Large diffeomorphism

For example, a two-dimensional real torus has a SL(2,Z) group of large diffeomorphisms by which the one-cycles

of the torus are transformed into their integer linear combinations.

More generally, for a surface S, the structure of self-homeomorphisms up to homotopy is known as the mapping class group.

It is known (for compact, orientable S) that this is isomorphic with the automorphism group of the fundamental group of S. This is consistent with the genus 1 case, stated above, if one takes into account that then the fundamental group is Z2, on which the modular group acts as automorphisms (as a subgroup of index 2 in all automorphisms, since the orientation may also be reverse, by a transformation with determinant −1).

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