Handlebody decomposition of $L(2,1)\times S^1$ I wish to know the handlebody decomposition of $L(2,1)\times S^1$ in terms of Kirby diagrams, where $L(2,1)\cong RP^3$. And if possible, is there a general recipe for getting the handlebody decomposition of any 4-dimensional manifold, e.g. giving the cell decomposition of the manifold?
 A: The question has been essentially answered in the comments, but let me say a few words in any case, to take this off the unanswered queue.
One way to do this is to start with a Heegaard diagram for $\Bbb{RP}^3$ (left) and thicken it up to a Kirby diagram for $\Bbb{RP}^3 \times I$ (right). This gives a $1$-handle $a$ attached along a pair of balls in $\partial D^4 \subset D^4$ with a $2$-handle whose feet is the portion of the red curve visible on $\partial D^4$, with blackboard framing. One may alternatively think of deleting the pair of balls and pasting the boundaries by a reflection about a mirror perpendicular to the axis joining the centers of the pair of balls. The red curve represents the element $2$ in the fundamental group of $D^4 \cup a$.

The above is simply a thickening of the standard cell structure of $\Bbb{RP}^3$. To build $\Bbb{RP}^3 \times S^1$ from $\Bbb{RP}^3 \times I$ we attach a new $1$-handle $b$ whose feet are a pair of balls located on $\Bbb{RP}^3 \times \{0\}$ and $\Bbb{RP}^3 \times \{1\}$ respectively. These should be imagined as situated inside and outside the plane of the paper. Finally, attach a $2$-handle whose core is the $2$-cell of the subcomplex $\Bbb{RP}^1 \times S^1$. This corresponds to the word $aba^{-1}b^{-1}$, which is the class of the blue curve tracing out the feet of the $2$-handle.

I find it convenient to write $1$-handles as a circle-with-dot, which indicates carving out a disk bounded by an unknot (or more generally, slice disks). One way to translate a usual Kirby diagram to the dotted circle notation is to bring the two feet of a $1$-handle close until they become squished like a pancake, and replacing them with a dotted circle afterwards. I have done this procedure to the above diagram in stages.

Cleaning up the final diagram, we have the following. All the curves have blackboard framing. The red $2$-handle and $a$ form a thickened copy of $\Bbb{RP}^2$ inside, and the blue $2$-handle and $a, b$ form a thickened copy of $T^2$ inside. The blue and red curves have linking number $1$, confirming the fact that the unoriented intersection number of $\Bbb{RP}^1 \times S^1$ and $\Bbb{RP}^2 \times \mathrm{pt}$ in $\Bbb{RP}^3 \times S^1$ is $1$.

