| title=Configuration spaces are not homotopy invariant| journal= Topology |volume= 44| year=2005|issue= 2|pages=375–380|doi= 10.1016/j.top.2004.11.002}}
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| * H. Seifert and W. Threlfall, ''A textbook of topology'' Pure and Applied Mathematics 89, Translated from the German edition of 1934, Academic Press Inc. New York (1980)
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| * H. Tietze, ''[http://www.maths.ed.ac.uk/~aar/haupt/tietze.pdf Ueber die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten],'' Monatsh. fuer Math. und Phys. 19, 1–118 (1908) (<math>\S</math>20) [http://www.maths.ed.ac.uk/~aar/haupt/Tietze1908.pdf English translation] (2008) by John Stillwell.
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| * M. Watkins, [http://www.maths.ex.ac.uk/~mwatkins/lensspaces.pdf "A Short Survey of Lens Spaces"] (1990 undergraduate dissertation)
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| == External links ==
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| *[http://www.map.mpim-bonn.mpg.de/Lens_spaces Lens spaces] at the Manifold Atlas
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| *[http://www.map.mpim-bonn.mpg.de/Lens_spaces:_a_history Lens spaces: a history] at the Manifold Atlas
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| *[http://www.map.mpim-bonn.mpg.de/Fake_lens_spaces Fake lens spaces] at the Manifold Atlas
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| [[Category:3-manifolds]]
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| [[Category:Manifolds]]
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