First, http://pi.math.cornell.edu/~hatcher/AT/AT.pdf is the site of the book.
Exercise 1.3.27 is :
Exercise 1.3.27. For a universal cover $p : \tilde X \to X $ we have two actions of $\pi_1(X, x_0)$ on the fiber $p^{-1}(x_0)$, namely the action given by lifting loops at $x_0$ and the action given by restricting deck transformations to the fiber. Are these two actions the same when $X=S_1 \vee S_1$ or $X=S_1 \times S_1$? Do the actions always agree when $\pi_1(X,x_0)$ is abelian?
My interpretation of the two actions are as :
(1) The first action is given in p.69 and defined as follows :
Let $[\gamma] \in \pi_1(X ,x_0)$ and let $\tilde x \in p^{-1}(x_0)$. Then there is a unique lift $\delta$ of $\bar{\gamma}$ starting at $\tilde x$. We define $[\gamma] \tilde x$ to be the point $\delta (1)$. (Hatcher used $\bar{\gamma}$, not $\gamma$, to make the action a left action. Also, as in the definition of an action in p.71, every action in this book means a left action. )
(2) The second action is : Fix $\tilde x _0 \in p^{-1}(x_0) $. Let $[\gamma] \in \pi_1(X ,x_0)$ and let $\tilde x \in p^{-1}(x_0)$. Since $p$ is a simply-connected covering, there is a unique deck transformation $\tau_\gamma$ sending $\tilde x _0$ to $\tilde \gamma (1)$, where $\tilde \gamma$ is the lift of $\gamma$ starting at $\tilde x _0$ (This is Proposition 1.39. In fact $\pi_1(X, x_0)$ is isomorphic to $G(\tilde X)$, the group of deck transformations, via the isomorphism sending $[\gamma]$ to $\tau_\gamma$.) Now we define $[\gamma]\tilde x$ to be the point $\tau_\gamma (\tilde x)$.
Assuming my understand is correct, what I am confused is these two actions are trivially different. Am I wrong with the definitions of the actions?
Best Answer
These are indeed not the same in most cases (including when $X=S^1\vee S^1$ or $X=S^1\times S^1$) for the rather trivial reason that the first action is defined using $\bar{\gamma}$ and the second is defined using $\gamma$ (so at least when acting on the basepoint $\tilde{x}_0$, the action of $[\gamma]$ by the first definition corresponds to the action of $[\gamma]^{-1}$ by the second). I'm guessing that what Hatcher had in mind, though, is the version of the first action which is defined using a lift of $\gamma$, not a lift of $\bar{\gamma}$. Of course, that makes the first action a right action rather than a left action, so the actions cannot possibly be the same when $\pi_1(X,x_0)$ is nonabelian (given that the actions are faithful). But for an abelian group, right and left actions are the same, and so this still leaves a nontrivial question of whether they coincide when $\pi_1(X,x_0)$ is abelian.