# Are the Morse inequalities sharp for 5-manifolds

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?