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)

Tools of homotopy

 Preliminary exam prep

Let $X,Y$ be topological spaces and $A$ a subspace of $X$. Recall that a path in $X$ is a continuous map $\gamma:I\to X$, and it is closed (or a loop), if $\gamma(0)=\gamma(1)$. When $X$ is pointed at $x_0$, we often require $\gamma(0)=x_0$, and call such paths (and similarly loops) based.

Definitions

Definition:

• $X$ is connected if it is not the union of two disjoint nonempty open sets.
• $X$ is path connected if any two points in $X$ have a path connecting them, or equivalently, if $\pi_0(X)=0$.
• $X$ is simply connected if every loop is contractible, or equivalently, if $\pi_1(X)=0$.
• $X$ is semi-locally simply connected if every point has a neighborhood whose inclusion into $X$ is $\pi_1$-trivial.

Path connectedness and simply connectedness have local variants. That is, for $P$ either of those properties, a space is locally $P$ if for every point $x$ and every neighborhood $U\owns x$, there is a subset $V\subset U$ on which $P$ is satisfied.

Remark: In general, $X$ is $n$-connected whenever $\pi_r(X)=0$ for all $r\leqslant n$. Note that 0-connected is path connected and 1-connected is simply connected and connected. Also observe that the suspension of path connected space is simply connected.

Definition:

• A retraction (or retract) from $X$ to $A$ is a map $r:X\to A$ such that $r|_A = \id_A$.
• A deformation retraction (or deformation retract) from $X$ to $A$ is a family of maps $f_t:X\to X$ continuous in $t,X$ such that $f_0 = \id_X$, $f_1(X) = A$, and $f_t|_A = \id_A$ for all $t$.
• A homotopy from $X$ to $Y$ is a family of maps $f_t:X\to Y$ continuous in $t,X$.
• A homotopy equivalence from $X$ to $Y$ is a map $f:X\to Y$ and a map $g:Y\to X$ such that $g\circ f \simeq \id_X$ and $f\circ g \simeq \id_Y$.

Definition: A pair $(X,A)$, where $A\subset X$ is a closed subspace, is a good pair, or has the homotopy extension property (HEP), if any of the following equivalent properties hold:

• there exists a neighborhood $U\subset X$ of $A$ such that $U$ deformation retracts onto $A$,
• $X\times \{0\}\cup A\times I$ is a retract of $X\times I$, or
• the inclusion $i:A\hookrightarrow X$ is a cofibration.

In some texts such a pair $(X,A)$ is called a neighborhood deformation retract pair, and HEP is reserved for any map $A\to X$, not necessarily the inclusion, that is a cofibration. For more on cofibrations, see a previous blog post (2016-07-31, "(Co)fibrations, suspensions, and loop spaces").

Definition: There is a functor $\pi_1:\text{Top}_*\to \text{Grp}$ called the fundamental group, that takes a pointed topological space $X$ to the space of all pointed loops on $X$, modulo path homotopy.

This may be generalized to $\pi_n$, which takes $X$ to the space of all pointed embeddings of $S^n$.

Definition: Let $G,H$ be groups. The free product of $G$ and $H$ is the group
\[
G*H = \{a_1\cdots a_n\ :\ n\in \Z_{\>0}, a_i\in G\text{ or }H, a_i\in G(H)\implies a_{i+1}\in H(G)\},
\]
with group operation concatenation, and identity element the empty string $\emptyset$. We also assume $e_Ge_H=e_He_G=e_G=e_H=\emptyset$, for $e_G$ ($e_H$) the identity element of $G$ ($H$).

The above construction may be generalized to a collection of groups $G_1*\cdots*G_m$, where the index may be uncountable. If every $G_\alpha=\Z$ (equivalently, has one generator), then $*_{\alpha\in A} G_\alpha$ is called the free group on $|A|$ generators.

Theorems

Theorem: (Borsuk-Ulam) Every continuous map $S^n\to \R^n$ takes a pair of antipodal points to the same value.

Theorem: (Ham Sandwich theorem) Let $U_1,\dots,U_n$ be bounded open sets in $\R^n$. There exists a hyperplane in $\R^n$ that divides each of the open sets $U_i$ into two sets of equal volume.

Volume is taken to be Lebsegue measure. The Ham sandwich theorem is an application of Borsuk-Ulam (see Terry Tao's blog post for more).

Theorem: If $X$ and $Y$ are path-connected, then $\pi_1(X\times Y)\cong \pi_1(X)\times \pi_1(Y)$.

Now suppose that $X = \bigcup_\alpha A_\alpha$ is based at $x_0$ with $x_0\in A_\alpha$ for all $\alpha$. There are natural inclusions $i_\alpha:A_\alpha\to X$ as well as $j_\alpha:A_\alpha\cap A_\beta \to A_\alpha$ and $j_\beta:A_\alpha\cap A_\beta \to A_\beta$.

Both $i_\alpha$ and $j_\alpha$ induce maps on the fundamental group, each (and all) of the $i_{\alpha*}:\pi_1(A_\alpha)\to \pi_1(X)$ extending to a map $\Phi:*_\alpha \pi_1(A_\alpha)\to \pi_1(X)$.

Theorem: (van Kampen)

• If $A_\alpha\cap A_\beta$ is path-connected, then $\Phi$ is a surjection. 
• If $A_\alpha\cap A_\beta\cap A_\gamma$ is path connected, then $\ker(\Phi) = \langle j_{\alpha*}(g)(j_{\beta*}(g))^{-1}\ |\ g\in \pi_1(A_\alpha\cap A_\beta,x_0)\rangle$.

As a consequence, if triple intersections are path connected, then $\pi_1(X) \cong *_\alpha A_\alpha /\ker(\Phi)$. Moreover, if all double intersections are contractible, then $\ker(\Phi)=0$ and $\pi_1(X)\cong *_\alpha A_\alpha$.

Proposition: If $\pi_1(X)=0$ and $\widetilde H_n(X)=0$ for all $n$, then $X$ is contractible.

References: Hatcher (Algebraic topology, Chapter 1), Tao (blog post "The Kakeya conjecture and the Ham Sandwich theorem")