Let G be a discrete or say for sake of simplicity a finite group. In Hatcher's book Algebraic Topology on p 89 the construction of universal bundle $EG$ carries structure of a $\Delta$-complex whose $n$-simplices are the ordered $(n + 1)$ tuples
$[g\_0, ... ,g\_n]$ of elements of $G$. As
quotient space the model of the classifing space $BG=EG/G$ inherits structure $\Delta$-complex where a $n$-simplex $BG$ can be written uniquely in the form $[g\_1 \vert g\_2\vert ... \vert g\_n]:= 
G \cdot [e\_G, g\_1, g\_1g\_2,..., g\_1,.., g\_n]$. Note that by construction $EG$ is realized even as a simplicial complex, while $BG$ inherits only structure of honest $\Delta$-complex due to more subtle identificiations by passing to quotient.


My question is if there is a rather "canonical" way to choose a trivializing cover $\{U\_i\}\_{i \in I}$ of the universal principal $G$-bundle $EG \to BG$, ie a family of open subsets of $BG$ such that $EG \vert \_{U\_i} \cong U\_i \times G$ with relatively good controllable associated transition functions $h\_{ij}: U\_i \cap U\_j \to G$ dictating the patching structure $ U\_i \cap U\_j \times G \to U\_i \cap U\_j \times G, (u,g) \mapsto (u, h\_{ij}(u)g)$.


The question is closely cross posted from [MSE](https://math.stackexchange.com/questions/4608049/construct-a-trivializing-cover-of-universal-principal-g-bundle-eg-to-bg)  where it not received any resonance.



Construct a 'nice' trivializing cover of universal principal $G$-bundle $EG \to BG$
I think that this works.


EDIT: No, it doesn't. See John Rognes's comment.


Notation: For a point $x\in EG$ we may symbolically write $x=\sum\_{j=0}^nt\_jg\_j$, where $g\_0,\dots ,g\_n$ is an ordered tuple of distinct elements of $G$ and $t\_j\ge 0$ and $\sum\_jt\_j=1$. Here the face relations are accounted for by agreeing that if $t\_j=0$ for some $j$ then $g\_j$ can be omitted from the tuple and the term $t\_jg\_j$ can be omitted from the expression, without changing the point $x$.


Invariant open cover of $EG$: For $p\ge 0$ let $U\_p\subset EG$ consist of the points $x=\sum\_{j=0}^nt\_jg\_j$ such that there is a (necessarily unique) subset $S(x)\subset \{0,n\}$ of cardinality $p+1$ such that


(1) $t\_j>t\_k$ whenever $j\in S(x)$ and $k\notin S(x)$, and


(2) $t\_j>0$ whenever $j\in S(x)$.


Thus for each $n$-simplex $\sigma=[g\_0,\dots ,g\_n]$ the set $U\_p\cap\sigma$ is as follows. If $p\le n$ then it is the disjoint union of convex open subsets, one for each $p$-dimensional face $\tau\subset\sigma$, namely the open star, in the barycentric subdivision of $\sigma$, of the barycenter of $\tau$. It is empty if $p>n$.


This is well-defined (compatible with the face relations), and open. The union of the sets $U\_p$ is all of $EG$. For each $p$ the set $U\_p$ is invariant under the action of $G$.


Local trivializations: Now let's show that the bundle is trivial over the image of $U\_p$, by describing a cross-section. Let $V\_p\subset U\_p$ consist of those points $x=\sum\_jt\_jg\_j\in U\_p$ such that when $j\_0$ is the smallest element of the set $S(x)$ then $g\_j=e$. This is open in $U\_p$, and every point in $U\_p$ is uniquely expressible as $gx$ for some $g\in G$ and $x\in V\_p$.I think


