Let \(X\) be a finite connected graph whose fundamental group is the free group on generators (e.g. the wedge sum \(S^1 \vee S^1\) or the theta graph with 3 edges). (1) Construct the universal covering space \(\widetilde{X}\) of \(X\) as the tree \(T\) of non-backtracking reduced paths from a basepoint \(x_0 \in X\). (2) Prove that \(\widetilde{X}\) is a connected, acyclic tree (hence simply connected: \(\pi_1(\widetilde{X}) = 0\)). (3) Explain why for a 3-valent graph (like the theta graph), the universal cover is the infinite 3-regular tree \(T_3\).
Solution
Choose a base vertex \(x_0\in X\).
1. Define the vertices of \(\widetilde X\) to be the reduced edge-paths in \(X\) starting at \(x_0\), including the empty path.For a reduced path \(\gamma\) ending at a vertex \(v\) and an oriented edge \(e\) issuing from \(v\), join \(\gamma\) by an edge to the reduced path obtained from \(\gamma e\) after cancelling a terminal backtrack if one occurs.Map this edge homeomorphically to \(e\) and map each vertex-path to its endpoint.
2. The resulting map \(p:\widetilde X\to X\) is a covering map.
At a vertex represented by \(\gamma\) ending at \(v\), the incident edges of \(\widetilde X\) are in bijection with the oriented edge germs at \(v\). Hence the star of \(\gamma\) maps homeomorphically onto the star of \(v\). These stars give evenly covered neighborhoods.
3. The graph \(\widetilde X\) is connected and has no cycles.
Every vertex \(\gamma=e_1\cdots e_n\) is joined to the empty path by successively deleting its terminal edges, so \(\widetilde X\) is connected.
If a nontrivial reduced closed edge-path existed in \(\widetilde X\), projecting it to \(X\) would give a nontrivial reduced word which, starting from some reduced path \(\gamma\), returns after successive reduction to the same path \(\gamma\). Cancelling the common initial word would force the projected reduced word to represent the identity in the free edge-path groupoid of a graph, hence to be empty, a contradiction.
4. Therefore \(\widetilde X\) is a tree. In particular it is contractible and simply connected, so \(p\) is the universal covering map.
5. If every vertex of \(X\) has valence \(3\), then every vertex of \(\widetilde X\) also has valence \(3\), because a covering preserves vertex stars. A finite connected \(3\)-regular graph contains a cycle, so its universal cover is infinite. Hence \(\widetilde X\) is the infinite \(3\)-regular tree \(T_3\).