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I asked this question long back but could not get a satisfying answer. Starting with a finite CW complex $X$ of dimension $n\geq 3$. I have embedded it inside $R^{2n}$. Now take a tubular neighborhood $U$ of $X$ in $R^{2n}$. I want to claim that there exists a proper Morse function on $U$ such that indexes of all critical points $\leq n$.

If I try to proceed by induction on $n$, and when I glue in a new $n+1$ cell can I glue in thickening with exactly one new critical point of index n+1?

Giving a handle body structure with the index of handles $\leq n$ is equivalent to give a Proper Morse function on the regular neighborhood(open manifold) of the CW complex(to which the handle body is homotopic) in $R^{2n}$?

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    $\begingroup$ Your question is confusing. You ask that the CW complex be embedded in euclidean space, but then your question appears to have nothing to do with the embedding. Why are you asking for the embedding? You are vague on several points. What is the domain of the Morse function? The regular neighbourhood is either a manifold with boundary (perhaps corners) or perhaps you want it to be an open manifold? $\endgroup$
    – Ryan Budney
    Oct 4, 2021 at 16:44
  • $\begingroup$ The statement written in the first paragraph is my main question. One can give an answer by creating a handle body that I don't want. Since creating a handle body means it will be embedded in some higher R^N which will create a problem for my setup. The domain of the Morse function is the tubular nbd U mentioned. above. The regular nbd is open manifold. I apologize for sloppy writing. $\endgroup$
    – piper1967
    Oct 4, 2021 at 16:56
  • $\begingroup$ Is $n$ the dimension of the CW-complex? It appears to be undefined at present. $\endgroup$
    – Ryan Budney
    Oct 4, 2021 at 17:05
  • $\begingroup$ Yes, n is the dimension of the CW complex. $\endgroup$
    – piper1967
    Oct 4, 2021 at 17:16
  • $\begingroup$ Does h-cobordism technique solves the problem under simply-connectedness assumption? It seems that if you have any ($2n\geq 6$)-manfiold (with boundary) of homotopy type of $n$-complex, then there is a handle decomposition with indexes at most $n$. Let your decomposition starts with 0-handle. You first reorder handles, then use Whitney's trick to relate everything with the Morse complex linear algebra and finally cancel all unwanted handles using acyclicity of the complex in degrees $>n$. $\endgroup$
    – Bad English
    Oct 4, 2021 at 19:21

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