Let $k > 2$. Is there an embedding of $S^1$ in $\#_k \mathbb{RP}^2$ such that $S^1$ is a retract of $\#_k \mathbb{RP}^2$?
I know that this is not correct when $k=1$ (homotopy argument), and this is correct when $k=2$.
But I have no idea in the case that $k>2$. Thanks.