Visualising $\mathbb CP^2$: a problem of attaching cells with a dimension gap >1

For the uninitiated

Morse theory, as many other early algebraic-topology widgets, leads to a picture of smooth manifolds as being built up from 'cells', copies of $\mathbb{D}^n$ for varying $n$, 'glued' to each other by the usual topological tools; giving rise to (in some sense) a more natural picture of homology as 'coming from' cellular homology.

Example

As an example, consider the torus $\mathbb{T}^2$: we begin with empty space, attach a 0-cell ($\mathbb{D}^0=$ a point), attach a 1-cell ($\mathbb{D}^1=$ a line) to your point (both ends of the line are attached to the point, creating a circle), attach another 1-cell (in the same way, to the same point, creating a sort of figure 8).

The hardest bit to visualise is next: attaching a 2-cell ($\mathbb{D}^2=$ a disk, which we will think of as its homeomorphism equivalent, a square). Begin by twisting your figure 8 so that one circle is in the xy-plane, the other in the xz-plane, now attach the top and bottom of your square (coloured red in picture below) to the xy circle (creating a 'curling round' tube) and the left and right edges (coloured blue below) of your square (now a tube) either side of the xz circle, completing the torus.

Problem

The above takes some thinking, but a little reading around shows that this is fairly easy to see. What makes it so easy is that the cells we are attaching are of adjacent dimensions, that is; we may easily identify the boundary of one with the entirety of another. Where it gets harder to visualise is when the dimensions of the cells we are attaching to one another differ by >1- the canonical example of this is the complex projective plane $\mathbb{CP}^2$, a 4-manifold built by attaching a disk to a point (making a sphere) and then attaching a 4-ball to that sphere.

The latter attaching map (wherein points are identified with their images), I know, may be thought of as the Hopf fibration $\partial \mathbb{D}^4=S^3 \to \mathbb{CP}^1=S^2 $, but I have no way of visualising this, particularly with regard to the interior of the 4-disk.

How does a 4 cell wrap around a 2 cell without producing a singularity of some kind? Is this analagous in some sense to Dehn surgery in which one uses a thickening? Is there a right way to think about this or can it only really be thought of 'intellectually'?

1) Near a point of $S^2$ the picture looks like $\mathbb{C}$ sitting inside, naturally $\mathbb{C}^2$ (because $\mathbb{C}P^2\cong(\mathbb{C}^3\setminus{0})/\mathbb{C}^{\times}$ and $S^2=\mathbb{C}P^1$ is the image of a hyperplane in $\mathbb{C}^3$).

2) Think first about $S^2$: one attaches a disk to a point, contracting the circle on the disk's boundary. Now, to make $\mathbb{C}P^2$ one takes 4-disk and do pretty much the same but not for one circle but for all, well, fibers of the Hopf fibration at once — or, in other words, for all points of $S^2$ at once. So, since there were no singularities after gluing $S^2$, there will be no singularities here either.

(Not sure if it's an answer, but only hope it helps.)

Tom, I'm not sure I see how it's getting any harder in passing from a torus to a projective space. In your $\mathbb CP^2$ case, you have $\mathbb CP^1$ sitting inside of it, and the boundary of a regular (tubular) neighbourhood of the $\mathbb CP^1$ is $S^3$. And $D^4$ has $S^3$ as its boundary, so the attaching map is tautological. The normal bundle is the missing data and that's what your CW-decomposition is ignoring.

This is essentially what always happens. Perhaps the conceptual hump you're dealing with is that you're asking for CW-decompositions of manifolds. Morse functions generically only build homotopy-equivalences to CW-complexes, they do not put CW-structures on the manifold without some significant work. Moreover, CW-decompositions ignore some of the most essential properties of the manifold, like smooth structures.

If instead you work with handle decompositions, what I state in my first paragraph is basically a generality -- critical points amount to handle attachments and the gluing instructions are always given in a direct way from the flow lines of the Morse function's (suitably normalized) gradient. So the handle decomposition is on the given manifold -- unlike the CW-case where you only have a homotopy-equivalence to a CW-complex.