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 and
S2 x S1
The three-dimensional lens spaces
L(p;q)
L(5;1)
L(5;2)
L(7;1)
L(7;2)
There is a complete classification of three-dimensional lens spaces, by fundamental group and Reidemeister torsion.
The three-dimensional lens spaces
L(p;q)
S3
\Z/p
p
q
S3
\Complex2
Z/p
S3
(z1,z2)\mapsto(e2\pi ⋅ z1,e2\pi ⋅ z2)
L(p;q)
This can be generalized to higher dimensions as follows: Let
p,q1,\ldots,qn
qi
p
S2n-1
Cn
L(p;q1,\ldotsqn)
S2n-1
Z/p
(z1,\ldots,zn)\mapsto
2\piiq1/p | |
(e |
⋅ z1,\ldots,
2\piiqn/p | |
e |
⋅ zn).
L(p;q)=L(p;1,q).
The fundamental group of all the lens spaces
L(p;q1,\ldots,qn)
\Z/p\Z
qi
Lens spaces are locally symmetric spaces, but not (fully) symmetric, with the exception of
L(2;1)
The three dimensional lens space
L(p;q)
a0
ap-1
ai
ai+q
ai+1
ai+q+1
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
L(p;q1)
L(p;q2)
q1q2\equiv\pmn2\pmod{p}
n\inN
q1\equiv\pm
\pm1 | |
q | |
2 |
\pmod{p}
q1\equiv\pm
\pm1 | |
q | |
2 |
\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, 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 spaces – 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.
\S