# Large diffeomorphism

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{{Short description|Class of diffeomorphism}}
{{unreferenced|date=June 2014}}
In [mathematics](/source/mathematics) and [theoretical physics](/source/theoretical_physics), a '''large diffeomorphism''' is an equivalence class of [diffeomorphism](/source/diffeomorphism)s 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](/source/torus) has a [SL(2,Z)](/source/special_linear_group) group of large diffeomorphisms by which the 1-cycles <math>a,b</math> of the torus are transformed into their integer linear combinations. This group of large diffeomorphisms is called the [modular group](/source/modular_group).

More generally, for a [surface](/source/Surface_(topology)) ''S'', the structure of [self-homeomorphism](/source/homeomorphism)s up to [homotopy](/source/homotopy) is known as the [mapping class group](/source/mapping_class_group). It is known (for [compact](/source/compact_space), [orientable](/source/orientable) ''S'') that this is isomorphic with the [automorphism group](/source/automorphism_group) of the [fundamental group](/source/fundamental_group) of ''S''. This is consistent with the [genus](/source/genus_(mathematics)) 1 case, stated above, if one takes into account that then the fundamental group is ''Z''<sup>2</sup>, on which the modular group acts as automorphisms (as a subgroup of [index](/source/Index_of_a_subgroup) 2 in all automorphisms, since the orientation may also be reverse, by a transformation with determinant &minus;1).

==See also==

*[Large gauge transformation](/source/Large_gauge_transformation)

Category:Diffeomorphisms
Category:Theoretical physics

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Adapted from the Wikipedia article [Large diffeomorphism](https://en.wikipedia.org/wiki/Large_diffeomorphism) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Large_diffeomorphism?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.