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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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4  
What is a $k$-fold projective plane? –  Mariano Suárez-Alvarez Dec 11 '11 at 2:50
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I mean k RP2's glued to each other –  mandegar Dec 11 '11 at 3:12
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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
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yes, exactly. thanks –  mandegar Dec 11 '11 at 3:19

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