Composition of infinite deformation retracts

[finsly]

I'm trying to give an answer to the following problem, I hope someone could come in help! Consider a smooth $n$-dimensional manifold $M$ with smooth (nonempty) boundary $\partial M$, and suppose given a function $f: M\setminus \partial M \to \mathbb{R}$ (which one can assume to be differentiable) satisfying the property that there exists $A > 0$ such that for any $A \le \alpha \le \beta$, one has that the sublevel $\left\{F\le -\beta\right\}$ is a deformation retract of $\left\{F\le -\alpha \right\}$. The question is: is it true that $\partial M$ is a deformation retract of $\left\{F\le -A\right\}\cup \partial M$ (i.e., is it true that a composition of infinitely many of such deformation retracts is a deformation retract)?

 

[Tinyboss]

I don't have a full answer for you, but as a rule of thumb, infinite compositions of maps don't necessarily retain the properties of the individual maps. I think in this case a compactness argument might work, although I think either I'm missing something from your statement, or it's incomplete. Do we know what $f\big|_{\partial M}$ is? I was assuming it's identically zero, but I realize the problem doesn't say, nor does it say anything about what happens on the levels between zero and A.

 

[finsly]

First of all, thank you for your reply. Next, you're right, I forgot an hypothesis that could be crucial: $f(p)\to -\infty$ as $p$ approaches the boundary $\partial M$. Could this do any difference?
Maybe, (but I don't know if this makes any sense...) an idea could be to work with the extended function $\hat{f}: M \to \mathbb{R}^*$, where $\mathbb{R}^*:=\mathbb{R}\cup \left\{\infty\right\}$ (the Alexandroff compactification of $\mathbb{R}$), $\hat{f}(p):=f(p)$ if $p \in M\setminus \partial M$ and $\hat{f}:=\infty$ if $f \in \partial M$ (hoping that this $\hat{f}$ inherits some regularity from $f$...). In this way, $\partial M$ would become the level $\left\{f=\infty\right\}$...

 

[lavinia]

Slice the north polar ice cap off of a sphere to get a manifold with boundary. Then remove the South pole. Let f be the reciprocal of the minimum of the distances along a great circles to the South pole and to the edge of the removed polar cap. This function is continuous and f(p) -> -∞ as p approaches the edge of the removed ice cap.

But but the set,

f < - the distance of the meridian where both distances are the same

does not deform onto the edge circle of the ice cap.it seems that you need to assume that f(p) -> -∞ if and only if p approaches the boundary.

 

[finsly]

I really apologize with all of you for the incompleteness of the provided hypothesis. Actually, the manifold $M$ is simply connected as well as its boundary $\partial M$, and these restrictions seems to exclude the latter counterexample (if I'm not wrong).
And (finally) these are all the hypothesis I have...