am currently working through J.Rosenbergs construction of classifying Spaces for the +-construction of higher K-Theory. He defines a CW-complex EG as the direct limit of the inductively defined Spaces $$ X_n = X_{n-1} \ast G$$ where $X_0 = G$ with the discrete topology and $\ast$ is the join of topological spaces. My questions are:
How do I show that $X_n$ is a CW-complex for every $n$? I know the cell-decomposition: every $X_n^j$ arises from $X^{j-1}_n$ by attaching the following cells:
- for $j>0$ the $j$-cells of $X_{n-1}$ and the $j-1$-cells of $X_{n-1}$ joint with $g \in G$ or
- for $j = 0$ the $0$-cells of $X_{n-1}$ disjoint with a copy of $G$.
but how do I show the Pushout, that formally shows that every $X_n^j$ arises from $X_n^{j-1}$ by attaching $j$-cells?
The obvious $G$-operation on $EG$ is free and properly discontinuous. I want to show that $BG=EG/G$ is also a CW-complex, so I want to show that it even is a $G$-CW-complex. Do I know for $G$-CW-complexes $X$, that $X/G$ is again a CW-complex?
I would be grateful for every help.
Yes ,If you have $G$- CW complex then $X/G$ is a genuine CW - complex.Because let $X$ and $Y$ be two $G$ -spaces and $f : X \rightarrow Y$ be a $G$-map and $C_f$ be cofibre of it. Then $C_f/G$ is the cofibre of $X/G \rightarrow Y/G$.