Covering spaces · Qual Corpus

A covering map $p\colon\tilde X\to X$ is a continuous surjection such that every point of $X$ has an open neighbourhood $U$ for which $p^{-1}(U)$ is a disjoint union of open sets, each mapped homeomorphically onto $U$ by $p$. Paths and homotopies in $X$ lift uniquely to $\tilde X$ once the starting point of the lift is chosen, and $p_*\colon\pi_1(\tilde X,\tilde x_0)\to\pi_1(X,x_0)$ is injective. The lifting criterion: for $Y$ path-connected and locally path-connected, a map $f\colon(Y,y_0)\to(X,x_0)$ lifts to $(\tilde X,\tilde x_0)$ if and only if $f_*\pi_1(Y,y_0)\subseteq p_*\pi_1(\tilde X,\tilde x_0)$. For $X$ path-connected, locally path-connected, and semilocally simply connected, isomorphism classes of based path-connected covering spaces correspond to subgroups $H\le\pi_1(X,x_0)$; the deck transformation group of the cover corresponding to $H$ is isomorphic to $N(H)/H$, where $N(H)$ is the normalizer of $H$, and the cover is normal if and only if $H\trianglelefteq\pi_1(X,x_0)$. If a group $G$ acts on $Y$ so that every point has a neighbourhood $U$ with $gU\cap U=\varnothing$ for all $g\ne1$, then $Y\to Y/G$ is a covering map. Applying the correspondence to graphs shows that every subgroup of a free group is free.

Covering space (Tag D-ANO2D)

Let $X$ be a topological space. A covering space of $X$ is a topological space $\tilde X$ together with a continuous map $p\colon \tilde X\to X$ such that every $x\in X$ has an open neighborhood $U$ for which $\inverseof{p}(U)$ is a union of pairwise disjoint open sets $V_i\subseteq\tilde X$, each mapped homeomorphically onto $U$ by $p$. The sets $V_i$ are the sheets of $\tilde X$ over $U$.

Let $p_1\colon\tilde X_1\to X$ and $p_2\colon\tilde X_2\to X$ be covering spaces of $X$. An isomorphism of covering spaces is a homeomorphism $f\colon\tilde X_1\to\tilde X_2$ with $p_2\circ f = p_1$:

[@Hat02].

Semilocally simply connected space (Tag D-EPQ54)

A topological space $X$ is semilocally simply connected if every $x\in X$ has a neighborhood $U$ such that the homomorphism $\pi_1(U, x)\to\pi_1(X, x)$ of fundamental groups induced by the inclusion $U\hookrightarrow X$ is trivial.

[@Hat02].

What does it mean for a space to be semilocally simply connected?

Universal cover (Tag D-BX3WD)

Let $X$ be a topological space. A universal cover of $X$ is a covering space $p\colon \tilde X\to X$ with $\tilde X$ simply connected [@Hat02].

Let $X$ be path-connected and locally path-connected. Then $X$ has a universal cover if and only if $X$ is semilocally simply connected [@Hat02], and a universal cover is unique up to isomorphism of covering spaces [@Hat02]. A universal cover covers every path-connected covering space of $X$ [@Hat02], and its deck transformation group is isomorphic to $\pi_1(X)$ [@Hat02].

Homotopy lifting property for covering spaces (Tag T-SKJB2)

A lift of the initial slice of a homotopy extends uniquely across the whole homotopy.Suggest a slogan

Let $p\colon\tilde X \to X$ be a covering space, $F\colon Y \cross I \to X$ a homotopy, and $\tilde F_0\colon Y\to \tilde X$ a lift of $F_0 = F(\wait, 0)$. Then there is a unique homotopy $\tilde F\colon Y\cross I\to \tilde X$ with $\tilde F(y, 0) = \tilde F_0(y)$ for all $y\in Y$ and $p\circ\tilde F = F$ [@Hat02]:

Path lifting property (Tag D-OISBB)

Let $p\colon E\to B$ be a continuous map and $I=[0,1]$. The map $p$ has the path lifting property if for every path $\gamma\colon I\to B$ and every $e_0\in \inverseof{p}(\gamma(0))$ there is a path $\tilde\gamma\colon I\to E$ with $p\circ\tilde\gamma=\gamma$ and $\tilde\gamma(0)=e_0$.

Identifying $\pt\times I$ with $I$, this is the homotopy lifting property of $p$ with respect to the one-point space $Y=\pt$.

Let $p\colon\tilde X\to X$ be a covering space. For every path $\gamma\colon I\to X$ and every $\tilde x_0\in \inverseof{p}(\gamma(0))$ there is a unique path $\tilde\gamma\colon I\to\tilde X$ with $p\circ\tilde\gamma=\gamma$ and $\tilde\gamma(0)=\tilde x_0$ [@Hat02].

Lifting criterion for covering spaces (Tag T-EF5IX)

A map lifts to a cover exactly when its fundamental group lands in the covering subgroup.Suggest a slogan

Let $p\colon(\tilde X, \tilde x_0)\to(X, x_0)$ be a covering space and $f\colon(Y, y_0)\to(X, x_0)$ a map with $Y$ path-connected and locally path-connected. Then a lift $\tilde f\colon(Y, y_0)\to(\tilde X, \tilde x_0)$ with $p\circ\tilde f = f$ exists if and only if $f_*(\pi_1(Y, y_0)) \subseteq p_*(\pi_1 (\tilde X, \tilde x_0))$ [@Hat02]:

If $Y$ is connected, two lifts of $f$ that agree at one point are equal [@Hat02].

