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Let $C$ be a compact 2-dimensional cylinder $[0,1]\times S^1$. Let $A$, $A'$ be the two connected components of its boundary.

Let $Q$ be a square. Let $a$, $a'$ be a pair of opposite edges of $Q$.

Consider a smooth imbedding $f\colon Q\to C$ such that $$f(a)\subset A,\, \, f(a')\subset A'.$$ Assume moreover that $f(\partial Q\setminus (a\cup a'))$ in contained in the interior of $C$.

Question. Is it true that $C\setminus f(Q)$ is connected?

A reference will be helpful.

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    $\begingroup$ What does $f(\partial C)$ mean? Isn't $f$ defined on $Q$? $\endgroup$
    – Yaakov Baruch
    Commented Sep 30 at 17:11
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    $\begingroup$ I would assume it was supposed to mean $f(Q\backslash (a\cup a'))$... $\endgroup$
    – Aleksei Kulikov
    Commented Sep 30 at 17:12
  • $\begingroup$ @YaakovBaruch: Sorry, corrected. $\endgroup$
    – asv
    Commented Sep 30 at 17:25
  • $\begingroup$ Take a look at the theory of surfaces, and embedded curves in those surfaces. There is a basic theory of how cutting along embedded curves affects the surface: its genus and number of boundary components. You can derive the theory from the classification of surfaces, or simply from Poincare Duality. $\endgroup$
    – Ryan Budney
    Commented Sep 30 at 18:53
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    $\begingroup$ @RyanBudney; Do you have a reference to this theory? A more focused information would be helpful. $\endgroup$
    – asv
    Commented Sep 30 at 18:55

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