# deck transformations and covering spaces

Let $p:\tilde X\rightarrow X$ be a universal covering space, and let $H\leq G$ where $G$ is the group of covering transformations. Let $q:\tilde X \rightarrow \tilde X/G$ be the quotient map which is regular covering space. Is $\tilde X/H\rightarrow X$ a covering space with appropriate map? If yes, what is this map?

## 1 Answer

I think there were some useful informations around page 70 in Hatcher's Algebraic Topology - book.

Corollary: If $p\colon (\tilde{X},\tilde{x_0})\to (X,x_0)$ is the universal covering, then for every $H\le \pi_1(X,x_0)$ the map $\tilde{X}/H\to X$ is the covering corresponding to $H.$

Proof: The fundamental group $\pi_1(\tilde{X}/H,x_H)$ is canonically isomorphic to $H$, where $q(\tilde{x})=x_H$, $q\colon \tilde{X}\to \tilde{X}/H$. The projection $\tilde{X}/H\to \tilde{X}/\pi_1(X,x_0)$ is a covering and $\tilde{X}/\pi_1(X,x_0)=X$, where $H$ acts on $\tilde{X}$ as a subgroup of $\pi_1(X,x_0)$. (cf. source)

So you might consider the projection.