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Let $f:M\to N$ be a smooth map between smooth 4-manifolds with boundary. When does $f$ map boundary of $M$ to boundary of $N$ upto homotopy i.e. when there is a map $F:M\to N$ homotopic to $f$ such that $F(\delta(M))\subseteq\delta(N)$? Can I impose some kind of condition on $M$, $N$ or $f$ such that $F$ maps boundary to boundary?

I apologize if my question is too naïve or too basic. Any kind of suggestions will be helpful. Thanks@RyanBudney for reformulating the question.

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    $\begingroup$ Could you define a submersion of manifolds with boundary? Is the self-map $x\to x/2$ of the unit disk a submersion? $\endgroup$
    – Igor Belegradek
    May 28, 2022 at 12:30
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    $\begingroup$ I deleted my comment as it was wrong (sorry!) (a smooth submersion between two boundaryless smooth manifolds of the same dimension is a local diffeomorphism, see Proposition 4.8. of Lee's Introduction to Smooth Manifolds ). The counterexample (for manifold with boundary) is given in the @IgorBelegradek comment. $\endgroup$
    – Random
    May 28, 2022 at 15:39
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    $\begingroup$ Without a refinement to your question, the answer is "almost never". The title of your question is different than the question in the body of your post. Both could be made more interesting by adding "up to 1-parameter families", i.e. up to homotopy for the title question, and up to 1-parameter families of submersions for the body question. But you should decide on what question you want to ask. $\endgroup$
    – Ryan Budney
    May 28, 2022 at 19:51
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    $\begingroup$ 1) The unit disc in the plane, $z \longmapsto z^2$. 2) Take the identity map on a manifold, and slightly perturb it away from the boundary. 3) Let the manifold be $[-1,1]$ and take the map $t \longmapsto t^2$. $2$-manifolds have quite a rich collection of homotopy classes of maps. $3$-manifolds even more rich. $\endgroup$
    – Ryan Budney
    May 29, 2022 at 4:01
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    $\begingroup$ I suppose it depends on what question you want to see discussed. Your question as-asked is trivial, so people do not discuss it. But if you ask it up to homotopy, or up to 1-parameter families of submersions, etc, this is a classical problem. For maps of surfaces it's now a fairly easy question to answer. The terminology is "peripheral systems". For 3-manifolds, this question largely goes back to the initial work leading up to the JSJ-decomposition, and Waldhausen's work, and the answer is analogous to the surface case. $\endgroup$
    – Ryan Budney
    May 29, 2022 at 4:12

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