Hatcher's Algebraic Topology Exercise 1.3.28 is :

Exercise 1.3.28. Show that for a covering space action of a group $G$ on a simply-connected space $Y$, $\pi_1(Y/G)$ is isomorphic to $G$. [If $Y$ is locally is path-connected, this is a special case of part (b) of Proposition 1.40.]

Proposition 1.40 is :

Proposition 1.40. Given a covering space action of a group $G$ on a space $Y$,

(a) The quotient map $p : Y \to Y/G$ is a normal covering space.

(b) $G$ is the group of deck transformations of the covering space if $Y$ is path-connected.

(c) $G$ is isomorphic to $\pi_1(Y/G)/p_*(\pi_1(Y))$ if $Y$ is path-connected and locally path-connected.

I think Exercise 1.3.28 is a special case of part (c) of Proposition 1.40, not (b). But there is no mention in the errata list in Hatcher's homepage : http://pi.math.cornell.edu/~hatcher/ . Am I wrong?

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    $\begingroup$ Your interpretation seems correct to me, since $p_*(\pi_1(Y))$ is going to be trivial if $Y$ is simply-connected. Probably just missing from the errata. (If you're studying for your algebraic topology midterms too right now, good luck!) $\endgroup$ – Jack Crawford Aug 25 '19 at 8:20

This community wiki solution is intended to clear the question from the unanswered queue.

You are right. Part (b) does not allow to conlude that $G \approx \pi_1(Y/G)$.


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