examples of mapping class group

An example of this concept is to take the 2-sphere S2, then one can calculate that




For the genus one orientable surface, i.e. the torus T=S1×S1, it is known that its (extended) mapping class groupPlanetmathPlanetmath


but usually by the (non-extended) mapping class group, that is, the group of isotopyPlanetmathPlanetmath classes of homeomorphismsMathworldPlanetmath that preserve orientationsPlanetmathPlanetmath (the Dehn’s twists) is just


In these two examples we see that * is an extension of by 2, trivial for the 2-sphere and non trivial for the torus.

For the projective planeMathworldPlanetmath P2 we have


And what about the Klein bottle?

Title examples of mapping class group
Canonical name ExamplesOfMappingClassGroup
Date of creation 2013-03-22 15:41:19
Last modified on 2013-03-22 15:41:19
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 8
Author juanman (12619)
Entry type Example
Classification msc 57R50
Synonym first homeotopy group
Related topic isotopy
Related topic group
Related topic Group
Related topic Isotopy