Just a heads-up that this question may be a little vague as I am new to algebraic topology.
Let $X$ be a topological space and $\pi_1(X,x_0)$ be its fundamental group. What general properties of $X$, if any, can be implied by $\pi_1(X,x_0)$ being abelian or even cyclic? I know that fundamental groups of even many simple spaces are often hard to compute and there is no general recipe for attacking the problem. I have seen a table of the higher homotopy groups of higher-dimentional spheres. I can even understand why some of the diagonals are the way that they are.
I know the following theorems:
Let $[S^1,X]:=\{[f]~|~f\text{ a continuous map from }S^1\text{ to }X\}$.
Then let $\Psi: \pi_1(X,x_0)\to[S^1,X]$ be defined by $[f]\mapsto[f]$.
Theorem 1: $\Psi$ is surjective if and only if $X$ is 0-connected (path-connected).
Theorem 2: $\Psi([f])=\Psi([g])$ if and only if $[f]$ and $[g]$ are conjugate in $\pi_1(X,x_0)$
Theorem 3: If $\pi_1(X,x_0)$ is abelian, then every class of functions is only conjugate with itself, so $\Psi$ is injective by Theorem 2.
Theorem 4: If $X$ is 1-connected (simply connected), then $\Psi$ is bijective.
By Theorems 1 and 3 we can determine that Theorem 4 does not hold in reverse, as a space can be path-connected and have an abelian fundamental group without being simply connected. An example would be $X=S^1$, which satisfies $\pi_0(S^1)$ being trivial with $\pi_1(S^1)\cong\mathbb Z$ being abelian and nontrivial. This is what made me wonder what you can infer in the backwards direction about $X$ from its fundamental group being abelian (and of order $\geq 6$ as the rest are always abelian). I also wonder if it being cyclic adds any additional information about $X$.