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?

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    $\begingroup$ I thought about this at some point but didn't really get anywhere. The problem is that the 3-handles actually do matter, already when you want to get a description of $H_2(X)$, for example. The best I could do is to describe $H_2(X)/torsion$ and the intersection form on this group for closed $X$. (This is written down in an appendix to my thesis, but probably also in other places.) [to be continued...] $\endgroup$ Feb 9 '17 at 14:33
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    $\begingroup$ To get $\pi_2(X)$ with all its structrure ($\pi_1$-module, equivariant intersection form) the obvious thing to try is to produce an infinite Kirby diagram for the universal cover and take it from there. But this looked like a combinatorial nightmare and I soon stopped trying. Of course, this doesn't mean that it's impossible to get information in this way, only that I didn't manage to do so. I'd still be interested in an answer to the question, though. $\endgroup$ Feb 9 '17 at 14:33

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)


A step towards what you seem to be imagining may be in the following paper (available here):

J. Faria Martins, The fundamental crossed module of the complement of a knotted surface, Trans. Amer. Math. Soc. 361 (2009), no. 9, 4593–4630

As the title states this looks at the case of the complement of a knotted surface. I do not know if that helps.


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