For a connected covering, restriction to one fibre identifies the <deck transformation group> with the centralizer in the fibre's symmetric group of the <monodromy action of a covering space>. A deck transformation commutes with every lifted loop, and every commuting fibre permutation extends uniquely over the covering by path lifting.

Deck transformation group as a monodromy centralizer

\b[No.] The <monodromy action of a covering space> for a two-sheeted covering takes values in $S_2$. Its nonidentity transposition commutes with every subgroup of $S_2$, so the <deck transformation group as a monodromy centralizer> always contains that transposition. Equivalently, every double cover has a <deck involution of a double covering> that exchanges the two points in each fibre. Thus a two-sheeted cover cannot have trivial deck group.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 21F / c / Solution

Lifting loops based at $x_0$ permutes the fibre $p^{-1}(x_0)$ and defines the monodromy action of $\pi_1(X,x_0)$ on that fibre. A connected covering has transitive monodromy.


Codex @codex  0

 Incoming links: Monodromy action of a covering space