Large diffeomorphism

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> and <a href="/facts/Theoretical_physics/Cs2Y12rQ">theoretical physics</a>, a large diffeomorphism is an equivalence class of <a href="/facts/Diffeomorphism/855N5YYj">diffeomorphisms</a> 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 <a href="/facts/Torus/J9SkaV1W">torus</a> has a <a href="/facts/Special_linear_group/rsahah9w">SL(2,Z)</a> group of large diffeomorphisms by which the 1-cycles 
 
 
 
 a
 ,
 b
 
 
 {\displaystyle a,b}
 
 of the torus are transformed into their integer linear combinations. This group of large diffeomorphisms is called the <a href="/facts/Modular_group/wL6l14Hk">modular group</a>.
More generally, for a <a href="/facts/Surface_(topology)/jK1b1KIG">surface</a> S, the structure of <a href="/facts/Homeomorphism/jMfWg3Mn">self-homeomorphisms</a> up to <a href="/facts/Homotopy/1nWrPxdj">homotopy</a> is known as the <a href="/facts/Mapping_class_group/buQm001q">mapping class group</a>. It is known (for <a href="/facts/Compact_space/d0cgXJH7">compact</a>, <a href="/facts/Orientable/1SEQM2Pt">orientable</a> S) that this is isomorphic with the <a href="/facts/Automorphism_group/65vImKwe">automorphism group</a> of the <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a> of S. This is consistent with the <a href="/facts/Genus_(mathematics)/o1Yhf8ty">genus</a> 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 <a href="/facts/Index_of_a_subgroup/ewY2A71Y">index</a> 2 in all automorphisms, since the orientation may also be reverse, by a transformation with determinant −1).

Large diffeomorphism open-in-new

Large diffeomorphism