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Large diffeomorphism
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{{Short description|Class of diffeomorphism}} {{unreferenced|date=June 2014}} In [[mathematics]] and [[theoretical physics]], a '''large diffeomorphism''' is an equivalence class of [[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]] has a [[special linear group|SL(2,Z)]] 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]]. More generally, for a [[Surface (topology)|surface]] ''S'', the structure of [[homeomorphism|self-homeomorphism]]s up to [[homotopy]] is known as the [[mapping class group]]. It is known (for [[compact space|compact]], [[orientable]] ''S'') that this is isomorphic with the [[automorphism group]] of the [[fundamental group]] of ''S''. This is consistent with the [[genus (mathematics)|genus]] 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 of a subgroup|index]] 2 in all automorphisms, since the orientation may also be reverse, by a transformation with determinant −1). ==See also== *[[Large gauge transformation]] [[Category:Diffeomorphisms]] [[Category:Theoretical physics]] {{topology-stub}} {{theoretical-physics-stub}}
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