# Group cochains invariant under the action of the symmetric group

Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups $$C^k = \{f: G^k \to A\}$$ and the coboundary map $$\delta : C^k \to C^{k+1}$$ $$(\delta f)(g_1, \ldots, g_{k+1}) = f(g_2, \ldots, g_{n+1}) + \cdots + (-1)^i f(g_1, \ldots, g_i g_{i+1}, \ldots, g_{k+1}) + \cdots + f(g_1, \ldots, g_k)$$ for the group cohomology of $G$ with coefficients in $A$.

To each $k$-tuple $(g_1,\ldots, g_k)$ we can associate a labeling of the (oriented) edges of the standard $k$-simplex as follows. The oriented edge from vertex $i$ to vertex $j$, with $i<j$, is labeled by the product $g_{i+1}g_{i+2}\cdots g_j$, and the edge from $j$ to $i$ is labeled by the inverse of the $i$-to-$j$ label. The canonical action of the permutation group $S_{k+1}$ on the $k$-simplex leads to an action of $S_{k+1}$ on $G^k$. Define an action of $S_{k+1}$ on $C_k$ by $$(\sigma f)(g_1, \ldots, g_k) = (-1)^{|\sigma|} f(\sigma(g_1, \ldots, g_k)) .$$ ($|\sigma|$ denotes the parity of the permutation.)

One can show (unless I've made a foolish mistake) that any group cohomology class can be represented by a $k$-cochain which is invariant under the action of $S_{k+1}$. (Sketch of proof: consider a more parsimonious model for the classifying space $BG$.) So, for example, for 1-cochains we want $$f(g) = - f(g^{-1})$$ and for 2-cochains we want $$f(g,h) = f(h, (gh)^{-1}) = f((gh)^{-1}, g) = -f(h^{-1}, g^{-1}) = -f(g^{-1}, gh) = -f(gh, h^{-1}) .$$

My question: Where can I find a reference for this fact about group cohomology?

(The reason I care: "In the wild" one frequently comes across $k$-simplices with edges labeled as above, but these $k$-simplices don't come equipped with an isomorphism to the standard $k$-simplex. (i.e. there is not preferred ordering of the vertices.) To work with standard group cochains one would need to break the symmetry and choose such an isomorphism. I'd like to avoid doing that.)

• I guess the point is the isomorphism between cohomology defined with homogeneous and inhomogeneous cochains (you find it, for instance, in Neukirch-Schmidt-Wingberg's "Cohomology of Number Fields", Chap I, §2) - I can try to write down a precise answer if you do not see what I mean. By the way, don't you have a degree shifting in your definition of $C^k=\mathrm{Map}(G^{k+1},A)$? Also, I guess you have a typo in the definition of $\delta$, the intermediate term should be $(-1)^if(g_1,\dots,g_{i-1},g_{i+1},\dots,g_{k+1})$, shouldn't it? – Filippo Alberto Edoardo Jan 9 '14 at 16:49
• I think there are different conventions, related by the "upper triangular" change of coordinates $g_i' = g_1 g_2 \cdots g_i$. The version I gave agrees with the Wikipedia article, these notes, and various other sources. I don't have access to the book you mention, so if you wanted to expand on the homogeneous/inhomogeneous cochain remark, I would be interested to hear more. – Kevin Walker Jan 9 '14 at 18:53
• I don't know a published reference, but I addressed the analogous question for singular cohomology of arbitrary spaces here. – Eric Wofsey Jan 9 '14 at 21:50

in Yamagami's paper "Polygonal presentations of semisimple tensor categories" there is a complete appendix about the wedge resolution of the trivial $G$-module $\mathbb Z$ of a discrete groups $G$.