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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?

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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 That paper is currently withdrawn from the arxiv; quoting the abstract, `I am withdrawing the paper because it is unclear to me if such a manifold exists.'

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  • $\begingroup$ Thanks a lot! I wouldn't have guessed that this question is so delicate... $\endgroup$ – Stephan Mescher Oct 3 '13 at 13:43
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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.

Many things have happened from the time Kirby made the above statement and I have not kept up with all the developments. In any case, the answer to your question is not straightforward.

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