Lens Space - Classification of 3-dimensional Lens Spaces

Classification of 3-dimensional Lens Spaces

Classifications up to homeomorphism and homotopy equivalence are known, as follows. The three-dimensional spaces and are:

1. homotopy equivalent if and only if for some ;
2. 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.