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?