Lens space

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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-manifolds, but in general can be defined for higher dimensions.

In the 3-manifold case, a picturesque description of a lens space is that of a space resulting from gluing two solid tori together by a homeomorphism of their boundaries. Of course, to be consistent, we should exclude the 3-sphere and S^2 \times S^1, both of which can be obtained as just described; some mathematicians include these two manifolds in the class of lens spaces.

The three-dimensional lens spaces L(p,q) were introduced by Tietze in 1908. They were the first known examples of 3-manifolds which were not determined by their homology and fundamental group alone. J.W. Alexander in 1919 showed that the lens spaces L(5;1) and L(5;2) were not homeomorphic even though they have isomorphic fundamental groups and the same homology.

There is a complete classification of three-dimensional lens spaces.

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[edit] Definition

Sit the 2n − 1-sphere S2n − 1 inside \mathbb C^n as the set of all n-tuples of unit absolute value. Let ω be a primitive pth root of unity and let q_1,\ldots,q_n be integers coprime to p. Let the set of powers \zeta_p=\{1,\omega,\ldots,\omega^{p-1}\} act on the sphere by

\omega\cdot(z_1,\ldots,z_n)=(\omega^{q_1}z_1,\ldots,\omega^{q_n}z_n).

The resulting orbit space is a lens space, written as L(p;q_1,\ldots,q_n).

We can also define the infinite-dimensional lens spaces as follows. These are the spaces L(p;q_1,q_2,\ldots) formed from the union of the increasing sequence of spaces L(p;q_1,\ldots,q_n) for n=1,2,\ldots. As before, the q_1,q_2,\ldots must be coprime to p.

[edit] Three-dimensional lens spaces

Three-dimensional lens spaces arise as quotients of S^3 \subset \mathbb{C}^2 by the action of the group that is generated by elements of the form \begin{pmatrix}\omega&0\\0&\omega^q\end{pmatrix}.

Such a lens space L(p;q) has fundamental group \mathbb{Z}/p\mathbb{Z} for all q, so spaces with different p are not homotopy equivalent.

[edit] Alternative Definitions of Three-dimensional Lens Spaces

Often 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 L(p,q).

Another related definition is to view the solid ball as the following solid polyhedron: construct a planar regular p sided polygon. Put two points n and s directly above and below the center of the polygon. Construct a polyhedron by joining each point of the regular p sided polygon to n and s. Fill in the polygon to make it solid and give the triangles on the boundary the same identification as above.

[edit] Classification of 3-dimensional lens spaces

Classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces L(p;q1) and L(p;q2) are:

  1. homotopy equivalent if and only if q_1 q_2 \equiv \pm n^2 \pmod{p} for some n \in \mathbb{N};
  2. homeomorphic if and only if q_1 \equiv \pm q_2^{\pm1} \pmod{p}.

The invariant that gives the homotopy classification of 3-dimensional lens spaces is the torsion linking form. The homeomorphism classification is more subtle: 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). Przytycki and Yasuhara have shown this gives the homeomorphism classification of lens spaces.

[edit] See also

[edit] References

  • G. Bredon, Topology and Geometry, Springer Graduate Texts in Mathematics 139, 1993.
  • A. Hatcher, Algebraic Topology available online, Cambridge University Press, 2002.
  • A. Hatcher, Notes on basic 3-manifold topology, available online (explains classification of L(p,q) up to homeomorphism)
  • H. Tietze, Ueber die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten [1], Monatsh. fuer Math. und Phys. 19, 1-118 (1908) (\S20)
  • M. Watkins, "A Short Survey of Lens Spaces" (1990 undergraduate dissertation)
  • Przytycki, Yasuhara. Symmetry of Links and Classification of Lens Spaces. Geom. Ded. Vol 98. No. 1. (2003)