Let $X$ be a finite CW complex then with one zero cell. Then (up to homotopy) the two skeleton of X is the same as a group presentation, via the Cayley complex construction. For a while I had been searching for some planar description of the second homotopy group, which would allow a concrete combinatorial description of the fundamental 2-groupoid of X (up to equivalence).


I found many discussions close to what I needed, before stumbling on the (IMHO beautiful) book "Techniques of geometric topology" by Roger Fenn. In Chapter 2 he gives a description of $\pi\_2(X)$ of a 3-complex in terms of certain diagrams modulo local relations. Each relation in the 2-complex gives a "relation spider" and the second homotopy group of $X$ is the group of isotopy classes of planar diagrams generated by these spider diagrams modulo certain "universal" local relations (analogous to $gg^{-1} = 1$ in the $\pi\_1$ case) and relations given by the 3-cells of $X$. (The spider diagrams are roughly dual to Van Kampen diagrams.)


My questions are:


1) are spider diagrams Fenn's invention? Perhaps this way of thinking about $\pi\_2$ was folklore?


2) what are other sources describing $\pi\_2$ (or even better the fundamental 2-groupoid) concretely (ideally diagrammatically) for small dimensional complexes?


I am aware that all of this can be viewed as a concrete example (for $n = 2$) of the dictionary between n-groupoids and n-types. However because of the applications I have in mind I am only looking for "concrete" sources!



Second homotopy groups of 3-complexes and Fenn's spiders.
I am not sure (see the notes ["Diagrams and groups"](https://old.i2m.univ-amu.fr/~short/Papers/crmshort.pdf) by Hamish Short) but I think these are related to Igusa's pictures. 


There is a nice paper by Loday on the idea of homotopical syzygies (J.-L. Loday, 2000, [Homotopical Syzygies](http://irma.math.unistra.fr/~loday/PAPERS/2000Loday%28syzygy%29.pdf), in Une dégustation 
topologique: Homotopy theory in the Swiss Alps, volume 265 of Contemporary Mathematics, 99–127, AMS.) which may help and also the paper by Kapranov and Saito (M. Kapranov and M. Saito, 1999, [Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions](https://faculty.math.illinois.edu/K-theory/0192/K3.pdf), in Higher homotopy structure in topology and mathematical physics (Poughkeepsie, N.Y. 1996), volume 227 of Contemporary Mathematics, 191–225, AMS.) which is worth reading. 


The situations in these papers relate to when the 3-complex is to be constructed from its 2-skeleton by killing the $\pi\_2$ but they are I think relevant.


