In an effort to blog a bit more, and to spur myself to work harder, I’m going to try to blog more about the work I’m doing this summer.

Briefly, I’m exploring properties of subsets of the modular group, defined as the set of 2×2 matrices with determinant 1. I work with an eye toward the problem of generating congruence subgroups, which is not adequately solved. It is possible to check whether a given group is a subgroup, but it has shown to be difficult to work in the opposite direction.

This week I’m delving into Topology. Any modular subgroup can be described as a polygon on the hyperbolic plane, which in turn can be turned into a Riemann surface by identifying corresponding edges (see for example Wohlfahrt). Of particular interest is the nature of the covering of that surface. Read beneath the fold for diffeomorphisms.

The following is adapted from Schlichenmaier, pp.13-14:

First, suppose we are given two differentiable manifolds and an isomorphism between them:
$$!(M,T), (M’,T’)$$
$$!f : M rightarrow M’$$

Now we take a point in the first manifold, and its image in the second. We take a neighbourhood around each of these two points and define the function that maps each neighbourhood to coordinates in the reals:
$$! a in U subset M quad ; quad b = f(a) in V subset M’ $$
$$! phi : U rightarrow mathbb{R}^n quad ; quad psi : V rightarrow mathbb{R}^n $$
Here $$(U,phi)$$ and $$(V,psi)$$ are known as “coordinate patches”.

Now, we can define a function that takes values in $$mathbb{R}^n$$, maps them into $$U$$, then $$V$$, then back to $$mathbb{R}^n$$, like so:
$$! f_{UV} : mathbb{R}^n rightarrow mathbb{R}^n $$
$$! f_{UV} = psi circ f circ phi^{-1} $$
This of course is defined on all the coordinates of the pre-image of $$V$$ in $$U$$ by $$f$$. It gives us $$n$$-dimensional real values that are coordinates of the image of $$U$$ in $$V$$.

Now, if the mapping $$ f_{UV} $$ is differentiable for all possible choices of $$a$$ and $$b$$, $$f$$ is called a differentiable map. If this is true of the inverse function as well, then $$f$$ is a diffeomorphism.

Diffeomorphisms help us classify manifolds. It’s easy to see that we can define an equivalence relation based on diffeomorphisms, and therefore divide manifolds into equivalence classes. The special case occurs when we’re talking about compact two-dimensional orientable manifolds (which includes all 1-dimensional complex manifolds). For these, there is exactly one diffeomorphic class for each genus. This is why we can classify such surfaces by genus, and why we get bad jokes like the following:

An employee at a coffee shop filled a topologist’s order, a coffee and donut. The topologist looked up, distraught, and asked, “But which is which?”

In higher dimensions, this is not true. Schlichenmeier gives the example of the 7-sphere, which has 28 inequivalent structures. Fortunately, I am only dealing with Riemann surfaces (1-dimensional complex) here :)

Finally, when dealing with complex manifolds, if all of our $$ f_{UV} : mathbb{C}^n rightarrow mathbb{C}^n $$ are holomorphic (analytic), and $$f$$ is bijective and holomorphic in both directions, then we call the map $$ f : M rightarrow M’ $$ an analytic isomorphism.

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