Mapping Class Group
In geometric topology especially, one considers the quotient group obtained by quotienting out by isotopy, called the mapping class group:
The MCG can also be interpreted as the 0th homotopy group, . This yields the short exact sequence:
In some applications, particularly surfaces, the homeomorphism group is studied via this short exact sequence, and by first studying the mapping class group and group of isotopically trivial homeomorphisms, and then (at times) the extension.
Read more about this topic: Homeomorphism Group
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