Construct a 'nice' trivializing cover of universal principal $G$-bundle $EG \to BG$

Question

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 where it not received any resonance.

Answer 1

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

Answer 2

Given that the map $\def\E{{\sf E}}\def\B{{\sf B}}\E G→\B G$ is the geometric realization of a simplicial covering map (namely, the nerve of the functor $\def\sq{/\!/} G\sq G→*\sq G$), the canonical trivializing open cover is quite easy to construct: take the open stars of vertices in the geometric realization of the second subdivision of the nerve of $*\sq G$. This geometric realization is canonically homeomorphic to $\B G$ via the barycentric map.

The resulting canonical open cover trivializes the covering map $\E G→\B G$ and has additional desirable properties. For example, it is a good cover: every finite intersection is empty or homeomorphic to $\def\R{{\bf R}}\R^n$. This construction works for any discrete group $G$.