Number of free homotopy classes < number of homotopy group elements

Question

I'm looking for a 'simple' path-connected space $X$ such that for some homotopy group $\pi_n(X)$ there are fewer free homotopy classes $[S^n,X]$ than elements of $\pi_n(X)$.

From this question we know that no such example can exist for $n=1$. EDIT: this is false! Thanks to Moishe Cohen. Ideally I would like an example for $n=2$.

Thanks for the help.

Answer 1

As pointed out by the OP, this is equivalent to having a nontrivial action of $\pi_1(X)$ on $\pi_n(X)$.

Answer 2

Just to close this question:

Consider $n\ge 2$. Start with an $(n-1)$-connected space $Y$ such that $\pi_n(Y)$ is finite, $\cong {\mathbb Z}_k$, say, attach $n+1$-cell to $S^n$ along a degree $k$ map, $k>1$. Take $X= S^1\vee Y$.

Then $[S^n, X]\cong \pi_n(Y)\cong {\mathbb Z}_k$ is finite, while $\pi_n(X)$ is infinite, isomorphic to the countably infinite direct sum $$ \oplus_{i\in {\mathbb Z}} {\mathbb Z}_k.$$ The fundamental group of $X$ acts on this group as the shift.