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.
Contents
- Definition
- Alternative definitions of three dimensional lens spaces
- Classification of 3 dimensional lens spaces
- References
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
The three-dimensional lens spaces
There is a complete classification of three-dimensional lens spaces, by fundamental group and Reidemeister torsion.
Definition
The three-dimensional lens spaces
is free. The resulting quotient space is called the lens space
This can be generalized to higher dimensions as follows: Let
In three dimensions we have
The fundamental group of all the lens spaces
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
- homotopy equivalent if and only if
q 1 q 2 ≡ ± n 2 ( mod p ) for somen ∈ N ; - homeomorphic if and only if
q 1 ≡ ± q 2 ± 1 ( mod 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 & 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.