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examples of mapping class group
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(Example)
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An example of this concept is to take the 2-sphere $S^2$ then one can calculate that $${\cal{M}}(S^2)=1,$$ but $${\cal{M}}^*(S^2)={\mathbb{Z}}_2.$$
For the genus one orientable surface, i.e. the torus $T=S^1\times S^1$ it is known that its (extended) mapping class group $${\cal{M}}^*(T)=GL_2({\mathbb{Z}}),$$ but usually by the (non-extended) mapping class group, that is, the group of isotopy classes of homeomorphisms that preserve orientations (the Dehn's twists) is just $${\cal{M}}(T)=SL_2({\mathbb{Z}}).$$
In these two examples we see that $\cal{M}^*$ is an extension of $\cal{M}$ by ${\mathbb{Z}}_2$ trivial for the 2-sphere and non trivial for the torus.
For the projective plane ${\mathbb{R}}P^2$ we have $${\cal{M}}({\mathbb{R}}P^2)={\cal{M}}^*({\mathbb{R}}P^2)=1$$
And what about the Klein bottle? $${\cal{M}}(K)={\mathbb{Z}}_2$$ $${\cal{M}}^*(K)={\mathbb{Z}}_2\oplus{\mathbb{Z}}_2$$
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"examples of mapping class group" is owned by juanman.
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Cross-references: Klein bottle, projective plane, extension, orientations, preserve, homeomorphisms, classes, isotopy, group, mapping class group, torus, surface, orientable, genus, calculate
This is version 5 of examples of mapping class group, born on 2006-02-17, modified 2006-06-03.
Object id is 7631, canonical name is ExampleOfMappingClassGroup.
Accessed 2580 times total.
Classification:
| AMS MSC: | 57R50 (Manifolds and cell complexes :: Differential topology :: Diffeomorphisms) |
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Pending Errata and Addenda
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