# Handlebody decomposition of an open 4-manifold

Let $M$ be the fake $CP^2$ (namely the closed topological 4-manifold which is homotopy equivalent but not homeomorphic to the complex projective plan). It is well-known that $M$ admits no smooth structures. However $M' = M - pt$ is smoothable by a theorem of Quinn. The question is: does someone know an explicit handlebody decomposition of $M'$?

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Here is a sketch. Start with a 4-ball, and then attach a (+1) framed 2-handle along a trefoil knot. If you've chosen the correct trefoil, the boundary of the resulting manifold is the Poincare homology sphere, say P. Then Freedman tells you that P is the boundary of a (topological) contractible manifold W, which you glue on to make the exotic $CP^2$, commonly known as Ch (for Chern). The reason that you don't need Quinn is that the construction of W is done by making a manifold $W'$ that is proper homotopy equivalent to $S^3 \times [0,\infty)$, and then using the proper h-cobordism theorem to recognize that the end of $W'$ is homeomorphic to $S^3 \times (1,\infty)$, from which you see that you can compactify $W'$ to a manifold by adding in a point.
In the case at hand, you can be more explicit. One way (this is what happens in Freedman's original paper) is to consider the building block $P \times I$, and then do (spin) surgery on a circle to kill the fundamental group, resulting in a compact manifold $P'$. The embedding theorems of Freedman find a (topological, locally flat) wedge of spheres in $P'$, which can be surgered out to give a compact, simply-connected homology cobordism from $P$ to itself. Stacking infinitely many of these gives $W'$. Presumably, although I've never done this, you can use techniques of Bizaca to build a handlebody picture of $W'$ from this description.
An alternate approach, which might be more amenable to drawing pictures, would come from Freedman's older paper, A Fake $S^3 \times R$. In this paper, which precedes his disk embedding theorem (but has many of the basic ideas, including reimbedding techniques) he constructs what I've called $W'$ by embedding Casson handles. There is also a Bourbaki exposition of this paper by Siebenmann that is helpful in trying to read it.