Examples of spaces with non-abelian $\pi_2(X, A)$

Question

It is well known that $\pi_n(X)$ are abelian for all $n\geq2$, but this only follows in relative homotopy groups for $n\geq3$. I am writing some notes on higher homotopy groups, and was searching for some simple counter-example to show $\pi_2(X, A)$ can be non-abelian, but every example I've found is either too pathological, or too complicated because they are asking the space for other additional properties. What's the simplest counter-example you can think of for this?

Answer 1

There is a homotopy long exact sequence of a pair $(X, A)$ as follows

$$\dots \to \pi_{n+1}(X, A) \to \pi_n(A) \to \pi_n(X) \to \pi_n(X, A) \to \pi_{n-1}(A) \to \dots$$

In particular, we have

$$\dots\to \pi_2(X) \to \pi_2(X, A) \to \pi_1(A) \to \pi_1(X) \to \dots$$

Note that if $\pi_1(X) = 0$ and $\pi_2(X) = 0$, then $\pi_2(X, A) \cong \pi_1(A)$. Therefore, any pair $(X, A)$ where $X$ is $2$-connected and $A$ has non-abelian fundamental group will have $\pi_2(X, A)$ non-abelian. For example, take $X = S^3$ and $A = S^1\vee S^1$.