# Large diffeomorphism

In mathematics and theoretical physics, a **large diffeomorphism** is an equivalence class of diffeomorphisms under the equivalence relation where diffeomorphisms that can be continuously connected to each other are in the same equivalence class.

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. This group of large diffeomorphisms is called the modular group.

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 *Z*^{2}, 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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