Def: Let $E$ and $X$ be topological spaces, and let $q:E\to X$ be a continuous map. An open subset $U\subseteq X$ is said to be evenly covered by $q$ if $q^{-1}[U]$ is a disjoint union of connected open subsets of $E$, called the sheets of the covering over $U$, each of which is mapped homeomorphically onto $U$ by $q$.

Note that the sheets are connected, disjoint and open implies that they are components of $q^{-1}[U]$, and the fact $q$ restricts to a homeomorphism from each sheet to $U$ implies $U$ is connected.

Def: A covering map is a continuos surjective map $q:E\to X$ such that $E$ is connected and locally path-connected, and every point of $X$ has an evenly covered neighbourhood. If $q:E\to X$ is a covering map, we call $E$ a covering space of $X$, and $X$ the base of the covering.

Obs: We see that if $q:E\to X$ is a covering map, then $X$ and $E$ are path-connected, and $X$ is locally path-connected.

Elementary Properties of Covering Maps:

• Every covering map is a local homeomorphism, an open map, and a quotient map.
• An injective covering map is a homeomorphism.
• A finite product of covering maps is a covering map.
• The restriction of a covering map to a saturated, connected, open subset is a covering map onto its image.
• If $q:E\to X$ is a covering map and $A\subseteq X$ is a locally path-connected subset, then the restriction of $A$ to each component of $q^{-1}[A]$ is a covering map onto its image.

Prop: Suppose $q:E\to X$ is a covering map.

• If $X$ is $T_2$, then $E$ is too.
• If $X$ is an $n$-manifold, then $E$ is too.
• If $E$ is an $n$-manifold and $X$ is Hausdorff, then $X$ is an $n$-manifold.

Prop: Let $X$ be a CW complex, and let $q:E\to X$ be a covering map. $E$ has CW decomposition for which each cell is mapped homeomorphically by $q$ onto a cell of $X$.

Prop: A proper local homomorphism between connected, locally path-connected, and compactly generated Hausdorff spaces is a covering map.

Def: If $q:X\to Y$ is any surjective continuous map, a section of $q$ is a continuous map $\sigma :Y\to X$ such that $q\circ \sigma ={\text{Id}}_Y$. If $U\subseteq Y$ is any open subset, a local section of $q$ over $U$ is continuous map $\sigma :U\to X$ such that $q\circ \sigma ={\text{Id}}_U$.

Existence of Local Sections: Let $q:E\to X$ be a covering map. Given any evenly covered map subset $U\subseteq X$, any $x\in I$, and any $e_0$ in the fiber over $x$, there exists a local section $\sigma :U\to E$ such that $\sigma (x)=e_0$.

Prop: For every covering map $q:E\to X$, the cardinality of the fibres $q^{-1}\{x\}$ is the same for all fibres.

If $q:E\to X$ us a covering map, the cardinality of any fiber is called the number of sheets of the covering.

Prop: A covering map is proper iff it is finite-sheeted.

Prop: Let $q:E\to X$ is a covering map. $E$ is compact iff $X$ is compact and $q$ is finite-sheeted.

Prop: Let $M$ and $N$ be connected manifolds of dimension $n$, and suppose $\tilde{M}\to M$ is a $k$-sheeted covering map. There is a connected sum $M\mathrm{\#}N$ that admits a $k$-sheeted covering by a manifold $\tilde{M}\mathrm{\#}N\mathrm{\#}\cdots \mathrm{\#}N$ (connected sum of $\tilde{M}$ with $k$ disjoint copies of $N$)

Prop: Every nonorientable compact surface of genus $n\ge 1$ has a two sheeted covering by an orientable one of genus $n-1$.

Prop: Let $X$ be a CW complex, and $q:E\to X$ be a covering map. Then $E$ has CW decomposition for which each cell is mapped homeomorphically by $q$ onto the cell $X$.

Lifting Properties

Def: If $q:E\to X$ is a covering map and $\phi :Y\to X$ is any continuous map, a lift of $\phi$ is a continuous map $\tilde{\phi }:Y\to E$ such that $q\circ \tilde{\phi }=\phi$:

\usepackage{tikz-cd}
\usepackage{amsfonts, amsmath, amssymb}

\begin{document}
\begin{tikzcd}[row sep=2cm, column sep=2cm]
     & E\arrow[d,"q"] \\
     Y\arrow[r,"\varphi"']\arrow[ur,dashed,"\widetilde\varphi"] & X
   \end{tikzcd}
\end{document}

Unique Lifting Property: Let $q:E\to X$ be a covering map. Suppose $Y$ is connected, $\phi :Y\to X$ is continuous, and ${\tilde{\phi }}_1,{\tilde{\phi }}_2:Y\to X$ are lifts of $\phi$ that agree at some point in $Y$. Then ${\tilde{\phi }}_1$ is equal to ${\tilde{\phi }}_2$.

Homotopy Lifting Theorem: Let $q:E\to X$ be a covering map, and let $Y$ be a locally connected space. Suppose ${\phi }_0,{\phi }_1:Y\to X$ are continuous functions, $H:Y\times I\to X$ is a homotopy from ${\phi }_0$ to ${\phi }_1$, and ${\tilde{\phi }}_0:Y\to E$ is any lift of ${\phi }_0$. Then there exists a unique lift $\tilde{H}$ satisfying ${\tilde{H}}_0={\tilde{\phi }}_0$. If $H$ is stationary on some subset $A\subseteq Y$, then so is $\tilde{H}$.

