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Given a compact smooth manifold $M$ denote by $b_i(M)$ the $i$-th Betti number and denote by $q_i(M)$ the minimal number of generators for $H_i(M)$. Let $f$ be a Morse function on $M$. The Morse inequalities say that the number of critical points of index $k$ equals at least $b_k(M)+q_k(M)+q_{k-1}(M)$.

If $M$ is simply connected one can ask whether the Morse inequality is sharp, i.e. whether exists always a Morse function such that the inequalities become an equality for every $k$. Smale showed that if $\dim(M)\geq 6$, then one can always find such $f$. By the resolution of the Poincare conjecture the Morse inequality is sharp for simply connected 3-manifolds. In the 4-dimensional setting it's a famous open problem whether the Morse inequalities are sharp. Doing some literature search I could not find anything though on the status of 5-manifolds. Is it know that the Morse inequalities are (not) sharp? Or is that an open problem?

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Presumably you mean closed. Otherwise a non-trivial h-cobordism would not have the minimal number of critical points.

For closed simply connected manifolds, it seems to me that this is Corollary 2.2.2 of Barden (Simply Connected Five-Manifolds, Annals of Mathematics Vol. 82, No. 3 (Nov., 1965), pp. 365-385)

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  • $\begingroup$ Thanks. Great find. Of course I meant "closed". It's a metatheorem in topology that when it comes to manifolds one always forgets at least one adjective! $\endgroup$ – Stefan Friedl Oct 14 at 20:56
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    $\begingroup$ @StefanFriedl Not such a hard find. Look who is standing in front of me in this photo: msp.org/gtm/1999/02/photos/photo02.html $\endgroup$ – Danny Ruberman Oct 14 at 21:01
  • $\begingroup$ Haha, very good. The photo is one of great classics of geometric topology. $\endgroup$ – Stefan Friedl Oct 15 at 20:12
  • $\begingroup$ the photo is fantastic $\endgroup$ – Bruno Martelli Oct 19 at 8:35

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