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.
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.
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.
