I was just reading the proof that a graph is a tree iff it is simply connected in Hatcher, on pg.85, which is given below:
The quotient map $X \to X/T$ is a homotopy equivalence by Proposition $0.17.$ The quotient $X/T$ is a graph with only one vertex, hence is a wedge sum of circles, whose fundamental group we showed in Example 1.21 to be free with basis the loops given by the edges of $X/T$ , which are the images of the loops $f_α$ in $X$.
The statement of Prop.0.17 is as follows:
If the pair $(X,A)$ satisfies the homotopy extension property and $A$ is contractible, then the quotient map $q: X \to X/A$ is a homotopy equivalence.
My questions about the proof are as follows:
1-Why the quotient map is a homotopy equivalence by proposition 0.17? Why is the pair $(X,T)$ satisfies the homotopy extension property?
2- I want to see a lot of examples of quotienting a graph by its maximal spanning tree please and see how the above proof is correct.
Can anyone clarify these points to me please? Thanks in advance!
As for 1, note that $T$ is a subcomplex of $X$ by definition, i.e. $(X, T)$ is a CW-pair and thus has the homotopy extension property by proposition 0.16 (see also example 0.14).
As for 2, here's one class of examples you could consider: Take $X = \mathbb{R}$ with the usual CW-structure, i.e. we identify $X_0$ with $\mathbb{Z}$ and then glue in one 1-cell from with end points $n$ and $n + 1$ for each $n$. This is a graph and a maximal spanning tree is just $\mathbb{R}$ itself, so this shows that $\mathbb{R}$ is contractible (it might be helpful to go through the proof an construct an explicit contracting homotopy with the given recipe). Now expand $X$ by taking a larger set of 1-cells: For instance, glue in an edge from 2 to 5, or from $2n$ to $2n + 2$ (for all $n$), or glue in infinitely many edges from 0 to 1, etc. The subcomplex $\mathbb{R} \subset X$ is still a maximal spanning tree, and all the additional edges you glued in will form loops on the unique 0-cell after contraction.
For another flavor of example, take a piece of paper, draw a few nodes (your 0-cells) and then draw edges between those nodes (your 1-cells), however many you want, but making sure the result is connected. Find a maximal spanning tree and then work through the contraction by hand.
There really isn't any more to it :)