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!

pairof spaces $(X,A)$ with a set $C$ of base points. This enabled us to prove a 2-d van Kampen theorem, which gave new info on second relative homotopy groups. There are lots of pictures in the new book reviewed (pdf of the book on the web page of the book). $\endgroup$ – Ronnie Brown Oct 18 '12 at 16:00