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Let $M$ be a connected compact closed 4 manifold. Then $H_4(M)=\mathbb{Z}$. If we assume it is smooth, from Morse theory we know that $M$ has a CW structure. But can we find a CW structure of $M$ with only one 4-cell?

Moreover, assume we drop the smoothable condition of $M$. I would like to know under what circumstances $M$ is homotopy equivalent to a CW complex with only one 4-cell? Thank you.

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    $\begingroup$ Yes, you can find a decomposition with a single $4$-cell, a single $0$-cell and other cells in dimensions 1,2,3. $\endgroup$ – Jim Conant Apr 27 '16 at 19:31
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    $\begingroup$ This was already asked (by myself) in more generality: mathoverflow.net/questions/120799/… . In particular, it answers both your questions! $\endgroup$ – Chris Gerig Apr 27 '16 at 22:49
  • $\begingroup$ @ChrisGerig I read your link and mathoverflow.net/questions/42234/rugged-manifold. But I cant find the paper online, so Im not sure whether it suits in my case. $\endgroup$ – Larry So Apr 28 '16 at 14:36
  • $\begingroup$ The paper does not quite suffice when $M$ is a non-smoothable 4-manifold. From my link, it is known if $M$ is furthermore simply-connected. $\endgroup$ – Chris Gerig Apr 28 '16 at 17:32
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For the smooth case:

Via Morse theory the claim is equivalent to having a Morse function with only one maximum, or only one minimum, or to have a handle decomposition with only one 0-handle.

Assume you have several 0-handles. By connectedness they have to be connected via 1-handles. Smale's handle cancellation says that a k-handle can be canceled against a k+1-handle if the belt sphere of the former intersects the attaching sphere of the latter in one point only. This condition is trivially true for k=0. So you can cancel all but one 0-handle.

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I think the topological case is handled by this:

Brown, Morton A mapping theorem for untriangulated manifolds. 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 92–94

According to the review on MathSciNet, it shows that an $n$-manifold $M$ has the form $M = X \cup_\alpha D^m$ where $\dim(X) < n$. I don't know if $X$ can be taken to be a CW complex.

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  • $\begingroup$ Unfortunately, it doesn't help (or isn't known) to get a CW-complex when $M$ is a non-smoothable 4-manifold. $\endgroup$ – Chris Gerig Apr 28 '16 at 17:34

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