Sounds like a fun puzzle, but I don't see how the resulting space can avoid having pinch-points that don't look like R^2 locally. The kind of space I'm picturing is what you get from shrinking the equator of a sphere to a point: the result is a bouquet of two spheres, and in the vicinity of the point where they meet, the space isn't a 2-manifold. How can a theta-curve on a torus avoid this fate? Am I totally on the wrong track, or does my puzzlement reflect what's cool about the puzzle? Jim Propp On 3/29/12, Dan Asimov <dasimov@earthlink.net> wrote:
In the torus T^2, find an embedded theta curve C (i.e., 2 points joined by each of 3 otherwise disjoint arcs) such that if C is identified to a point, the quotient space is topologically a sphere S^2.
--Dan
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