TITLE: Existence of Morse functions on simply connected manifolds
QUESTION [13 upvotes]: 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?

REPLY [14 votes]: 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 other elliptic surfaces in the same homotopy type) has a handle decomposition with no 1 or 3 handles. 
In the opposite direction, Rasmussen 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  That paper is currently withdrawn from the arxiv; quoting the abstract, `I am withdrawing the paper because it is unclear to me if such a manifold exists.'