It's easy to derive a presentation of the fundamental group of a 4-manifold if you have a Kirby diagram: The 1-handles are generators and the 2-handles are the relations. The 3- and 4-handles, which are invisible in the Kirby diagrams, don't contribute.


*Is there anything like this for the homotopy 2-type?*


I'm imagining an algorithm that takes a Kirby diagram and spits out a groupoid, or a crossed module. Probably, the 1-handles generate the 1-morphisms, the 2-handles the 2-morphisms, and the higher handles are relations? How can the relations be visualised, though?



Given a Kirby diagram of a 4-manifold, what's its homotopy 2-type?
Yes: given a CW-complex $X$ the fundamental crossed module $\Pi\_2(X,X^1)=(\partial \colon \pi\_2(X,X^1) \to \pi\_1(X^1))$, where $X^1$ is the 1-skeleton, represents the homotopy 2-type of $X$ (this can be stated in several ways). Moreover $\Pi\_2(X,X^1)$ can be calculated combinatorially: $(\partial \colon \pi\_2(X^2,X^1) \to \pi\_1(X^1))$ is a totally free crossed module, and when you attach three handles you solely need to impose relations on $\pi\_2(X^2,X^1)$ in order to get to $\pi\_2(X,X^1)$ 


Now if you have a Kirby diagram (i.e. a handlebody decomposition), then squashing the handles along their core yields a CW-complex. So in theory the fundamental crossed module of the associated CW-complex can be calculated combinatorially. Except that it might be a non-trivial exercise to determine the attaching maps of the 3-handles in such that way that the relations on $\pi\_2(X^2,X^1)$ become transparent. (For complements of knotted surfaces this is quite a doable thing.)


Some discussion is in J. Faria Martins, The fundamental crossed module of the complement of a knotted surface, Trans. Amer. Math. Soc. 361 (2009), no. 9, 4593–4630 <https://arxiv.org/abs/0801.3921>


and also in: *Higher lattices, discrete two-dimensional holonomy and topological phases in (3+1) D with higher gauge symmetry* (<https://arxiv.org/abs/1702.00868>) section 3.4
and *On 2-Dimensional Homotopy Invariants of Complements of Knotted Surfaces* (<https://arxiv.org/abs/math/0507239>)


