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

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# Composition of infinite deformation retracts

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