Covering spaces induce injections on $\pi_1$ (Tag PR-R5EN3)

Covering maps inject fundamental groups, with image detected by closed lifts of loops.Suggest a slogan

Let $p\colon(\tilde X, \tilde x_0)\to(X, x_0)$ be a covering space. The induced map $p_*\colon \pi_1(\tilde X, \tilde x_0) \to \pi_1(X, x_0)$ is injective, and its image consists of the classes of loops at $x_0$ whose lifts to $\tilde X$ starting at $\tilde x_0$ are loops [@Hat02].

Number of sheets of a covering space (Tag PR-BHJGY)

The number of sheets of a connected cover is the index of its induced fundamental-group subgroup.Suggest a slogan

Let $p\colon(\tilde X, \tilde x_0)\to(X, x_0)$ be a covering space with $X$ and $\tilde X$ path-connected. The number of sheets of $p$, that is, the cardinality of $p^{-1}(x_0)$, equals the index \[ [\pi_1(X, x_0) : p_*(\pi_1(\tilde X, \tilde x_0))] \] [@Hat02].

Deck transformation (Tag D-4VGAW)

Let $p\colon \tilde X\to X$ be a covering space. A deck transformation of $p$ is a homeomorphism $f\colon\tilde X\to\tilde X$ with $p\circ f = p$. The deck transformations of $p$ form a group $\Deck(p)$ under composition.

Monodromy action on a fiber (Tag D-6BK54)

Let $p\colon \tilde X \to X$ be a covering space and $x_0 \in X$. For $\tilde x\in \inverseof{p}(x_0)$ and a loop $\gamma$ in $X$ based at $x_0$, let $\tilde\gamma$ be the unique lift of $\gamma$ with $\tilde\gamma(0) = \tilde x$. The monodromy action is the right action of $\pi_1(X, x_0)$ on $\inverseof{p}(x_0)$ given by \[ \tilde x \cdot [\gamma] \coloneqq \tilde\gamma(1) .\]

The endpoint $\tilde\gamma(1)$ depends only on the homotopy class $[\gamma]$, by the homotopy lifting property of covering spaces. The stabilizer of $\tilde x$ is $p_*\pi_1(\tilde X, \tilde x)$. If $X$ is path-connected, the action is transitive if and only if $\tilde X$ is path-connected, and then $\tilde x\cdot[\gamma]\mapsto p_*\pi_1(\tilde X, \tilde x)\,[\gamma]$ is a bijection from $\inverseof{p}(x_0)$ onto the set of right cosets of $p_*\pi_1(\tilde X, \tilde x)$ in $\pi_1(X, x_0)$.

[@Hat02].

Galois, normal, and regular covering spaces (Tag D-MWD2L)

Let $p\colon\tilde X\to X$ be a covering space, let $x_0\in X$, and let $\Deck(\tilde X)$ be its group of deck transformations. The covering $p$ is Galois, also called normal or regular, over $x_0$ if $\Deck(\tilde X)$ acts transitively on the fiber $\inverseof{p}(x_0)$: for all $\tilde x_1,\tilde x_2\in \inverseof{p}(x_0)$ there is $\psi\in\Deck(\tilde X)$ with $\psi(\tilde x_1)=\tilde x_2$.

Universal cover and deck groups (Tag PR-UBJ6P)

A universal cover factors through every connected cover, and its deck group is the fundamental group.Suggest a slogan

Let $X$ be path-connected, locally path-connected and semilocally simply connected. Then $X$ has a universal cover $p\colon\hat X \to X$ [@Hat02]. If $q\colon C\to X$ is a covering map with $C$ path-connected, then there is a covering map $\tilde p\colon \hat{X} \to C$ with $q\circ\tilde p = p$ [@Hat02]:

In particular, $\hat X$ is unique up to isomorphism of covering spaces.

The deck transformation group of the universal cover is isomorphic to $\pi_1(X)$. More generally, for a covering map $q\colon C\to X$ with $C$ path-connected and $H\coloneqq q_*\pi_1(C)\leq\pi_1(X)$, the deck transformation group of $q$ is isomorphic to $N_{\pi_1(X)}(H)/H$ [@Hat02].

Free and properly discontinuous group actions (Tag D-2PNEG)

Let $G$ be a group acting on a topological space $X$ by homeomorphisms, $G\actson X$.

The action is free if $g(x) = x$ for some $x\in X$ implies $g = e$.
The action is properly discontinuous if every $x\in X$ has a neighborhood $U$ such that for all $g_1, g_2\in G$, $g_1(U) \intersect g_2(U) \neq \emptyset$ implies $g_1 = g_2$.

A properly discontinuous action in this sense is free. Other sources call an action properly discontinuous under the weaker condition that every $x\in X$ has a neighborhood $U$ with $U \intersect g(U) \neq \emptyset$ for only finitely many $g\in G$. An action satisfying the condition in the definition is called a covering space action in [@Hat02].

Covering space actions and their quotients (Tag T-O44GI)

A properly discontinuous group action produces a normal cover whose deck group is the acting group.Suggest a slogan

Let a group $G$ act on a space $Y$ such that each $y\in Y$ has a neighborhood $U$ with $g_1U\cap g_2U = \emptyset$ whenever $g_1\neq g_2$ in $G$. Then

the quotient map $p\colon Y \to Y/G$, $p(y) = Gy$, is a normal covering space;
if $Y$ is path-connected, then $G$ is the group of deck transformations of $p$;
if $Y$ is path-connected and locally path-connected, then $G\cong \pi_1(Y/G) / p_*(\pi_1(Y))$

[@Hat02].

Every subgroup of a free group is free (Tag C-ILVEB)

Subgroups of free groups are free.Suggest a slogan

Let $F$ be a free group and let $H \leq F$ be a subgroup. Then $H$ is free.

[@Hat02].

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