Second homotopy groups of 3-complexes and Fenn's spiders. - MathOverflow

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

2012-10-17T08:46:16Z

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!

Answer by Tim Porter for Second homotopy groups of 3-complexes and Fenn's spiders.

2012-10-17T16:49:40Z

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.

Answer by Ronnie Brown for Second homotopy groups of 3-complexes and Fenn's spiders.

2012-10-17T20:08:39Z

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".