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In the 3-manifold case, a lens space can be visualized as the result of gluing two solid tori together by a homeomorphism of their boundaries. Often the 3-sphere and , both of which can be obtained as above, are not counted as they are considered trivial special cases.
The three-dimensional lens spaces were introduced by Heinrich Tietze in 1908. They were the first known examples of 3-manifolds which were not determined by their homology and fundamental group alone, and the simplest examples of closed manifolds whose homeomorphism type is not determined by their homotopy type. J. W. Alexander in 1919 showed that the lens spaces and 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 three-dimensional lens spaces are quotients of by -actions. More precisely, let and be coprime integers and consider as the unit sphere in . Then the -action on generated by the homeomorphism
is free. The resulting quotient space is called the lens space .
This can be generalized to higher dimensions as follows: Let be integers such that the are coprime to and consider as the unit sphere in . The lens space is the quotient of by the free -action generated by
In three dimensions we have
The fundamental group of all the lens spaces is independent of the .
Alternative definitions of three-dimensional lens spaces
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 a0 to ap-1, 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 ai with ai+q and ai+1 with ai+q+1. The resulting space is homeomorphic to the lens space .
Another related definition is to view the solid ball as the following solid bipyramid: construct a planar regular p sided 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.
Classification of 3-dimensional lens spaces
Classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces and are:
- homotopy equivalent if and only if for some ;
- homeomorphic if and only if .
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.
The invariant that gives the homotopy classification of 3-dimensional lens spaces is the torsion linking form.
The homeomorphism classification is more subtle, and is given by Reidemeister torsion. This was given in (Reidemeister 1935) as a classification up to PL homeomorphism, but it was shown in (Brody 1960) to be a homeomorphism classification. In modern terms, lens spaces are determined by simple homotopy type, and there are no normal invariants (like characteristic classes) or surgery obstruction.
A knot-theoretic classification is given in (Przytycki & Yasuhara 2003): 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.
Another invariant is the homotopy type of the configuration spaces – (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 products.
- Glen Bredon, Topology and Geometry, Springer Graduate Texts in Mathematics 139, 1993.
- Brody, E. J. (1960), "The topological classification of the lens spaces", Annals of Mathematics, 2, 71 (1): 163–184, doi:10.2307/1969884, JSTOR 1969884
- Allen Hatcher, Algebraic Topology, Cambridge University Press, 2002.
- Allen Hatcher, Notes on basic 3-manifold topology. (Explains classification of L(p,q) up to homeomorphism.)
- Przytycki, Józef H.; Yasukhara, Akira (2003), "Symmetry of Links and Classification of Lens Spaces", Geometriae Dedicata, 98 (1): 57–61, doi:10.1023/A:10240, MR 1988423
- Reidemeister, Kurt (1935), "Homotopieringe und Linsenräume", Abh. Math. Sem. Univ. Hamburg, 11 (1): 102–109, doi:10.1007/BF02940717
- Salvatore, Paolo; Longoni, Riccardo (2005), "Configuration spaces are not homotopy invariant", Topology, 44 (2): 375–380, doi:10.1016/j.top.2004.11.002
- 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)
- Heinrich Tietze, Ueber die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatsh. fuer Math. und Phys. 19, 1–118 (1908) (20) English translation (2008) by John Stillwell.
- Matthew Watkins, "A Short Survey of Lens Spaces" (1990 undergraduate dissertation)