I am reading through John Lee's book on topological manifolds and in the middle of attempting a question on page 274. The question asks the reader to assume $n$ to be an integer greater than $2$ and to construct a polygonal presentation whose geometric realization has a fundamental group which is cyclic of order $n$.
Does anyone have any clues on this one?
Corollary 7.3 in Bredons book "Geometry and topology" states that if a group G acts properly discontinuously on the simply connected and locally arcwise connected space X then $G = \pi_1(X/G)$ where $X/G$ is the set of orbits of all points under the group action with the topology induced by the projection map.
For an example of an application of this, look up lens spaces.