# 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?

## 1 Answer

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)

• 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! – Stefan Friedl Oct 14 '20 at 20:56
• @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 – Danny Ruberman Oct 14 '20 at 21:01
• Haha, very good. The photo is one of great classics of geometric topology. – Stefan Friedl Oct 15 '20 at 20:12
• the photo is fantastic – Bruno Martelli Oct 19 '20 at 8:35