### Group Cocycles and Simplices

#### Posted by Urs Schreiber

**Christoph Wockel** asks me to forward the following question to the $n$-Café:

What is a reference for a generalisation of the following canonical cocycle to higher dimensions?

Let $G$ be a connected (locally contractible) topological group and take a section $a : G \to P G \,,$ continuous on a neighbourhood of the neutral element, of the endpoint evaluation map $ev : P G \to G$ from the space $P G$ of continuous paths starting at the neutral element of $G$. Then the assignment $(g,h) \mapsto [a(g)+g \cdot a(h)-a(g h)]$ is a $\pi_1(G)$-valued group 2-cocycle on $G$, describing the universal cover of $G$ as a central extension. Moreover, this cocycle is universal for discrete groups.

A similar construction works in higher dimensions, yielding for each $(n-1)$-connected group $G$ a $\pi_n(G)$ valued group $(n+1)$-cocycle. Moreover, this cocycle is universal for discrete groups. I’ve been searching for a reference for this, but did not succeed.

Thanks for any hints.

My remark: notice that the construction this question is about is closely related to the construction of Čech cocycles for characteristic classes by Brylinski and McLaughlin which I talked about here.

## Re: Group Cocycles and Simplices

You’re saying there’s a god-given $\pi_n(G)$-valued $(n+1)$-cocycle on a sufficiently nice topological group $G$.

Instead of focusing on the specific cocycle, let’s think about its cohomology class. Then you’re saying that if $G$ is sufficiently nice, there’s a god-given element of

$H^{n+1}(G,\pi_n(G))$

This is group cohomology. But this is isomorphic to

$H^{n+1}(K(G,1),\pi_n(G))$

where $K(G,1)$ is the 1st Eilenberg–Mac Lane space of $G$.

But cohomology is represented by Eilenberg–Mac Lane spaces, so the above group is isomorphic to

$[K(G,1),K(\pi_n(G),n+1)]$

where the square brackets mean ‘the set of homotopy classes of maps’.

I would like to keep chewing away on this until I see that the god-given element you’re talking about is something like an identity map in disguise! But I seem to be stuck right now.

If we take advantage of your assumption that the topological group $G$ is $(n-1)$-connected, then we can use the Hurewicz theorem, which says that in this case

$\pi_n(G) \cong H_n(G)$

I’m not sure how this helps, but I thought I should mention it.