Many authors state the conditions of a 1-cocycle $(g_{ij})$ as
- $g_{ii} = 1$,
- $g_{ij} = g_{ji}^{-1}$,
- $g_{ij}g_{ik}^{-1}g_{jk} = 1$.
Note that in this case the first two conditions are implied by the third. If we look instead at a 2-cocycle $(g_{ijk})$, the cocycle condition is $$ g_{ijk} g_{ij l}^{-1} g_{ikl} g_{jkl}^{-1} = 1. $$ This condition doesn't for example imply that $g_{iii} = 1$. In this paper by Karoubi, right before Remark 1.1 he defines a "completely normalised" 2-cocycle to be a 2-cocycle as above such that
- $g_{ijk}=1$ when two of $i,j,k$ are equal, and
- $g_{\sigma(i)\sigma(j)\sigma(k)} = (g_{ijk})^{\varepsilon(\sigma)}$ where $\sigma$ is a permutation of $(i,j,k)$ with signature $\varepsilon(\sigma)$.
These two conditions are similar to the two extra conditions in the definition of a 1-cocycle I stated above except now they aren't a consequence of the Cech 2-cocycle condition. I have seen some authors state similar conditions as being part of the definition of a 2-cocycle, for example page 3 of these notes by Hitchin. This confused me until I read Remark 1.1 in the paper I linked:
Remark 1.1. One can prove (see [21] for instance) that a Cˇech cocycle in any dimension is cohomologous to a completely normalized one. Moreover, if every open subset of X is paracompact, any cohomology class may be represented by a completely normalized Cˇech cocycle.
The paper [21] cited is Karoubi's own paper "Resolutions symetriques" which is unfortunately in French. So my questions are:
- Is this result (or a similar one) the reason why some can authors add these "normalisation" conditions to the definition of a cocycle?
- How might one "normalise" a cocycle, that is find a cohomologous cocycle that is normalised? Even just an English reference would be great.
Thanks in advance!