That could explain why I didn't find this anywhere, I suppose. So where does my purported homeomorphism go belly up? Immersed in 3-space, once the sweep inceeds the tube, a nodal singularity develops where the surface self-intersects. But this doesn't behave like a winding point at all --- one revolution around it leaves me back where I started (quite so). In any case, immersion is irrelevant to the homeomorphism: continuity at the winding points would be lost as soon as I tried to depart from radius zero. And another "proof" bites the dust! Still, nice exercise ... WFL On 4/19/14, Dan Asimov <dasimov@earthlink.net> wrote:
Your 2-sheeted Riemann sphere with 2 "winding points" can be thought of as a map from the unknown surface S to the sphere having 2 points that are each "ramified" of (I presume) index 2 (locally the map looks like z |-> z^2).
This can be identified by viewing it as 2 spheres, each of which is missing 2 points, but with 2 of those 4 missing points replaced. This means the Euler characteristic is X(S) = 2(X(S^2) - 2) + 2 = 2*0 + 2 = 2. Checking that it's locally Euclidean, this nails it as a sphere, the only surface S with X(S) = 2.
Or am I misunderstanding what a "winding point" is?
--Dan
On Apr 19, 2014, at 6:42 AM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
This material is surely elementary, but I couldn't find it anywhere: maybe someone can cite a reference?
A 2-sheet Riemann sphere with 2 winding points is homeomorphic to a torus: for if its tube radius remains fixed while the swept radius approaches zero, the torus approaches a double sphere with winding points at opposite ends of a diameter --- whence they may be deformed to arbitrary locations, and the sphere to a topological sphere.
In a similar fashion, s sheets with t winding points are homeomorphic to a sphere with (s t)/4 handles, ie. genus = (s t)/4 , at any rate for s, t even; otherwise it perhaps suffices to take the ceiling ...
I would probably have remained unaware of this idea if I had not once been experimenting with a simple Maple graphics demo program to plot a torus, and noticed that what I had casually assumed must be a simple sphere instead appeared mysteriously fuzzy and uneven. On this occasion at any rate, Maple was innocent of blame, and indeed had provided valuable insight!
Fred Lunnon
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