A way to construct a K(G,1) cw-complex for an arbitrary G

Question

I am currently reading the appendix 1.B of Hatcher's. There he described a way to construct a $\Delta$-complex which is $K(G,1)$ for an arbitrary group $G$. Later in the chapter he used the fact that it is possible to construct a $K(G,1)$ CW-complex (In the proof of theorem 1.B.8 using proposition 1.B.9) in the Graphs of Groups part, but i did not see a part where he described such construction. My question is how to make such a construction, or is there a way to deduce it from the $\Delta$-complex construction?

Answer 1

As mentioned in the comments, every $\Delta$-complex is a $CW$-complex. In fact, $CW$-complex structures are in some sense just less restrictive $\Delta$-complex structures. Just as $\Delta$-complex structures are less restrictive simplicial complex structures.

To be more precise, I will show how to obtain a $CW$-structure from a $\Delta$-complex structure. Let $X$ be a $\Delta$-complex with $\Delta$-complex structure given by $\{\sigma_\alpha:\Delta^{n(\alpha)}\to X\}_{\alpha\in \mathcal{A}}$. The $i$-cells of the induced $CW$-structure are given by $\{\text{im}(\sigma_\alpha) \mid n(\alpha)=i\}$, and the attaching map for a given $i$-cell $\text{im}(\sigma_\alpha)$ when $i>0$ is given by $\sigma_\alpha|_{\partial\Delta^i}$. It is not difficult to check that this satisfies the definition of a $CW$-structure. We of course need to choose a homeomorphism $(\Delta^i, \partial\Delta^i)\overset{\cong}{\to}(D^i,S^{i-1})$ to use the exact same definition of $CW$-complex structure as Hatcher.