On Sun, Apr 04, 2004 at 07:28:02AM -0400, William P. Thurston wrote:
As your reasoning suggests, there are 4 regular homotopy classes of embeddings (or immersions) of a torus in R^3.
Are you sure? After thinking about it for a while, I think there should be 4 regular homotopy classes of immersions, but only 3 of them are represented by embeddings. The regular homotopy classes of immersions are distinguished by taking the standard meridian and longitude, and seeing whether the band obtained by taking a regular neighborhood in the surfaces is twisted by an even or odd number of times. In the standard embedding of the torus, both the meridian (the (1,0) curve) and longitude (0,1) are untwisted, while the diagonal (1,1) is twisted. In general, curves (p,q) where p and q are both odd (and relatively prime, to give a simple curve on the torus) are twisted an odd number of times. But two such curves cannot form a basis for the homology of the torus. I believe another way to say this is that of the 4 different spin structures on the torus, 3 of them are even (and bound 3-manifolds) while one of them is odd. This also resolves another point I was having difficulty reconciling; namely, the subgroup of SL(2,Z) I found before, generated by the matrices ( 1 2 ) (-1 0 ) ( 0 1 ) ( 0 1 ) ( 0 -1 ) ( 1 0 ) seemed to be of index 3 in SL(2,Z). Namely, it seems to be the inverse image of a subgroup of order 2 in SL(2,Z_2) =~ S_3. (I haven't checked that these matrices generate that subgroup.) Peace, Dylan