Using the Hatcher's notation on proposition 1.39. Let $G(\tilde{X}):=\{ \varphi: \tilde{X} \to \tilde{X} \; \mathrm{Isomorphism} \}$ be the Deck transformation group for the covering space $p: \tilde{X} \to X$.

Question: Could someone please explain to me why $$G(\tilde{X})\cong \pi_1(X,x_0)/p_*(\pi_1(\tilde{X},\tilde{x_0}))$$ holds just when the covering $\tilde{X}$ is a normal covering? Please provide a counterexample when this does not hold when the covering is not normal.

Many thanks.

  • 2
    $\begingroup$ If the covering is not normal, how can you talk about a group isomorphism? $\endgroup$ – Connor Malin Mar 9 at 12:21
  • $\begingroup$ Such a simple but really precise and correct answer. Thanks! $\endgroup$ – rarc Mar 10 at 8:31

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