Chromotopy

February 07, 2010, by Eric Peterson

Apparently sometime last semester I agreed to give a talk for the undergraduate math honors society here at UIUC. I’d completely forgotten, until their chair reminded me to come up with a topic a week or two ago. It’s easiest to give a talk about something you’ve been thinking about already, so I tried slapping two things together:

Probably the most important thing I learned in the past year was to think of C-bundles over a base space X as being equivalent, via a homotopy colimit functor, to $\Pi_\infty X$ -indexed homotopy coherent diagrams in C, where C is something like a quasicategory built from a simplicially enriched and tensored model category.
I also spent two years of my undergraduate life thinking about certain classes of affine group scheme extensions, so I also have an unusual affection for the extension problem in group cohomology.

These ideas are even related – given two groups H and N, you can think of them as homotopy 1-types (sometimes written BH and BN), and an extension of groups $0 \to N \to G \to H \to 0$ is “the same” as a principal BN-bundle over BH. Principal BN-bundles over BH are in turn “the same” as the pseudofunctors $BH \to \mathrm{Aut}\;BN$ classifying them as per the above, and it turns out that what “pseudofunctor” means translates into concrete algebraic conditions. Namely, such a pseudofunctor is specified by a pair of functions $u: H \times H \to N$ and $\alpha: H \to \mathrm{Aut}\;N$ , satisfying

$u(e, e) = e$ ,
$\alpha_e(n) = n$ ,
$\alpha_{xy}(n)u(x, y) = u(x, y)\alpha_x(\alpha_y(n))$ , and
$u(x, yz)u(y, z) = u(xy, z)\alpha_z(u(x, y))$ .

Such a pair is called a “Dedecker cocycle,” and the general framework goes by the name “Schreier theory,” due to Otto Schreier, who discovered some ghost of it in 1926. I find the topological flavor of this and its ability to cope with the nonabelian case quite interesting!

So that’s what I claimed I’d talk about. I think the idea for giving a single talk to undergraduates is to get across some cute facts that make use of relatively digestable material – but my idea of what “cute” and “digestable” mean are skewed quite far from the norm. To correct for this, the plan is to address only the abelian case, use lots of analogies and pictures, explicitly describe the 2-cocycles u associated to the extensions

$\mathbb{Z}/4 \to \mathbb{Z}/2,$ $\mathbb{Z}/2 \oplus \mathbb{Z}/2 \to \mathbb{Z}/2,$

and show that the former’s cocycle is related to the sum-and-carry algorithm in binary. It’s probably asking a little much from them, but it’s an honors society, and giving a talk that everyone understands bottom to top would end up being boring. I typed up some notes here, and I’m eager to take suggestions. Additionally, Blanco, Bullejos, and Faro wrote about this recently in some detail, so if you want all the gripping details, you can check out their arXiv posting.