Path Lifting Property: Let $q:E\to X$ be a covering map. Suppose $f:I\to X$ is any path, and $e\in q^{-1}\{f(0)\}$. Then there exists a unique lift $\tilde{f}:I\to E$ such that ${\tilde{f}}_e(0)=e$.

Whenever $q:E\to X$ is a covering map, we use the following notation for lifts of paths: if $f:I\to X$ is a path and $e\in q^{-1}\{f(0)\}$, then ${\tilde{f}}_e:I\to E$ is the unique lift of $f$ such that ${\tilde{f}}_e(0)=e$.

Monodromy Theorem: Let $q:E\to X$ be a covering map. Suppose $f$ and $g$ are paths in $X$ with the same initial point and same terminal point, and ${\tilde{f}}_e$, ${\tilde{g}}_e$ are lifts with the same initial point $e\in E$. Then:

• ${\tilde{f}}_e\sim {\tilde{g}}_e$ iff $f\sim g$
• If $f\sim g$, then ${\tilde{f}}_e(1)\sim {\tilde{g}}_e(1)$.

Injectivity Theorem: Let $q:E\to X$ be a covering map. For any point $e\in E$, the induced homomorphism $q_{\ast }:{\pi }_1(E,e)\to {\pi }_1(X,q(e))$ is injective.

Def: The injectivity theorem shows that the fundamental group of a covering space is isomorphic to a certain subgroup of the fundamental group of the base. We call this the subgroup induced by the covering.

The General Lifting Problem

A natural problem arises when considering covering maps, and that is when a function has a lift, this is called as the lifting problem.

Lifting Criterion: Suppose $q:E\to X$ is a covering map. Let $Y$ be a connected and locally path-connected space, and let $\phi :Y\to X$ be a continuous map. Given any points $y_0\in Y$ and $e_0\in E$ such that $q(e_0)=\phi (y_0)$, the map $\phi$ has a lift $\tilde{\phi }:Y\to E$ satisfying $\tilde{\phi }(y_0)=e_0$ iff the subgroup ${\phi }_{\ast }[{\pi }_1(Y,y_0)]$ of ${\pi }_1(X,\phi (y_0))$ is contained in $q_{\ast }[{\pi }_1(E,e_0)]$, i.e., ${\phi }_{\ast }[{\pi }_1(Y,y_0)]\le q_{\ast }[{\pi }_1(E,e_0)]\le {\pi }_1(X,\phi (y_0))$.

Obs: If $\phi :Y\to X$ has a lift $\tilde{\phi }:Y\to E$, then we get that ${\phi }_{\ast }[{\pi }_1(Y,y_0)]\le q_{\ast }[{\pi }_1(E,e_0)]$, without the additional hypothesis about $Y$.

Lifting Maps from Simply Connected Spaces: If $q:E\to X$ is a covering map and $Y$ is a simply connected and locally path-connected space, then every continuous function $\phi :Y\to X$ has a lift to $E$. Given any point $y_0\in Y$, the lift can be chosen to take $y_0$ to any point in the fiber over $\phi (y_0)$.

Lifting Maps to Simply Connected Spaces: Suppose $q:E\to X$ is a covering map and $E$ is simply connected. For any connected and locally path-connected space $Y$, a continuous map $\phi :Y\to X$ has a lift to $E$ iff ${\phi }_{\ast }$ is the zero homomorphism from some base point $y_0\in Y$. If this is the case, then the lift can be chosen to take $y_0$ to any point in the fiber over $\phi (y_0)$.

Interesting Theorems

Borsuk-Ulam Theorem $2$-dimensional Version: For any continuous map $F:{\mathbb{S}}^2\to {\mathbb{R}}^2$, there is a point $x\in {\mathbb{S}}^2$ such that $F(x)=F(-x)$. We get the nice corollary, there are antipodal points on earth with the same temperature and atmospheric pressure.

Ham Sandwich Theorem in ${\mathbb{R}}^3$: Given three disjoint, bounded, connected open subsets $U_1,U_2,U_3\subseteq {\mathbb{R}}^3$, there exists a plane that simultaneously bisects all three, in the sense that the plane divides ${\mathbb{R}}^3$ into two half-spaces $H^{+}$ and $H^{-}$ such that for each $i$ $U_i\cap H^{+}$ has the same Lebesgue measure as $U_i\cap H^{-}$.

Generalisation

Def: If $N$ and $M$ are topological spaces, et us say that a map $\pi :N\to M$ is a generalised covering map if it satisfies all the requirements of a covering map except that $N$ might not be connected: this means that $N$ is locally path-connected, $\pi$ is surjective and continuous, and each point $p\in M$ has a neighbourhood that is evenly covered by $\pi$.

Lemma: Suppose $N$ and $M$ are topological spaces and $\pi :N\to M$ is a generalised covering map. If $M$ is connected, then the restriction of $\pi$ to each component fo $N$ is a covering map.