# Existence of Morse functions on simply connected manifolds

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?

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 That paper is currently withdrawn from the arxiv; quoting the abstract, `I am withdrawing the paper because it is unclear to me if such a manifold exists.'
In dimension $3$ we know from Perelman's work that a simply connected $3$-manifold is a sphere. In dimension $4$ I quote Kirby in 1989
"It is not known if a simply connected $4$-manifold needs $1$-handles and/or $3$-handles but the Dolgachev surface is a good candidate for needing them." page 8, R. Kirby: The Topology of $4$-Manifolds, Lect. Notes in Math. vol. 1374.