# Errata in Hatcher's Algebraic Topology Exercise 1.3.28?

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?

• 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!) Aug 25, 2019 at 8:20

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