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?



Are the Morse inequalities sharp for 5-manifolds
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)


