Simple space

For example, any topological group is a simple space (provided it satisfies the condition on the homotopy type).

Most Eilenberg-Maclane spaces ${\displaystyle K(A,n)}$ are simple since the only nontrivial homotopy group is in degree ${\displaystyle n}$. This means the only non-simple spaces are ${\displaystyle K(G,1)}$ for ${\displaystyle G}$ nonabelian.

Every connected topological space ${\displaystyle X}$ has an associated (universal) simple space from the universal cover ${\displaystyle \pi :U_{X}\to X}$; indeed, ${\displaystyle \pi _{1}(U_{X})=*}$ and the universal cover is its own universal cover.

• Dennis Sullivan, Geometric Topology