I am reading Bogopolski's 'Introduction to group theory' and on page 127 the following maps are defined:
My confusion is about the expansion map. I get that the collapsing map is just collapsing trees in the forest to points. I have two questions:
1) From exp($\tilde{v}$) = v I would conclude that we simply copy vertices when expanding the graph $G/G_0$, however, expanding edges would introduce new vertices (since we need to take reduced paths in trees which were collapsed to points). This confuses me: does exp($\tilde{v}$)= v means that $\tilde{v}$ gets mapped to one $v$ or to all $v$ which are collapsed onto $\tilde{v}$?
2) I do not get the last line: clearly col $\circ$ exp $=$ id, but exp $\circ$ col might not be the identity. I drew multiple graphs, applied the collapsing map and then the expansion map, but always end up with the same graph as the one I started with.
Any help would be appreciated.
1) The expanding map is defined to have codomain $G$, which is the full original graph. However, not all vertices (nor edges) in the graph $G$ are in the range of $\exp$. (I'll demonstrate for vertices.) The definition of $G/G_0$ involved a choice of vertices $v_i \in T_i \subset G$; the vertex $\tilde{v}_i$ (which represents $T_i$ collapsed) will map to that particular chosen vertex $v_i$ only. The other vertices in $T_i$ will be in $G$, but will not be in the range of the expansion map.
In particular, this means that you shouldn't look at the expansion map as defining the graph which expands $G/G_0$; the codomain graph $G$ is defined before the expansion map is defined. In fact, if you had two different graphs $G$ and $H$ which both collapsed to the same graph $P$ (i.e., $G/G_0 \cong H/H_0 \cong P$), you can define two different expansion maps on $P$, one with codomain $G$ and one with codomain $H$. (You can even define two different expansion maps $G/G_0 \to G$ with the same codomain by changing the choice of vertices $v_i$.)
2) The answer to (1) should clarify this question. The codomain of exp$\circ$col is $G$, the same as its domain; so it's the same graph in the domain and codomain, which was verified by the examples you drew. However, the map applied to particular vertices (and edges) may not be the identity. (I'll demonstrate again for vertices.) If you start with some vertex $w_i \in T_i$ which is not the chosen vertex $v_i$, then exp$\circ$col$(w_i) = \exp(\tilde{v}_i) = v_i$, so the map exp$\circ$col does not return the input vertex $w_i$; thus it is not the identity.