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...