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\begin{document}
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\noindent\parbox[b]{3in}{\tiny
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}: \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \href{http://www.math.toronto.edu/~drorbn/classes/\#2526}{2025-26}:
  \href{http://drorbn.net/26-1301}{MAT 1301 Algebraic Topology}:
}~
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  \null\hfill\sheeturl
}~%
\raisebox{0.2em}{
  \qrcode[height=1.2em,level=L,nolink]{drorbn.net/26-1301}
}
\vskip 2mm 
\noindent{\Large\bf Everything You Always Wanted to Know About Covering Spaces}

\vskip -3mm
\noindent\rule{\linewidth}{1pt}
\vskip 2mm

\begin{multicols}{2}

\noindent{\bf Some famous coverings.}
\begin{enumerate}
\item $S^n\to\bbR\bbP^n$.
\item $S^3$ is a double cover of $SO(3)$.
\item $\bbH^2\to\Sigma_g$ for $g\geq 2$.
\item Lens spaces, for relatively prime $p,q\in\bbZ$, $S^3 = \{(z_1,z_2)\in\bbC^2\colon |z_1|^2+|z_2|^2=1\} \to L_{p,q} = S^3 / (z_1,z_2)\sim(e^{2\pi i/p}z_1,e^{2\pi iq/p}z_2)$.
\end{enumerate}

\end{multicols}

\vskip 2mm
{\bf Some Coverings of $8^a_b$} (from Hatcher's {\em Algebraic Topology}, page 58):
\[
  \def\covA{{$\langle a, b^2, bab^{-1} \rangle$}}
  \def\covB{{$\langle a^2, b^2, ab \rangle$}}
  \def\covC{{$\langle a^2, b^2, aba^{-1}, bab^{-1} \rangle$}}
  \def\covD{{$\langle a, b^2, ba^2b^{-1}, baba^{-1}b^{-1} \rangle$}}
  \def\covE{{$\langle a^3, b^3, ab^{-1}, b^{-1}a \rangle$}}
  \def\covF{{$\langle a^3, b^3, ab, ba \rangle$}}
  \def\covG{{$\langle a^4, b^4, ab, ba, a^2, b^2 \rangle$}}
  \def\covH{{$\langle a^2, b^2, (ab)^2, (ba)^2, ab^2a \rangle$}}
  \def\covI{{$\langle a^2, b^4, ab, ba^2b^{-1}, bab^{-2} \rangle$}}
  \def\covJ{{$\langle b^{2n}ab^{-2n-1}, b^{2n+1}ab^{-2n} \colon n\in\bbZ\rangle$}}
  \def\covK{{$\langle b^nab^{-n}  \colon n\in\bbZ\rangle$}}
  \def\covL{{$\langle a \rangle$}}
  \def\covM{{$\langle ab \rangle$}}
  \def\covN{{$\langle a, bab^{-1} \rangle$}}
  \def\covU{{$\langle \ \rangle$}}
  \import{}{figs/CoveringsOf8.pdf_t}
\]

\needspace{20mm}
\noindent{\bf Unbased Covering Spaces.}

Let $B$ be a topological space and let $\calC(B)$ be the category of covering spaces of $B$: The category whose objects are (unbased!) coverings $X\to B$ and whose morphisms are maps between such coverings that commute with the covering projections -- a morphism between $p_X:X\to B$ and $p_Y:Y\to B$ is a map $\alpha\colon X\to Y$ so that the diagram below is commutative:

\[ \xymatrix{X \ar[rr]^\alpha \ar[rd]_{p_X} & & Y \ar[ld]^{p_Y} \\ & B & } \]

Every topologists' highest hope is to find that their favourite category of topological objects is equivalent to some category of easily understood algebraic objects. The following theorem realizes this dream in full in the case of the category $\calC(B)$ of covering spaces of any reasonable base space $B$:

\begin{theorem} (Classification of covering spaces)
\begin{itemize}
\item If $B$ has base point $b_0$ and fundamental group $G=\pi_1(B,b_0)$, then the map which assigns to every covering $p\colon X\to B$ its fiber $p^{-1}(b_0)$ over the basepoint $b_0$ induces a functor $\calF$ from the category $\calC(B)$ of coverings of $B$ to the category $\calS(G)$ of $G$-sets -- sets with a right $G$-action and set maps that respect the $G$-action.
\item If in addition $B$ is connected, locally connected and semi-locally simply connected then the functor $\calF$ is an equivalence of categories. (In fact, this is iff).
\end{itemize}
\end{theorem}

If indeed the categories $\calC(B)$ and $\calS(G)$ are equivalent, one should be able to extract everything topological about a covering $p\colon X\to
B$ from its associated $G$-set $\calF(X)=p^{-1}(b_0)$. The following theorem shows this to be right in at least two ways:

\begin{theorem} For $B$ connected, locally connected and semi-locally simply connected and $X$ a covering of $B$:
\begin{itemize}
\item The set of connected components of $X$ is in a bijective correspondence with the set of orbits of $G$ in $\calF(X)$.
\item Let $x_0\in\calF(X)=p^{-1}(b_0)$ be a basepoint for $X$ that covers the basepoint $b_0$ of $B$. Then the fundamental group $\pi_1(X,x_0)$ is isomorphic via the projection $p_\star$ into $G=\pi_1(B,b_0)$ to the stabilizer group $\{h\in G\colon x_0h=x_0\}$ of $x_0$ in $\calF(X)$.
\end{itemize}
\end{theorem}

(Both assertions of this theorem can be sharpened to deal with morphisms as well, but we will not bother to do so).

\noindent{\bf Based Covering Spaces.}
There are similar theorems (call them Theorem 1' and Theorem 2') relating the category of based covering spaces with the category of based $G$-sets.

