Mapping class group
This article defines an association of a group to every topological space. The association is not necessarily functorial
The mapping class group of a topological space is defined as the quotient of its self-homeomorphism group by the normal subgroup comprising all homeomorphisms isotopic to the identity map. Equivalently, it can be thought of as the group of isotopy classes of homeomorphisms from the space to itself.
Note that the association of a mapping class group to every topological space is not functorial.