In Hatcher's "Algebraic Topology" (p. 92), the space $B\Gamma$ (for a graph of groups on a graph $\Gamma$) is defined to be a collection of spaces $BG_v$ for each vertex $v$, which are connected by certain mapping cylinders corresponding to the edge morphisms. If we replace $BG_v$ with any $K(G_v,1)$ space in the construction, call the resulting space $K\Gamma$. Then there is a comment:
"We leave it to the reader to check that the resulting space $K\Gamma$ is homotopy equivalent to the space $B\Gamma$ constructed above."
My question is: how does one prove this?
What I've got: I can't seem to extend the homotopy equivalences $BG_v\to K(G_v,1)$ to a homotopy equivalence of the entire network of mapping cylinders. I suppose it's enough to do it for the case of only one edge as long as the two homotopies that govern the homotopy equivalence are constant on both sides of the two mapping cylinders.
Given a group morphism $G_v\to G_w$, I can see that the following diagram commutes upto homotopy:
$BG_v\longrightarrow BG_w$
$\downarrow\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \downarrow$
$K(G_v,1)\to K(G_w,1)$
since the two compositions of maps induce the same map $\pi_1(BG_v)\to\pi_1(K(G_w,1))$. (See Proposition 1B.9 on page 90). But I don't know where to go from there.
Update: What I'm looking for is Hatcher's intended solution, which probably involves explicitly contructing a homotopy equivalence.
The question boils down to this. Suppose we have a commutative diagram
$\;\;\;\;\;f_1$
$A\longrightarrow B$
$\downarrow f\;\;\;\;\;\;\;\;\downarrow g$
$C\longrightarrow D$
$\;\;\;\;\;f_2$
where $f$ and $g$ are homotopy equivalences, $f^{-1}$ and $g^{-1}$ are their homotopy inverses. Suppose also that the diagram is homotopy commutative and so is the diagram with the homotopy inverses. Let $M_1$ be the mapping cylinder for $f_1$ and $M_2$ be the mapping cylinder for $f_2$.
Suppose we have the following homotopies:
Define $F:M_1\to M_2$ to be:
Define $G:M_2\to M_1$ to be:
We want to show that $GF\simeq 1$ using a homotopy that extends both $H_f$ and $H_g$
My attempt:
Computing $GF$ gives:
We can partially define $H:M_1\times I\to M_1$ as follows:
So what I need is a function $H:A\times[\frac{1}{2},1]\times I\to M_1$ that satisfies the following boundary conditions: