The math I didn't do

I was watching a YouTube of Zagier talking about partitions and modular forms and was reminded of a summer project I worked on in graduate school. At the heart of the Zagier's talk was a discussion of simple singularities ("branch points") in [I think] some kind of covering map. My thought was: why go to the bother of creating a modular form when there is a direct route into studying the global properties of the mapping based on analyzing the singularities?

So, for whatever it is worth claiming you thought about something but took it nowhere, here is a little info about that summer project. At the time, a professor named Charlie McArthy approved the project and I got a little grant. I hate to regret the past, since it got me to where I am in the present, but it would have probably been better to have him as a thesis advisor than William Pohl. Pohl promised me that we would do differential geometry using probabilistic "forms" but ended up assigning me a problem from integral geometry that, to be honest, was a dead end in terms of what most people are interested in. Of course I did like the Buffon needle problem but resolution of singularities and the impact on PDE's would probably been a wiser course. Sadly, at that time I wouldn't have dreamed of asking a Functional Analysis guy to be my advisor.

It starts like this: I was intrigued with the exponential mapping and the logarithm. The exp{i*t} mapping a line into a circle really appealed to me and especially the way you could invert the mapping using cutting and pasting with a little algebra. In particular, consider the following general construction: 

Suppose a manifold M with a nicely embedded submanifold S, such that the topology of the boundary of M\S is a Lie group G. We consider (M\S)xG. Now, away from the boundary of M\S, we have (manifold)x(group). But along the boundary we have GxG, so can this can be pasted onto a single copy of G using the group operation (g1,g2)-->(g1*g2).

One variation is when the group is the quotient group of another group. Then (g1,g2) can map to one g1*g2 in the super group or to the quotient via the quotient mapping applied to  g1*g2.

Claim: The result is a manifold, we'll call M'. So M'=M\SxG + G

Proof: just define the neighborhoods along the pasted in copy of G.

Claim: The mapping defined on the boundary of M\SxG extends naturally to a mapping from all of M' back to M. This covering map is the exponential when M is a circle and S is a point and we use Z (so M' is the real line) or Z2 (where this resolves the singularity of a "figure 8").

More interestingly, then M is the 2-sphere and S is a point, G is a circle, and the covering map is the Hopf fibration. I like the notation

S1 = e^R1

S2 ₌ e^S3

I guess of some interest is the fact that the exponential map and the Hopf fibration are the lowest dimension examples of something general. For example, when the construction involves a 3D manifold with an embedded circle. The boundary of what is left is a torus, with its usual group structure. I would be surprised if there were not some nice theorems relating possible group structures to coverings of a manifold - based on homology of geodesics. 

But then, I expect many of us have stories about the big fish that got away. I was high enough to see these possibilities but too stoned to do the heavy lifting of real mathematics.