Abstract:A quick exploration of some facts about 2-surfaces that have a very large number of toplogical handles. This is a topic that has probably received serious attention in the mathematical literature; however, I am unaware of such work, and so this treatment will appear to be shallow to anyone in the know.
Imagine a three-dimensional cube. Each of its six faces is a two dimensional surface. Imagine connecting a tube between opposite faces. This structure has a simple topological form: it is equivalent to three donuts joined together. [Need picture] Imagine solving some differential equation on the surface of this object: ones solution would have to fit periodic boundary conditions. Imagine now a similar but very different structure: a cubic lattice of cubes; the face of each cube joined to its neighbor by a tube. [Need picture]. This structure is in some certain sense 'topologocially complicated': it is multiply connected with a very high degree. Imagine smoothing off the corners of the cube (make the surface infinitely differentiable) and providing a metric on this surface.
Research Problem: Describe the form of the geodesics on this surface. Lets imagine the set of all geodesics passing through a point. Some of these will loop around in the local cubie, while others will pass through the 'tubes' to neighboring lattice cubies. Will all geodesics orbit around in a small neighborhood, or will some escape to infinity? What is the measure of each set? Is there something fractal or chaotic at play? Note that the lattice is three dimensional, in that we can talk about the cubie at lattice point i,j,k (where i,j,k are integers). Clearly, a collection of rays emanating from one point cannot visit each cell; nearby rays will most likely form caustics ...
Each face of a cubie is toplogically equivalent to a a surface with a hole cut in it, or more precisely, a ring with inner and outer radius [need picture]. The surface as a whole is obtained by sewing together neighboring rings. [need picture]. Homework: Describe the conditions that need to be applied to the metric between neighboring rings so that the metric is continuous and differentiable accross the boundry. To solve this problem, it is more convenient to think of the rings as overlapping, and one specifies a homoeomorphism between the two. How does the metric map? Homework: One needs to worry not only about smooth joints between neighboring rings, but also between three neighbors [need picture]. Show that the boundary conditions on the metrics of the three obey a bianchi-like identity.
With these tools in hand, it is now straightforward to construct 'random' toplogies. One can imagine joining together rings to N neighbors, where N follows some random distribution (e.g. a poisson distribution). One can also imagine that some of these join back to other nearest neighbors, creating a random network with some 'effective' higher dimensionalty. One can discover the dimension D of the space by counting the number of nearest neighbors, and then the number of next-nearest neighbors, etc. and taking the logarithm, where we keep in mind that the volume of a sphere in D dimensions goes as the radius to the D'th power.