Milnor's infinite-join model of EG

Let $G$ be a topological group. All products, quotient spaces, and actions below use the stated ordinary topologies.

For $N\ge 0$, let $J_N^{\mathrm{q}}=G^{\ast (N+1)}$ denote the ordinary quotient of ${\mathrm{\Delta }}^N\times G^{N+1}$ in which

$(t_0,\dots ,t_N;g_0,\dots ,g_N)\sim (t_0,\dots ,t_N;g_0^{\mathrm{\prime }},\dots ,g_N^{\mathrm{\prime }})$

exactly when $g_i=g_i^{\mathrm{\prime }}$ for every $i$ with $t_i>0$. We write its points as finite formal sums ${\sum }_{i=0}^N{t}_i{g}_i$. Appending a zero coordinate gives the usual inclusions of these finite quotient joins.

As a set, Milnor's infinite join is the increasing union

$EG=G\ast G\ast \cdots ={\bigcup }_{N\ge 0}{J}_N^{\mathrm{q}}\mathrm{.}$

Every point therefore has an expression

$x={\sum }_{i\ge 0}{t}_i{g}_i,t_i\ge 0,{\sum }_i{t}_i=1,$

with only finitely many $t_i$ nonzero; a label $g_i$ is ignored when $t_i=0$. We choose one of two ordinary topologies on this set, according to $G$:

• If $G$ is compact Hausdorff, $EG$ has the ordinary weak direct-limit topology of the compact finite quotient joins $J_N^{\mathrm{q}}$: a set is open exactly when its intersection with every $J_N^{\mathrm{q}}$ is open. These are compact Hausdorff stages with closed inclusions. This branch makes the circle and two-point-group models the standard weak CW unions of their finite joins.
• Otherwise, $EG$ has Milnor's ordinary coordinate-label strong topology: the coarsest topology for which every barycentric function $t_i:EG\to [0,1]$ and every partial label function $g_i:\{t_i>0\}\to G$ is continuous. A map from any ordinary topological space into this strong join is continuous exactly when all its weights and all its labels on their positive-weight loci are continuous. This is not the weak direct-limit topology; the subspace topology on a finite-stage set need not equal the quotient topology of $J_N^{\mathrm{q}}$.

In both branches the weights $t_i$ and partial labels $g_i$ are continuous; in the weak branch this follows by checking their restrictions to the finite quotient stages. For compact Hausdorff $G$, each finite strong join and finite quotient join agree because the latter is compact and the former Hausdorff. Neither branch is additionally kified. The noncompact strong branch is an explicit ordinary-Top exception to the standing CGWH convention; all bundle charts and homotopies use ordinary products, as required by the library's ordinary bundle definition. We do not silently replace an ordinary product by a k-product.

The diagonal right action is

$\left({\sum }_i{t}_i{g}_i\right)h={\sum }_i{t}_i(g_{i}h)\mathrm{.}$

Define $BG=EG\mathrm{/}G$ with the ordinary orbit-quotient topology, and let $p:EG\to BG$ be the orbit map. Each $t_i$ is invariant and hence descends to a continuous function, again denoted $t_i$, on $BG$. We use the identity-labelled vertex in coordinate $1$ as $e_0\in EG$ and its orbit as $b_0\in BG$.