# Covering

A mapping $p:X\rightarrow Y$ of a space $X$ onto a space $Y$ such that each point $y\in Y$ has a neighbourhood $U(y)$ the pre-image of which under $p$ is a union of open subsets that are mapped homeomorphically onto $U(y)$ by $p$. Equivalently: $p$ is a locally trivial fibre bundle with discrete fibre.

Coverings are usually considered on the assumption that $X$ and $Y$ are connected; it is also usually assumed that $Y$ is locally connected and locally simply-connected. Under these assumptions one can establish a relationship between the fundamental groups $\pi_1(X,x_0)$ and $\pi_1(Y,y_0)$: If $p(x_0)=y_0$, then the induced homomorphism $p^*$ maps $\pi_1(X,x_0)$ isomorphically onto a subgroup of $\pi_1(Y,y_0)$ and, by varying the point $x_0$ in $p^{-1}(y)$, one obtains exactly all subgroups in the corresponding class of conjugate subgroups. If this class consists of a single subgroup $H$ (i.e. if $H$ is a normal divisor), the covering is said to be regular. In that case one obtains a free action of the group $G=\pi_1(Y,y_0)/H$ on $X$, with $p$ playing the role of the quotient mapping onto the orbit space $Y$. This action is generated by lifting loops: If one associates with a loop $q:[0,1]\rightarrow Y$, $q(0)=q(1)=y_0$, the unique path $\bar{q}:[0,1]\rightarrow X$ such that $\bar{q}(0)=x_0$ and $p\bar{q}=q$, then the point $\bar{q}(1)$ will depend only on the class of the loop in $G$ and on $x_0$. Thus, each element of $G$ corresponds to a permutation of points in $p^{-1}(y_0)$. This permutation has no fixed points if $\gamma\neq 1$, and it depends continuously on $y_0$. One obtains a homeomorphism of $X$.

In the general case this construction defines only a permutation in $p^{-1}(y)$, i.e. there is an action of $\pi_1(Y,y_0)$ on $p^{-1}(y)$, known as the monodromy of the covering. A special case of a regular covering is a universal covering, for which $G=\pi_1(Y,y_0)$. In general, given any subgroup $H\subset\pi_1(Y,y_0)$, one can construct a unique covering $p:(X,x_0)\rightarrow (Y,y_0)$ for which $p^*(\pi_1(X,x_0))=H$. The points of $X$ are the classes of paths $q:[0,1]\rightarrow Y$, $q(0)=x_0$: Two paths $q_1$ and $q_2$ are identified if $q_1(1)=q_2(1)$ and if the loop $q_1q_2^{-1}$ lies in an element of $H$. The point $q(1)$ for the paths of one class is taken as the image of this class; this defines $p$. The topology in $X$ is uniquely determined by the condition that $p$ be a covering; it is here that the local simple-connectedness of $Y$ is essential. For any mapping $f$ of an arcwise connected space $(Z,z_0)$ into $(Y,y_0)$, its lifting into a mapping $\bar{f}:(Z,z_0)\rightarrow (X,x_0)$ exists if and only if $f^*(\pi_1(Z,z_0))\subset H$. A partial order relation can be defined on the coverings of $Y$ (a covering of a covering is a covering); this relation is dual to the inclusion of subgroups in $\pi_1(Y,y_0)$. In particular, the universal covering is the unique maximal element.

Examples. The parametrization $(\cos \phi,\sin \phi)$ of the circle defines a covering of the circle by the real line, $\phi\in \mathbb{R}$, often described in the complex form $e^{i\phi}$ and called the exponential covering. Similarly, the torus is covered by the plane. Identification of antipodal points on a sphere yields a covering by the sphere of a projective space of corresponding dimension. In general, free actions of discrete groups are a source of regular coverings (over the orbit space); not every such action yields a covering (the orbit space may be non-separable), but finite groups do.