On page 66 of Hatcher's Algebraic Topology, he discusses the universal cover of a space $X$ which is a cylinder with its edges glued to a circle by maps $z \mapsto z^m$ and $z \mapsto z^n$. He describes the universal cover $\tilde{X}$ as a graph like the one pictured crossed with $\mathbb{R}$ (here $m=4$ and $n=3$, I believe). enter image description here

The fundamental group of $X$ and the group of deck transformations of the universal cover $\tilde{X}$ are both $\langle a,b \mid a^m = b^n \rangle$.

My Question: I'm a little confused about how the deck transformations act (this part he discusses on page 76). If I look at the cylinder as two halves, $A$ and $B$, corresponding to the $m$- and $n$-fold gluings. I can view $A$ as the "four-way cross" sections of the graph (crossed with $\mathbb{R}$) and $B$ as the "tripod" sections of the picture. He describes the action of $b \in \pi_1(X, x)$ as a corkscrew motion about the center of the tripods. I am trying to figure out how this affects points in the $A$ sections of $\tilde{X}$. Maybe someone can help me understand this.


A deck transformation corresponding to the generator $b$, based at an axis of a tripod section, is simply rotation by 1/3 of a circle around this point, while sliding everything up by 1/3 of a unit (the "corkscrew" motion that Hatcher refers to). In the picture, it looks like this is not a symmetry of the given graph, but you have to view the size of the various "tripods" and "four-crosses" as being incidental. Every hub is a possible axis of symmetric "corkscrew" rotation for the graph.

As Whitman says, "There is no point so soft but it makes a hub for the wheel'd universe."


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