I started with a finite CW complex $X$ of dimension $n\geq 3$. I have embedded (local embedding) it inside $R^{2n}$. Now take a regular neighbourhood $U$ of $X$ in $R^{2n}$ which has the same homotopy type as $X.$ I want to claim there exist a proper Morse function such that indexes of all critical points $\leq n.$
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$\begingroup$ Maybe you could explain a bit more about what kind of embedding you have in mind. For instance, what does regular neighborhood mean? For PL embedding of complexes this is standard and your question sounds plausible, but is there a well-defined notion of regular neighborhood for a topological embedding? $\endgroup$– Danny RubermanCommented Aug 20, 2021 at 15:00
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$\begingroup$ If I embed the CW complex $X$ in $R^{2n}$ from Hatcher Appendix A.5. there is an open neighborhood $N(X,\epsilon)$ so that $X$ is a deformation retract of the neighborhood. Here I meant normal tubular neighborhood of the CW complex. $\endgroup$– piper1967Commented Aug 20, 2021 at 21:44
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$\begingroup$ In that proposition, X is embedded as a subcomplex, so I think you have a chance. Have a look at the proof, which suggests that the neighborhood N is built cell by cell. $\endgroup$– Danny RubermanCommented Aug 21, 2021 at 11:09
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