Suppose $G$ is a perfect group, and let us consider $BG$ and its plus construction (We consider $BG$, the classifying space, by the nerve construction). By following Hatcher’s book (Proposition 4.40, page 374) - for any $g \in G$, we have a loop and we glue a 2-cell along its boundary. Now we add in 3-cells, by the long exact sequence and the Hurewicz isomorphism as such. I'm able to follow Hatcher's algebraic approach, but I'm not able to see how we're adding it geometrically.
For instance, in the case when $g = [a,b]$, we attach a 2-cell $e_{g}^2$ so that it becomes a torus, and we also have 2-cells attached along “a” and “b”. How are we attaching the 3-cells to it?
By Milnor's paper "Geometric realization of a semi simplicial complex", we know that $BG$ is a CW complex whose $n$-cells are in 1-1 correspondence with the non-degenerate $n$-simplices of $BG$. Is the inclusion map $i : BG \rightarrow (BG)^{+}$ a cellular map? I believe it to be the case, but I'm not able to prove it.
The crucial fact is that if $A$ is a subcomplex of $Y$ and $f\colon A\to X$ is a cellular map, then the adjunction space $X\cup_f Y$ is a CW-complex such that $X$ is a subcomplex with a cell not in $X$ for every cell of $Y$ which is not in $A$. In particular, if $X'$ is obtained by attaching $n$-cells along cellular maps $\coprod_{i\in I} S^n_i\to X$, then $X$ is a subcomplex of $X'$.
In the context and notation of Hatcher Proposition 4.40 we can by cellular approximation assume that all attaching maps are cellular, hence $X$ is a subcomplex of $X'$ and $X'$ is a subcomplex of $X^+$, hence $X$ is also a subcomplex of $X^+$.