Lens space
A lens space is an example of a topological space, considered in mathematics. The term often refers to a
specific class of 3-manifolds, but in general can be defined for higher dimensions.
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.
There is a complete classification of three-dimensional lens spaces, by fundamental group and Reidemeister
torsion.
Contents
Definition
Properties
Alternative definitions of three-dimensional lens spaces
Classification of 3-dimensional lens spaces
See also
References
External links
Definition
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
coprime to and consider
as the unit sphere in
. The lens space
by the free
-action generated by
be integers such that the
are
is the quotient of
In three dimensions we have
Properties
The fundamental group of all the lens spaces
is
independent of the
.
Lens spaces are locally symmetric spaces, but not (fully) symmetric, with the exception of
which is
symmetric. (Locally symmetric spaces are symmetric spaces that are quotiented by an isometry that has no
fixed points; lens spaces meet this definition.)
Alternative definitions of three-dimensional lens spaces
The three dimensional lens space
is often defined to be a solid ball with the following identification:
first mark p equally spaced points on the equator of the solid ball, denote them
to
, 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
with
and
with
. 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 threedimensional spaces
and
are:
1. homotopy equivalent if and only if
2. homeomorphic if and only if
for some
;
.
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.
See also
Spherical 3-manifold
References
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 (https://doi.org/10.2307%2F1969884),
JSTOR 1969884 (https://www.jstor.org/stable/1969884)
Allen Hatcher, Algebraic Topology (http://www.math.cornell.edu/~hatcher/AT/ATpage.html),
Cambridge University Press, 2002.
Allen Hatcher, Notes on basic 3-manifold topology (http://www.math.cornell.edu/~hatcher/3M/3
Mdownloads.html). (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 (https://doi.org/10.1023%2
FA%3A10240), MR 1988423 (https://www.ams.org/mathscinet-getitem?mr=1988423)
Reidemeister, Kurt (1935), "Homotopieringe und Linsenräume", Abh. Math. Sem. Univ.
Hamburg, 11 (1): 102–109, doi:10.1007/BF02940717 (https://doi.org/10.1007%2FBF02940717)
Salvatore, Paolo; Longoni, Riccardo (2005), "Configuration spaces are not homotopy
invariant", Topology, 44 (2): 375–380, arXiv:math/0401075 (https://arxiv.org/abs/math/0401075),
doi:10.1016/j.top.2004.11.002 (https://doi.org/10.1016%2Fj.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 (ht
tp://www.maths.ed.ac.uk/~aar/haupt/tietze.pdf), Monatsh. fuer Math. und Phys. 19, 1–118 (1908)
(§ 20) English translation (http://www.maths.ed.ac.uk/~aar/haupt/Tietze1908.pdf) (2008) by John
Stillwell.
Matthew Watkins, "A Short Survey of Lens Spaces" (http://www.maths.ex.ac.uk/~mwatkins/lens
spaces.pdf) (1990 undergraduate dissertation)
External links
Lens spaces (http://www.map.mpim-bonn.mpg.de/Lens_spaces) at the Manifold Atlas
Lens spaces: a history (http://www.map.mpim-bonn.mpg.de/Lens_spaces:_a_history) at the
Manifold Atlas
Fake lens spaces (http://www.map.mpim-bonn.mpg.de/Fake_lens_spaces) at the Manifold
Atlas
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