This question is kind of dual to this question, where I asked if smashing with a space $Y$ always increases (which means ''$\geq$'') the connectivity of a space $X$ and the answer was "yes".
The connectivity of a pointed space $X$ is the maximal number $\operatorname{con}(X)$ such that $\pi_i(X)=0$ for all $0\leq i\leq\operatorname{con}(X)$.
Let $X$ and $Y$ be pointed CW complexes and equip the set $\operatorname{Hom_\bullet} (Y,X)$ of pointed continuous maps with the compact-open topology.
Is it true that $\operatorname{con}(\operatorname{Hom_\bullet} (Y,X))\leq \operatorname{con}(X)$ if $X$ is connected?
This is particullary interesting when $Y=S^1$ where $\operatorname{Hom_\bullet} (S^1,X)=\Omega(X)$ but then $\pi_n(\Omega(X))=\pi_{n+1}(X)$ and the statement is true in this case.