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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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    What is a $k$-fold projective plane?2011-12-11
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    I mean k RP2's glued to each other2011-12-11
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    this is a 2-manifold. for example Klein bottle is homeomorphic to 2RP22011-12-11
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    Glued to each other *how*? (and please ad this information to the body of the question itself)2011-12-11
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    @mandegar There are lots of ways to glue two $\mathbb{RP}^2$'s together. In this case, you are talking about the connected sum. So $k\mathbb{RP}^2:=\mathbb{RP}^2\#\cdots\#\mathbb{RP}^2$ $k$ times.2011-12-11
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    yes, exactly. thanks2011-12-11

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