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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 −1). |
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| | ==See also== |
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| | *[[large gauge transformation]] |
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| | [[Category:Diffeomorphisms]] |
| | [[Category:Theoretical physics]] |
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| | {{topology-stub}} |
Revision as of 19:14, 31 January 2014
In mathematics and theoretical physics, a large diffeomorphism is a diffeomorphism that cannot be continuously connected to the identity diffeomorphism (because it is topologically non-trivial).
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 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).
See also
Template:Topology-stub