Is it true that any simply connected closed manifold possesses a Morse functions that does not have critical points of index one?


If the dimension is at least 5, this is a consequence of the results from Milnor's "Lectures on the $h$-cobordism theorem", but what about dimensions 3 and 4?


If such a function does not exist in general in low dimensions, are there reasonable topological conditions for the existence of a Morse function without critical points of index one?



Existence of Morse functions on simply connected manifolds
It is still an open and very interesting question in dimension 4. Akbulut (The Dolgachev surface. Disproving the Harer-Kas-Kirby conjecture. Comment. Math. Helv. 87 (2012), no. 1, 187–241) showed that the Dolgachev surface (and [subsequently](http://arxiv.org/abs/0906.3107) other elliptic surfaces in the same homotopy type) has a handle decomposition with no 1 or 3 handles. 


In the opposite direction, [Rasmussen](http://arxiv.org/abs/1005.4674) gave an argument that an exotic homotopy $S^2 \times S^2$ detected by an Ozsvath-Szabo (or presumably Seiberg-Witten) invariant would require either 1 or 3-handles. That paper is currently withdrawn from the arxiv; quoting the abstract, `I am withdrawing the paper because it is unclear to me if such a manifold exists.'


