## Second homotopy groups of 3-complexes and Fenn’s spiders.

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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!

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The fundamental 2-groupoid of a CW-complex admits a presentation (in the category where it lives) analogous (indeed extending) to the presentation of the fundamental group. This is the philosophy of relations among relators discovered by Whitehead long ago. He used crossed modules, which are equivalent to 2-groups. – Fernando Muro Oct 17 at 9:40
Some references to early sources (van Kampen, Reidemmeister, ...) can be found in the review pages.bangor.ac.uk/~mas010/pdffiles/… . The book reviewed gives colimit methods for calculating second relative homotopy groups (in fact homotopy 2-types) and hence some calculations of $\pi_2$.  – Ronnie Brown Oct 17 at 9:44
I am not sure what Geordie and Fernando understand by "the fundamental 2-groupoid of a CW-complex".  I find 2-groupoids not easy to manage compared with crossed modules (over groupoids), and not useful for proving theorems compared with double groupoids with connections, and these two are well defined for  pairs with a set of base points, or more generally for filtered spaces.  – Ronnie Brown Oct 17 at 10:36
@Ronnie: by fundamental 2-groupoid I mean the 2-groupoid with objects points, 1-morphisms paths between points, and 2-morphisms paths between paths up to homotopy. (A truncation of the fundamental $\infty$-groupoid). I am a complete novice at these things (hence the question), and am sure you are right that other models are easier to work with. Thank you for the link to the review, which looks interesting. – Geordie Williamson Oct 18 at 7:38
@Geordie: Philip Higgins and I really made progress in 1974 when we considered the (strict!) homotopy double groupoid (with connections) of a pair of 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).  – Ronnie Brown Oct 18 at 16:00

## 2 Answers

I am not sure (see the paper at www.cmi.univ-mrs.fr/~hamish/Papers/crmshort.pdf) 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 , in Une dégustation topologique: Homotopy theory in the Swiss Alps , volume 265 of Con- temporary 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 alge- braic K-theory and in the space of Morse functions , 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.

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Thank you very much for these excellent references. I seems Igusa's pictures are exactly Fenn's diagrams. Igusa says that the observation that one obtains a description of $\pi_2$ in this way is "essentially due to J. H. C. Whitehead". Also, the paper of Loday is exactly what I was hoping would exist somewhere in the literature. – Geordie Williamson Oct 18 at 7:32
I wrote up the ideas of Whitehead on free crossed modules in  30.  On the second relative homotopy group of an adjunction space: an  exposition of a theorem of J.H.C. Whitehead'', _J. London Math. Soc._ (2)  22 (1980) 146-152. Analogous ideas were developed independently by Peiffer and Reidemeister; the book review pages.bangor.ac.uk/~mas010/pdffiles/… gives good historical background.  – Ronnie Brown Nov 5 at 11:00

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The web site Homological algebra programming by Graham Ellis gives methods of constructing resolution of groups; the basic idea is to construct inductively a universal cover of a $K(G,1)$ together with a contracting homotopy, each inductive step gives another "home" for a contracting homotopy. This method is a higher dimensional version of constructing a tree in a Cayley graph, and is more computational than the traditional "killing kernels".

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