Let $M\colon \partial\_- M \to \partial\_+ M$ be an oriented, compact cobordism. Assume that there is a handle decomposition with at most one 0-handle, and denote the handle bodies by $M\_i, i \in \{0,\dots 4\}$.


It is known ("On attaching 3-handles to a 1-connected manifold" by Bruce Trace) that if $M$ is simply connected and $\partial\_+ M$ is empty or connected, $M\_4$ is essentially determined by $M\_2$. Or put differently, $M$ is completely determined by $\partial\_- M$, $\partial\_+ M$ and $M\_2$.


What can we say for general $M$? Does the attaching map of a 3-handle contain nontrivial data? Are there Kirby diagrams with 3-handle attachments?
(I've never seen 3-handles represented in Kirby diagrams, probably because in many cases, only their number matters.)



Are there Kirby diagrams with 3-handles?
3-handles do matter. The reason why they're not usually there is because people often care about closed 4-manifolds; in this case, there is an essentially unique way of attaching all 3-handles to the boundary of (in your notation) $M\_2$, which is just $\#^n(S^1\times S^2)$ for some $n$. The result is due to Laudenbach and Poenaru, based on Cerf theory, if I recall correctly. You can find more precise references and more details in Gompf and Stipsicz's *4-manifolds and Kirby calculus*.


For open 4-manifolds, attaching 3-handles does create differences. A 3-handle is attached along a (0-framed) 2-sphere in a 3-manifold; such a sphere can be either separating or non-separating (in the boundary). Attaching a 3-handle along a separating one creates an extra boundary component (and creates $H\_3$); attaching along a non-separating one kills an $S^1\times S^2$ summand in the boundary.


