Deck transformation Calculations.

Question

Is there any source for the detailed calculations for the Deck Transformations stated on pg.70 in AT given below:

Answer 1

As a consequence of Theorem 2 in Chapter 2, Section 6 of E.H. Spanier's AT, if $p:\tilde X\rightarrow X$ is a covering projection, then we have an isomorphism $$\psi: G(p)\rightarrow N(p_*\pi_1(\tilde X,\tilde x_0))/p_*\pi_1(\tilde X,\tilde x_0)$$ defined by $\psi(f)=[p\circ w_f]$ where $w_f$ is a path from $\tilde x_0$ the $f(\tilde x_0)$. In particular, if $p$ is regular then $$G(p)\cong \pi_1(X,x_0)/p_*\pi_1(\tilde X,\tilde x_0).$$

1. For the regular covering projection $p:\Bbb R\rightarrow S^1$, $\pi_1(\Bbb R,0)=0$, so $G(P)\cong\pi_1(S^1,1)=\Bbb Z.$ The automorphism $n\in G(p)$ is $n(x)=x+n.$
2. For the regular covering projection $p:S^1\rightarrow S^1$ defined by $p(z)=z^n$, we have $p_*\pi_1(S^1,1)=n\Bbb Z$ hence $G(p)=\Bbb Z/n\Bbb Z=\Bbb Z_n.$ The automorphism $k\in G(p)$ is $k(z)=e^{\frac{2\pi ki}{n}}z.$