The Lens space L(p,q) is the quotient of S^3 by an action of \mathbb{Z}_p. The group acts on \mathbb{C}^2 by the p^{th} root of unity in the first coordinate and the q^{th} power of that in the second coordinate, and the action restricts to S^3. So why a "Lens" space, and how does one visualize it?

 I drew L(5,2) by drawing a lattice in S^3 invariant under the group action. Here's the picture (after stereographic projection):

 The spheres in the picture are images of great spheres on S^3. The region between any two adjacent spheres is a fundamental region: every point on the interior corresponds to exactly one point in the quotient, while the boundary has some identifications. This lens space can then be thought of as the region between, say, the red and yellow spheres, where the red boundary is identified with the yellow boundary. This identification is given by a rotation that matches up the missing wedges.

So why is it called a lens space? Because the region between the green and red spheres looks like a lens. Here's the side view:

To get a better feeling for the transformation, here's an animation showing one iteration of the map. More precisely, the group action is by (e^{\frac{2\pi i}{5}},e^{\frac{4\pi i}{5}}), and the video shows the picture under the transformation (e^{\frac{2\pi i t}{5}},e^{\frac{4\pi i t}{5}}), where t ranges from 0 to 1.

 Note: Please right-click and turn off automatic loop, and right click -> play to repeat. It makes more sense that way.

 

Note that each sphere is the image of the red one under the group action, including the location of the wedge. In the animation, it looks like the picture repeats itself several times, but look at where the wedge is. The animation can't stop until the wedges line up properly.

A technical note: the picture was constructed by looking at the Dirichlet region for the origin (the point (0,-i) in S^3. That is, I looked at the images of the origin under the group action. For each image, there is a great sphere whose points are equidistant from both the image and the origin. Each of these great spheres cuts S^3 in two pieces. The intersection of all the half-spheres containing the origin is the region between the green and red surfaces in the picture. I then took just the red sphere, cut out a wedge, and drew its orbit under the group action.

 

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Last Updated ( Tuesday, 12 May 2009 17:45 )