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[parent] examples of mapping class group (Example)

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

$\displaystyle {\cal{M}}(S^2)=1,$
but
$\displaystyle {\cal{M}}^*(S^2)={\mathbb{Z}}_2.$

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

$\displaystyle {\cal{M}}^*(T)=GL_2({\mathbb{Z}}),$
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
$\displaystyle {\cal{M}}(T)=SL_2({\mathbb{Z}}).$

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

$\displaystyle {\cal{M}}({\mathbb{R}}P^2)={\cal{M}}^*({\mathbb{R}}P^2)=1$

And what about the Klein bottle?

$\displaystyle {\cal{M}}(K)={\mathbb{Z}}_2$
$\displaystyle {\cal{M}}^*(K)={\mathbb{Z}}_2\oplus{\mathbb{Z}}_2$



"examples of mapping class group" is owned by juanman.
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See Also: isotopy, group, group, isotopy

Other names:  first homeotopy group
Keywords:  Dehn's twist, surface

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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 1818 times total.

Classification:
AMS MSC: 57R50 (Manifolds and cell complexes :: Differential topology :: Diffeomorphisms)
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