Simple space

In algebraic topology, a branch of mathematics, a simple space is a connected topological space that has a homotopy type of a CW complex and whose fundamental group is abelian and acts trivially on the homotopy and homology of the universal covering space. Though not all authors include the assumption on the homotopy type.
Examples.
Topological groups.
For example, any topological group is a simple space.
Eilenberg-Maclane spaces.
Most Eilenberg-Maclane spaces formula_1 are simple since the only nontrivial homotopy group is in degree formula_2. This means the only non-simple spaces are formula_3 for formula_4 nonabelian.
Universal covers.
Every connected topological space formula_5 has an associated universal space from the universal cover formula_6 since formula_7 and the universal cover of a universal cover is the universal cover itself.