# Mapping class group

From Topospaces

*This article defines an association of a group to every topological space. The association is not necessarily functorial*

## Definition

### Symbol-free definition

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.