Space with Abelian mapping class group

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

Stronger properties

 * Space with trivial mapping class group

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.