# understanding construction and definition of classifying space BG

Let $$G$$ be a discrete group. $$EG$$ is defined as the $$\triangle$$-complex (Hatcher p.102) whose $$n$$-simplices are given by $$[g_0,g_1,...,g_n]$$ glued together in the obvious way. Then define $$BG=EG/G$$. I am trying to understand this definition by a simple example. If I choose $$G= \mathbb{Z}/2\mathbb{Z}$$, then I think $$EG$$ should be a path with endpoints $$0$$ and $$1$$, and then $$BG$$ will identify the endpoints and give us $$S^1$$. This does not seem right since $$S^1 = K(\mathbb{Z},1)$$. What am I doing wrong here?

In your construction, $$E(\mathbb{Z}/2\mathbb{Z})$$ is $$S^\infty$$ with the antipodal $$\mathbb{Z}/2\mathbb{Z}$$-action.

You can think of $$n$$-simplices here as binary strings of length $$n+1$$. Nondegenerate simplices are given by binary strings that alternate strictly between $$0$$ and $$1$$.

• There are two $$0$$-simplices $$[0]$$ and $$[1]$$. So far this is $$S^0$$.
• There are two (nondegenerate) $$1$$-simplices $$[01]$$ and $$[10]$$ which are attached to their boundaries: $$[0]$$ at one end, $$[1]$$ at the other. This is $$S^1$$.
• There are two (nondegenerate) $$2$$-simplices $$[010]$$ and $$[101]$$, which are attached to $$[01]$$ and $$[10]$$ (and other degenerate simplices). This is $$S^2$$.

This pattern continues, and you may recognize the whole thing as the usual presentation of $$S^\infty$$ as a $$\mathbb{Z}/2\mathbb{Z}$$-CW complex.

The quotient of $$S^\infty$$ by the antipodal action is $$S^\infty / (\mathbb{Z}/2\mathbb{Z}) \simeq \mathbb{R}P^\infty$$, which is indeed $$K(\mathbb{Z}/2\mathbb{Z}, 1)$$.

• I still have 1 question: you say nondegenerate simplices are given by binary strings alternating between 0 and 1. Why is that? [0,1,0] has 3 faces, [0,1], [1,0] and [0,0], but so does [0,0,1]. Nov 20, 2020 at 2:42
• The $2$-simplex $[0,0,1]$ only has faces $[0,0]$ and $[0,1]$ (twice), so visually it's "collapsed" onto the edge $[0,1]$.
– JHF
Nov 20, 2020 at 2:48