# Space with Abelian mapping class group

From Topospaces

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

*This property of topological spaces is defined as the property of the following associated group: mapping class group having the following group property: Abelian group*

## Definition

A topological space is said to have **Abelian mapping class group** if its mapping class group is Abelian.

## Relation with other properties

### Stronger properties

## Facts

In a space with Abelian mapping class group, any two conjugate self-homeomorphisms are isotopic, and hence, in particular, homotopic. Thus, the problem of finding a *good* self-homeomorphism that is conjugate to an arbitrary self-homeomorphism, also solves the problem of finding a good self-homeomorphism that is homotopy-equivalent to the given one.