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Wikipedia / 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, both of which can be obtained as above, are not counted as they are considered trivial special cases.

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) 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.

Definition

be coprime as the unit sphere generated by the homeomorphism

is free. The resulting quotient space is called the lens space.

-action generated by

Properties

.

is given by

Lens spaces are locally symmetric spaces, three-dimensional real projective space, 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

is often defined to be a solid ball with the following identification: first mark p.

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

are:

  1. ;

  2. .

as in case 2., 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 as a classification up to PL homeomorphism, but it was shown in 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 : 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 spacess – 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

References

Notes

Bibliography