For a given path-connected space $X$, I recently learned that one could construct a CW complex $X_{1}$ by considering each generator $j: S^{q} \rightarrow X$ of $\pi_{q}(X)$ and setting $X_{1}=\bigvee_{(j,q)}S^{q}$, the wedge of spheres $S^{q}$, one for each generator $j$. The map $\gamma: X_{1} \rightarrow X$ defined by setting $\gamma$ to be $j$ on each $(j,g)th$ wedge component, induces a surjection of homotopy groups. This map however does not always induce an injection. I am having trouble understanding why this is so. Furthermore, I am unable to think of a specific example of this.
What are some examples of the spaces and the induced homomorphism above not being injective?
The reason this shouldn't always induce an injection is there may be nontrivial relations between the elements of the homotopy groups that we're ignoring in $X_1$. For instance, carrying out this procedure for the torus, we get $X_1 = S^1 \vee S^1$, but the fundamental group of the torus is abelian (a fact ignored in $X_1$).
Another factor that leads to non-injectivity is that the homotopy groups of spheres can be (and often are) nontrivial in higher degrees, in ways that aren't reflected in $X$; the procedure for $\Bbb{CP}^\infty$ gives $X_1 = S^2$, but $\pi_n(\Bbb{CP}^\infty)=0$ for $n \neq 2$, and $\pi_3(S^2) = \Bbb Z$.
For, say, $\Bbb{RP}^2$ we get some of both worlds. Its fundamental group is $\Bbb Z/2\Bbb Z$, so twice the generator of $S^1$ is sent to zero (our $X_1$ can't notice torsion). But $X_1$ also has copies of $S^k$ for generators of $\pi_n(\Bbb{RP}^2) = \pi_n(S^2)$ for $n>1$, and these (many) $S^k$s have extra homotopy that's not reflected in $\Bbb{RP}^2$, etc.