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In [[mathematics]] and [[theoretical physics]], a '''large diffeomorphism''' is a [[diffeomorphism]] that cannot be continuously connected to the identity diffeomorphism (because it is [[topology|topologically]] non-trivial).
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For example, a two-dimensional real [[torus]] has a [[special linear group|SL(2,Z)]] group of large diffeomorphisms by which the one-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]].
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More generally, for a [[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 &minus;1).
 
==See also==
 
*[[large gauge transformation]]
 
[[Category:Diffeomorphisms]]
[[Category:Theoretical physics]]
 
 
 
{{topology-stub}}

Latest revision as of 20:34, 17 November 2014

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