Is it true that any manifold homotopy equivalent to a k-dimensional CW-complex admits a proper Morse function with critical points all of index <= k? I believe this is not true, so I would like to see a counterexample.
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2 Answers
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Take a contractible $3$-manifold which is not homeomorphic to $\mathbb R^3$ -- like the Whitehead manifold.
If such a Morse function existed on the Whitehead manifold, it would be a Morse function with only one critical point, the minimum, and therefore the Whitehead manifold would be an open 3-ball. The proof of this has two steps: (1) $f$ can have at most one critical point, WLOG a minimum by homotopy-type considerations, and it can't have less than one critical point by the "edit" below. Step (2) if it has one critical point let it be $f(p)=0$, the minimum. By the Morse Lemma, $f^{-1}[0,\epsilon]$ is diffeomorphic to a closed 3-ball. The flow the the gradient of $f$ gives a diffeomorphism between $f^{-1}[\epsilon,\infty)$ and $S^2 \times [\epsilon, \infty)$. Pasting these two diffeomorphisms together gives you a diffeomorphism between the manifold and $\mathbb R^3$.
edit: Well, I guess there's the special case to consider that the Morse function could have no critical points but then you could deal with this by the argument that the Whitehead manifold isn't a product of a surface and $\mathbb R$, which by the classification of surfaces amounts to saying that the Whitehead manifold isn't homeo/diffeomorphic to $\mathbb R^3$.
Welcome to MO, Victor!
answered Oct 5, 2010 at 17:37
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$\begingroup$ Can one find a counterexample among interiors of compact manifolds with boundary? $\endgroup$– VictorOct 7, 2010 at 14:40
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1$\begingroup$ I believe it's unknown whether or not there are exotic smooth structures on open 4-balls that extend to smooth structures on a compact 4-ball so that would be a potential counter-example for $k=0$. Less hypothetically, Mazur manifolds are counter-examples. In high dimensions if your manifold and its boundary are simply-connected the minimal handle presentation theorem would kick in and say there are no counter-examples. But once you have $\pi_1$ in either the manifold or the boundary there are potential obstructions. I wrote up a little thing on Mazur manifolds on Wikipedia. $\endgroup$ Oct 7, 2010 at 16:12
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$\begingroup$ In full generality there's likely a natural Whitehead-torsion obstruction to doing what you want to do, given by the s-cobordism theorem. That would require more thought than I'm prepared for this morning. $\endgroup$ Oct 7, 2010 at 16:28
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$\begingroup$ Thank you Ryan. As usual you give a very interesting answer. In relation with that, don't you know what is known about the configuration spaces of these manifolds? And what about their embeddings into $R^3$, $R^4$, $R^5$...? $\endgroup$– VictorOct 8, 2010 at 16:44
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$\begingroup$ Mazur manifolds (at least, ones that can be represented as $S^1 \times D^3$ union a $2$-handle) are all smooth submanifolds of $S^4$. On the Wikipedia page the reason is sketched in the section titled "Mazur's Observation". When you refer to configuration spaces, are you trying to go after a Salvatore-Longoni type problem, looking at the homotopy-type of configuration spaces of a Mazur manifold, that kind of thing? $\endgroup$ Oct 8, 2010 at 21:01
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Suppose that $M$ has a proper Morse function $f\ge 0$ with all critical points of index at most $k$. Then homotopically $M$ can be made of low-dimensional cells: it has homotopical dimension at most $k$. But also $M$, relative to its boundary $M^{\ge c}$, can be made by attaching high-dimensional cells. Thus the pair $(M,M^{\ge c})$ is $(dim(M)-k-1)$-connected. So for example if $k\le dim(M)-3$ and $M$ is simply connected then $M$ must also be "simply connected at infinity".
answered Oct 6, 2010 at 3:59
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$\begingroup$ Does it mean that the one-point compactification of any manifold is always (dim(M)-k-1)-connected? By "any" I mean not only those which are interiors of compact manifolds with boundary. Perhaps my question should be formulated as "is the homotopy inverse limit of m-connected spaces m-connected provided all maps are m-connected"? $\endgroup$– VictorOct 7, 2010 at 14:37