# Retraction and the k-fold projective plane

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.

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What is a $k$-fold projective plane? – Mariano Suárez-Alvarez Dec 11 '11 at 2:50
I mean k RP2's glued to each other – mandegar Dec 11 '11 at 3:12
this is a 2-manifold. for example Klein bottle is homeomorphic to 2RP2 – mandegar Dec 11 '11 at 3:14
Glued to each other how? (and please ad this information to the body of the question itself) – Mariano Suárez-Alvarez Dec 11 '11 at 3:16
yes, exactly. thanks – mandegar Dec 11 '11 at 3:19