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

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.

• Is the symbol $\triangle$ on purpose? (instead of $\Delta$) Jan 16 at 22:15
• @MartinBrandenburg: nope, should be just the usual $\Delta$-complex Jan 16 at 23:57
• This model for $EG$ is not a simplicial complex (as asserted in the question), since a simplex $[g_0, \dots, g_n]$ is only determined by its ordered set of vertices, not just its unordered set of vertices. A simplicial complex with only finitely many vertices must be finite dimensional, while $EG$ for finite, nontrivial $G$ must be infinite dimensional. Jan 17 at 18:50

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

• what do you mean by "stars of $p$-dimensional faces"? sorry, I haven't heard about this terminology before Jan 17 at 0:05
• I'll rewrite that. Jan 17 at 1:01
• alright, so just to expand the vocabulary: in a geometric realization of a simplicial complex a "star of a $p$-simplex" is by definition the union of the inner of all $(p+1)$-simplices, which have this $p$-simplex as one their faces? is it correct? Jan 17 at 12:29
• Maybe the star of a simplex $\sigma$ consists of all of the simplices of any dimension that have $\sigma$ as a face (including $\sigma$ itself), plus the faces (of any dimension) of these; the link of $\sigma$ consists of all of the simplices in the star that are disjoint from $\sigma$; and what I mean by the open star is the difference set: realization of the star minus realization of the link. Jan 17 at 14:16
• @TomGoodwillie Can you really assume that the $g_0, \dots, g_n$ are all distinct? This would make $EG$ finite dimensional for finite $G$. For $n=2$ I think you have to allow $\sigma = [g_0, g_1, g_2]$ with $g_0 = g_2 \ne g_1$, in which case the edge $e = [g_0, g_2]$ is degenerate and $U_0$ does not seem to be open, since its preimage in $\sigma$ contains the barycenter of $e$, but not a full neighborhood around it. Jan 17 at 18:40

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 $$G$$.

• could you loose some words on how to interpret the notation $\def\sq{/\!/} G\sq G→*\sq G$, ie what do you mean by this $A \sq B$ notation? Is this just some kind of "categorified" version of the usual nerve functor associating to a group a simplicial set? A naive guess: here you replace a usual group $G$ by a "group object" in some kind of fibered category (at least this object looks like one comming from there) and this "bulked" nerve functor maps to some kind functor with additional simplicial structure. Is this the rough idea behind this notation? Jan 17 at 0:20
• @user7391733: The notation // is just the action groupoid: ncatlab.org/nlab/show/action+groupoid. The nerve functor is the usual functor from small categories to simplicial sets. Jan 17 at 2:31
• @DmitriPavlov What is the canonical homeomorphism from |Sd(Sd(X))| to |X| that you have in mind for X = *//G? There is no natural such homeomorphism, cf. Rudolf Fritsch, Zur Unterteilung semisimplizialer Mengen. I., Math. Z. 108 (1969), 329–367. Jan 17 at 7:43
• @JohnRognes: The canonical homeomorphism of Fritsch–Puppe (Theorem 4.6.4 in Fritsch–Piccinini's Cellular structures in topology). It is is functorial with respect to isomorphisms of simplicial sets, which is all what is necessary here. Jan 18 at 6:09