Covering spaces

 Preliminary exam prep

Let $X,Y$ be topological spaces.

Definition: A space $\widetilde X$ and a map $p:\widetilde X\to X$ are called a covering space of $X$ if either of two equivalent conditions hold:

• There is a cover $\{U_\alpha\}_{\alpha\in A}$ of $X$ such that $p^{-1}(U_\alpha)\cong \bigsqcup_{\beta\in B_\alpha} U_\beta$.
• Every point $x\in X$ has a neighborhood $U\subset X$ such that $p^{-1}(U) \cong \bigsqcup_{\beta\in B}U_\beta$.

We also demand that every $U_\beta$ is carried homeomorphically onto $U_\alpha$ (or $U$) by $p$, and the $U_\alpha$ (or $U$) are called evenly covered.

Some definitions require that $p$ be surjective. A universal cover of $X$ is a covering space that is universal with respect to this property, in that it covers all other covering spaces. Moreover, a cover that is simply connected is immediately a universal cover.

Remark: Every path connected (pc), locally path connected (lpc), and semi locally simply connected (slsc) space has a universal cover.

Theorem: (Lifting criterion) Let $Y$ be pc and lpc, and $\widetilde X$ a covering space for $X$. A map $f:Y\to X$ lifts to a map $\widetilde f:Y\to \widetilde X$ iff $f_*(\pi_1(Y))\subset p_*(\pi_1(\widetilde X))$.

Further, if the initial map $f_0$ in a homotopy $f_t:Y\to X$ lifts to $\widetilde f_0:Y\to \widetilde X$, then $f_t$ lifts uniquely to $\widetilde X$. This is called the homotopy lifting property. Next, we will see that path connected covers of $X$ may be classified via a correspondence through the fundamental group.

Theorem: Let $X$ be pc, lpc, and slsc. There is a bijection (up to isomorphism) between pc covers $p:\widetilde X\to X$ and subgroups of $\pi_1(X)$, described by $p_*(\pi_1(\widetilde X))$.

Example: Let $X=T^2$, the torus, with fundamental group $\Z\oplus \Z$. Below are some covering spaces of $p:\widetilde X\to X$ with the corresponding subgroups $p_*(\pi_1(\Z\oplus\Z))$.

Definition: Given a covering space $p:\widetilde X\to X$, an isomorphism $g$ of $\widetilde X$ for which $\id_X\circ p = p\circ g$, is called a deck transformation, the collection of which form a group $G(\widetilde X)$ under composition. Further, $\widetilde X$ is called normal (or regular) if for every $x\in X$ and every $\widetilde x_1,\widetilde x_2\in p^{-1}(x)$, there exists $g\in G(\widetilde X)$ such that $g(\widetilde x_1)=\widetilde x_2$.

For path connected covering spaces over path connected and locally path connected bases, being normal is equivalent to $p_*(\pi_1(\widetilde X))\leqslant  \pi_1(X)$ being normal. In this case, $G(\widetilde X)\cong \pi_1(X)/p_*(\pi_1(\widetilde X))$. This simplifies even more for $\widetilde X$ a universal cover, as $\pi_1(\widetilde X)=0$ then.

Theorem: Let $G$ be a group, and suppose that every $x\in X$ has a neighborhood $U\owns x$ such that $g(U)\cap h(U) = \emptyset$ whenever $g\neq h\in G$. Then:

• The quotient map $q:X\to X/G$ describes a normal cover of $X/G$.
• If $X$ is pc, then $G = G(X)$.

A group action satisfying the hypothesis of the previous theorem is called a covering space action.

Proposition: For any $n$-sheeted covering space $\widetilde X\to X$ of a finite CW complex, $\chi(\widetilde X) = n\chi(X)$.

References: Hatcher (Algebraic Topology, Chapter 1)