- viz 1 (available in VR)
- viz 2 (available in VR)
Lens Space Definition and Fundamental Group
Lens spaces are a rich family of topological spaces that test our ability to distinguish spaces from one other using topological invariants.
As we discussed in Living on a Torus, one of the main tools for distinguishing spaces from one another is to look at the group of closed paths in that space, also known as the fundamental group. Since the fundamental group is the same for homeomorphic (i.e. topologically equivalent) spaces, we know that spaces with different fundamental groups must be topologically unequivalent. However, as our study of lens spaces will teach us, the converse is not true. Spaces which are topologically different do not necessarily have different fundamental groups.
First of all, what is a lens space? There are several equivalent definitions, but I'll use one that lends itself well to visualization. There is a lens space for each pair of integers \(p, q\) that are relatively prime. The lens space is then, the combination of two solid pyramids whose base is a polygon with \(p\) sides, as shown in the visualization above. The upper faces of the top pyramid are glued to the bottom faces of the bottom pyramid \(q\) faces over. I have indicated this gluing the in the visualization by making the glued faces the same colour. We can denote this lens space as \(L(p;q)\).
In the demonstration above, we visualize the fundamental group of lens space. Clicking Append Loop multiple times will build up a path by serially appending a loop to itself. The appended loop starts and ends at the chosen base point, which is indicated by two dodecahedrons. Note that despite the fact that there are two dodecahedrons, they represent the same point, connected by the red portal.
Once the loop is appended, the combined path is continuously rearranged so that it is a series of vertical segments followed by a diagonal one leading back to the base point. Notice that during the rearrangement, the gluing of the faces is always respected. That is, a rope passing through a face always reemerges at the corresponding point at the face it is glued to.
It may seem that we could continue appending loops indefinitely and get an infinite number of distinct closed paths. However, something remarkable happens when we append \(p\) loops to each other. Once rearranged, the loop is a series of \(p\) distinct vertical segments that connect each face of the upper and lower pyramids. (Now, we see why we demanded that \(p\) and \(q\) be relatively prime. If they weren't, the vertical segments would loop back to the base point before cycling across all of the triangular faces). We can push the \(p\) vertical segments to the boundary of the polygon base, while respecting the triangle gluings. The beginning and end of the path, however, stay anchored to the base point, as they must for a path in the fundamental group. Once the segments are pushed to the boundary polygon, the resuling path lies entirely in the red triangles. At this point, it is a simple matter to gather the path up into the base point. What this means is that after concatenating \(p\) loops together, we end up back where we started!
A succinct way of saying all this is that the fundamental group for \(L(p;q)\) is the cyclic group of order \(p\).
The first thing to notice is that the fundamental groups for two spaces with different values of \(p\) are different and therefore we can conclude right away that lens spaces with different value of \(p\) are not homeomorphic.
The second thing to notice is that the fundamental group for a given \(p\) does not depend on the choice of \(q\). However, it is not at all clear that all lens spaces of a given \(p\) are all homeomorphic to each other. As a matter of fact, they aren't. What that means is that the fundamental group is not sufficiently refined to distinguish all of the lens spaces from each other.
In order to classify all of the distinct lens spaces from each other, mathematicians invented a more refined invariant known as Reidemeister torsion. This is a typical example of how mathematical progress is made in the field of algebraic topology. Mathematicians construct a ever wider variety of topological spaces and try to classify them with known invariants. If existing variants are not up to the task, then new, more refined invariants are created to do the job.
Lens Space Identities
Some lens spaces with different values of \(q\) are homeomorphic. We show one way of proving such relationships in the visualization above, taken from Chapter 1 of Thurston's Three-Dimensional Geometry and Topology. We can cut up the lens space into \(p\) tetrahedra that share a central vertical axis. We then rearrange these tetrahedra by rotating the edges coming from the boundary polygon 90 degrees to form a new central axis. The ordering of the tetrahedra around the new central axis is determined by making sure that only faces that are glued to each other are placed adjacent to each other. The result is again a lens space with the same value of \(p\), but with a potentially different value of \(q\). From the demonstration above, we see for instance that \(L(7;2)\) is homeomorphic to \(L(7;4)\).
When pulling the tetrahedra apart we need to define new gluing maps for the newly created faces. To represent these new gluing maps, adjacent faces are given the same colour.