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Edit: as explained in my comment on alesia's answer, I mistakenly did not ask below the question I intended (due to my misguided efforts to simplify it). Thus, I revised and reposted my question here.

Old question:

Are there any known examples of smooth manifolds $M$, $N$ of equal dimension such that a topological embedding $M\hookrightarrow N$ does not exist, but for each $\varepsilon > 0$ there exists a continuous map $f:M\to N$ whose fibers $f^{-1}(n)$ all have diameter smaller than $\varepsilon$ (with respect to some fixed Riemannian metric on $M$)?

For example, take $N=\mathbb{R}^2$ and $M$ to be a 2-torus with a point removed. Does there exist a sequence $f_k: M\to \mathbb{R}^2$ of continuous maps such that $$\sup_{x\in \mathbb{R}^2} \operatorname{diam} f_k^{-1}(x) \to 0$$ as $k\to \infty$ with respect to some fixed metric?

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  • $\begingroup$ If the fibers aren't points or empty, what does it mean for the fibers to be arbitrarily small? I mean, if they're not empty they have some fixed diameter and you wouldn't be able to make $\epsilon$ smaller than that. Are you talking about in an isotopy class of an embedding? $\endgroup$
    – Ryan Budney
    Commented Dec 8 at 18:36
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    $\begingroup$ @RyanBudney The Riemannian metric on $M$ induces a distance function, with respect to which we can define the diameter of an arbitrary subset. I am asking whether there exist $M$, $N$ such that $M$ does not embed in $N$ but there exists a sequence of continuous maps $f_k : M\to N$ such that $\sup_{n\in N} \text{diam} f_k^{-1}(n) \to 0$ as $k\to \infty$. For example, by an argument using top degree homology, this situation cannot occur if $N$ is an open manifold and $M$ is not. I think my question is related to 1979 work of Chapman and Ferry, but I do not see how to answer my question. $\endgroup$
    – Matthew Kvalheim
    Commented Dec 8 at 20:15
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    $\begingroup$ @RyanBudney and no, I do not think I am asking about an isotopy class of an embedding. In my question, it is not allowed for there to exist any embeddings at all of $M$ into $N$. $\endgroup$
    – Matthew Kvalheim
    Commented Dec 8 at 20:17
  • $\begingroup$ Did you check if Ferry's arguments from his 1979 AJM paper on $\epsilon$-homeomorphisms apply in your situation? (There is no formal application, of course.) $\endgroup$
    – Moishe Kohan
    Commented Dec 8 at 20:39
  • $\begingroup$ @MoisheKohan I had started to do so, but have not yet made much progress. $\endgroup$
    – Matthew Kvalheim
    Commented Dec 8 at 20:48

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Isn't the following an example?

Build $M$ starting from the plane by attaching an infinite sequence of handles whose size decreases to zero.

Take for $N$ the disjoint union of $M_k$, where $M_k$ is the manifold obtained instead of $M$ if one attaches only the first $k$ handles.

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    $\begingroup$ I believe so. Thank you, +1, accepted. However, now I realize that I did not ask the question I wanted to ask. I was trying to simplify Ferry's use of open covers in defining "small fibers" by using a Riemannian metric, which inadvertently permits your example due to noncompactness. I am actually only interested in examples for which, if $M$ is noncompact, $M$ is the interior of a compact manifold with boundary $L$ and the metric on $M$ is the restriction of one on $L$. I suppose it may be best for me to post this as a new question. Thank you for answering my misguided question and helping me. $\endgroup$
    – Matthew Kvalheim
    Commented 2 days ago

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