Lens space

The three-dimensional lens spaces ${\displaystyle L(p;q)}$ are quotients of ${\displaystyle S^{3}}$ by ${\displaystyle \mathbb {Z} /p}$-actions. More precisely, let ${\displaystyle p}$ and ${\displaystyle q}$ be coprime integers and consider ${\displaystyle S^{3}}$ as the unit sphere in ${\displaystyle \mathbb {C} ^{2}}$. Then the ${\displaystyle \mathbb {Z} /p}$-action on ${\displaystyle S^{3}}$ generated by the homeomorphism

${\displaystyle (z_{1},z_{2})\mapsto (e^{2\pi i/p}\cdot z_{1},e^{2\pi iq/p}\cdot z_{2})}$

is free. The resulting quotient space is called the lens space ${\displaystyle L(p;q)}$.

This can be generalized to higher dimensions as follows: Let ${\displaystyle p,q_{1},\ldots ,q_{n}}$ be integers such that the ${\displaystyle q_{i}}$ are coprime to ${\displaystyle p}$ and consider ${\displaystyle S^{2n-1}}$ as the unit sphere in ${\displaystyle \mathbb {C} ^{n}}$. The lens space ${\displaystyle L(p;q_{1},\ldots q_{n})}$ is the quotient of ${\displaystyle S^{2n-1}}$ by the free ${\displaystyle \mathbb {Z} /p}$-action generated by

${\displaystyle (z_{1},\ldots ,z_{n})\mapsto (e^{2\pi iq_{1}/p}\cdot z_{1},\ldots ,e^{2\pi iq_{n}/p}\cdot z_{n}).}$

In three dimensions we have ${\displaystyle L(p;q)=L(p;1,q).}$

The three dimensional lens space ${\displaystyle L(p;q)}$ 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 ${\displaystyle a_{0}}$ to ${\displaystyle a_{p-1}}$, 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 ${\displaystyle a_{i}}$ with ${\displaystyle a_{i+q}}$ and ${\displaystyle a_{i+1}}$ with ${\displaystyle a_{i+q+1}}$. The resulting space is homeomorphic to the lens space ${\displaystyle L(p;q)}$.

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.

Classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces ${\displaystyle L(p;q_{1})}$ and ${\displaystyle L(p;q_{2})}$ are:

1. homotopy equivalent if and only if ${\displaystyle q_{1}q_{2}\equiv \pm n^{2}{\pmod {p}}}$ for some ${\displaystyle n\in \mathbb {N} }$;
2. homeomorphic if and only if ${\displaystyle q_{1}\equiv \pm q_{2}^{\pm 1}{\pmod {p}}}$.

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 & Yasukhara 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 2005) 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.

• Spherical 3-manifold

• 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, arXiv:math/0401075, doi:10.1016/j.top.2004.11.002
• Herbert Seifert and William 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) (${\displaystyle \S }$20) English translation (2008) by John Stillwell.
• Matthew Watkins, "A Short Survey of Lens Spaces" (1990 undergraduate dissertation) Archived 2006-09-25 at the Wayback Machine

• Lens spaces at the Manifold Atlas
• Lens spaces: a history at the Manifold Atlas
• Fake lens spaces at the Manifold Atlas