Let $M$ and $N$ be two differential manifolds and there is a surjective submersion $f$ from $M$ to $N$ such that $f^{-1}(p)$ is compact and connected for any $p$ on $N$. Can we conclude that $f$ is proper, that is, the preimage of a compact set is compact?

It is also posted on https://math.stackexchange.com/q/1004543/13534

  • $\begingroup$ This looks a lot like a homework problem to me. I've voted to close. $\endgroup$ – Ryan Budney Nov 10 '14 at 20:17
  • $\begingroup$ Without tryng to judge one way or another whether it's homework, it is the type of thing that has been discussed at math.stackexchange, e.g., here: math.stackexchange.com/questions/7096/… If OP cannot construct an argument from that or similar threads, he should probably ask over there. (I happen to think it's a good question.) $\endgroup$ – Todd Trimble Nov 10 '14 at 21:20
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    $\begingroup$ @Todd Trimble, I am afraid the answers in that Math.StackExchange discussion get us no closer to an answer to this question. They seem to be mostly wrong as far as I can tell. $\endgroup$ – Ricardo Andrade Nov 10 '14 at 21:41
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    $\begingroup$ This has been asked also at math.stackexchange.com/questions/1004543/… (and then deleted by the OP; but I undeleted it) $\endgroup$ – Mariano Suárez-Álvarez Nov 11 '14 at 0:18
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    $\begingroup$ Please if after some time you do not get an answer in math.stackexchange.com it can be sensible to reask here, but always provide links to the other question in both sides, and please do not delete questions which have answers on which other users have spent time and energy. $\endgroup$ – Mariano Suárez-Álvarez Nov 11 '14 at 0:19

This is true in greater generality. If $M$ and $N$ are locally compact, and the fibers of $f$ are compact connected and $f$ is a quotient map then f is proper. This falls under the rubric of monotone light factorization. Look at the book of G.T. Whyburn and E. Duda titled Dynamic Topology. Monotone Light factorization says if a map between locally compact Hausdorff has the property, that all connected components of all fibers are compact, then the map can be factored as a proper map onto a space with connected fibers and this space maps to the co-domain with totally disconnected fibers. Thanks to Dan Petersen and Richard Andrade for correction. Addendum: Let a connected component of the fiber of a map be called a level.Suppose X and Y are locally compact Hausdorff spaces and F a continuous map such that all levels are compact .Suppose the equivalence relation R on X is given by saying x and y are equivalent if they lie in the same level, then R is a proper equivalence relation and the map from X to X/R is proper with connected fibers and F factors through X/R to Y as a map with totally disconnected fibers . In the case of the question of the OP the equivalence relation R and the one given by f coincide and therefore the map is proper

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    $\begingroup$ Nice answer, Mohan. $\endgroup$ – Todd Trimble Nov 10 '14 at 21:22
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    $\begingroup$ I don't think this completely answers the question. Consider e.g. $\mathbf R\setminus \{0\} \sqcup \{0\}$ with its natural map to $\mathbf R$, which is a nonproper map with compact connected fibers. To rule out a silly counterexample like this it seems one should use the submersion hypothesis (which your argument doesn't). $\endgroup$ – Dan Petersen Nov 10 '14 at 22:55
  • $\begingroup$ The spaces in Dan Petersen's example are locally connected. You may want to require that the map be (at least) a quotient map. $\endgroup$ – Ricardo Andrade Nov 10 '14 at 23:08
  • $\begingroup$ In Dan's case the intermediate space is just the space he started with .The identity map is proper. $\endgroup$ – Mohan Ramachandran Nov 10 '14 at 23:15
  • $\begingroup$ What I meant is that Dan's comment still provides a counter-example to the second sentence in your answer. On another note, the spaces before the periods in your answer are kind of confusing. The spaces should usually come after the periods. $\endgroup$ – Ricardo Andrade Nov 10 '14 at 23:16

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