ScalingStacks

Let $W$ be a smooth, oriented, connected, compact four-manifold (possibly with boundary). By standard Morse theory, $W$ can be decomposed into $k$-handles for $k=0,\dots,4$, arranged according to their index $k$. Furthermore, without loss of generality, we can arrange so that there is a unique 0-handle, and the number of 4-handles is either $0$ or $1$, according to whether $W$ has empty boundary or not.

Denote the numbers of $1$-, $2$- and 3-handles by $m$, $n$ and $p$, respectively. After attaching the 1-handles to the 0-handle we get the handlebody $\natural^{m}(S^{1}\times B^{3})$, with boundary $\#^{m}(S^{1}\times S^{2})$. (Here, $\natural$ denotes the boundary connected sum, and $\#$ the usual interior connected sum.) The attaching circles for the 2-handles form a link

with components $K_{1},\dots,K_{n}$. The link also has a framing, which specifies how the 2-handles are attached. Once these are attached, the boundary of the resulting manifold must be of the form $Y\#p(S^{1}\times S^{2})$. Attaching the 3-handles gets rid of the $p$ summands of $S^{1}\times S^{2}$, so the resulting boundary is some $3$-manifold $Y$. In the case $\partial W\neq\emptyset$, we stop here and we have $\partial W=Y$. In the case where $W$ is closed, we must have $Y=S^{3}$ and we attach the 4-handle (a four-ball) to $S^{3}$ at the last step to eliminate the boundary.

The handle decomposition allows us to represent $W$ by a Kirby diagram. This consists of drawing $\#^{m}(S^{1}\times S^{2})$ as $m$ pairs of spheres in ${\mathbb{R}}^{3}$, where we think of the spheres in each pair as identified to produce a 1-handle (and we also add the point at infinity to ${\mathbb{R}}^{3}$). We then draw a picture of the attaching link $K$ for the 2-handles, where the link can go through the 1-handles. The framing of $K$ can be specified by drawing parallel copies of the components of $K$. (The components that don’t go through the 1-handles can be viewed as living in $S^{3}$; for those, an alternative way to specify the framing is by an integer, which is the difference between the given framing and the Seifert framing.) To determine $W$, in principle we should also specify the attaching spheres for the 3-handles. These are usually not drawn in the Kirby diagram. In the case where $\partial W=\emptyset$, this leaves no ambiguity, because there is a unique way to fill $\#^{p}(S^{1}\times S^{2})$ by 3-handles and then by a 4-handle.

For example, we show here a Kirby diagram of $W=\mathbb{CP}^{2}\#\mathbb{CP}^{2}$ with one 1-handle and three 2-handles. For the attaching curve of the 2-handle that goes through the 1-handle, we specified the framing by drawing a parallel copy by a dashed curve; for the other 2-handles, we used numbers:

For more details about the subject, we refer to the book [10].