\noindent{\bf The Main Point.} 
Ok. Every math technician can spend some time and effort and understand the statements and (only then) the proofs of these two theorems. Your true challenge is to digest the following statement:
\[ \framebox{\parbox{0.5\linewidth}{\centering\bf
  Almost all there is to know about covering spaces follows from these two theorems
}} \]

In particular, the following facts are all simple algebraic corollaries of these theorems:

\begin{corollary} If $X$ is connected then its covering number (``number of decks'') is equal to the index of $H=p_\star\pi_1(X)$ in $G=\pi_1(B)$, and the decks of $X$ are in a non-canonical correspondence with the left cosets $H\backslash G$ of $H$ in $G$.
\end{corollary}

\begin{corollary} If $B$ is semi-locally simply connected, there exists a unique (up to base-point-preserving isomorphism) ``universal covering space $U$ of $B$'' (a connected and simply connected covering $U$).
\end{corollary}

\begin{corollary} The group of automorphisms of the universal covering $U$ is equal to $G=\pi_1(B)$.
\end{corollary}

\begin{corollary} $\pi_1(S^1)\cong\bbZ$.
\end{corollary}

\begin{corollary} $\pi_1(SO(3)) \cong \bbZ/2\bbZ$.
\end{corollary}

\begin{corollary} If $B$ is semi-locally simply connected, then for every $H < G = \pi_1(B)$ there is a unique (up to base-point-preserving isomorphism) connected covering space $X$ with $p_\star\pi_1(X)=H$.
\end{corollary}

\begin{corollary} If $B$ is semi-locally simply connected there is a bijection between conjugacy classes of subgroups of $G=\pi_1(B)$ and unbased connected coverings of $B$.
\end{corollary}

\begin{corollary} A connected covering $X$ is normal (for any $x_1,x_2\in p^{-1}(b)$ there is an automorphism $\tau$ of $X$ with $\tau x_1=x_2$) iff its group $p_\star\pi_1(X)$ is normal in $G=\pi_1(B)$.
\end{corollary}

\begin{corollary} If $X$ is a connected covering of $B$ and $H=p_\star\pi_1(X)$, then $\operatorname{Aut}(X)=N_G(H)/H$ where $N_G(H)$ is the normalizer of $H$ in $G$.
\end{corollary}

\begin{proposition} If $X_i$ for $i=1,2$ are connected coverings of $B$ with groups $H_i=p_{i\star}\pi_1(X_i)$ and if $H_1<H_2$ then $X_1$ is a covering of $X_2$ of covering number $(H_2:H_1)$. \end{proposition}

\noindent{\bf Steps in the proofs of Theorem 1 and 2.}

\begin{enumerate}\itemsep0em
\item Use path liftings to construct a right action of $G$ on $p^{-1}(b_0)$.
\item Show that this is indeed a group action and that morphisms of coverings induce morphisms of right $G$-sets.
\item Start the construction of an ``inverse'' functor $\calG$ of $\calF$: Use spelunking (cave exploration) to construct a ``universal covering'' $U$ of $B$ (meaning, for now, that $U$ is connected and simply connected), if $B$ is semi-locally simply connected.
\item Show that $\calF(U)=G$.
\item Use the construction of $U$ or the general lifting property for covering spaces to show that there is a left action of $G$ on $U$.
\item For a general right $G$-set $S$ set $\calG(S) = S\times_GU = \{(s,u)\in S\times U\}/(sg,u)\sim(s,gu)$ and show that $\calG(S)$ is a covering of $B$ and $\calF(\calG(S)) = S$.
\item Show that ${\mathcal G}$ is compatible with maps between right $G$-sets.
\item Understand the relationship between connected components and orbits.
\item Prove Theorem 2.
\item Use the existence and uniqueness of lifts to show that ${\mathcal G}\circ{\mathcal F}$ is equivalent to the identity functor (working connected component by connected component).
\end{enumerate}

\noindent\parbox{\dimexpr\linewidth-2in\relax}{
\noindent{\bf A Deep Thought Question.}
What does it at all mean ``${\mathcal G}\circ{\mathcal F}$ is equivalent
to the identity functor'' (and first, why can't it simply be the
identity functor)? And even harder, what does it at all mean for two
categories to be ``equivalent''? If you answer this question correctly,
you'll probably re-invent the notions of ``natural transformation between
two functors'' and ``natural equivalence'', that gave the historical
impetus for the development of category theory.

From the Wikipedia entry for \href{https://en.wikipedia.org/wiki/Natural_transformation}{Natural Transformation}:

\begin{quote}
Saunders Mac Lane, one of the founders of category theory, is said to have remarked, ``I didn't invent categories to study functors; I invented them to study natural transformations.'' Just as the study of groups is not complete without a study of homomorphisms, so the study of categories is not complete without the study of functors. The reason for Mac Lane's comment is that the study of functors is itself not complete without the study of natural transformations.
\end{quote}
}
\hfill\parbox{1.8in}{\centering
  \includegraphics[width=\linewidth]{OnYorkStreet_20260201_175742.jpg}
  \newline On York Street
}

\vfill
\noindent{\bf Tilings.} The plane $\bbR^2$ covers the torus and the Klein bottle:

\vskip 1mm
\noindent\includegraphics[width=0.49\linewidth]{../../Projects/Tilings/0/GroningenBikeParking.jpg}
\hfill\includegraphics[width=0.49\linewidth]{../../Projects/Tilings/oo/TPHonKingSt.jpg}

But regrettably the lovely maps onto the annulus and onto the M\"obius band aren't coverings:
\vskip 1mm
\noindent\includegraphics[width=0.49\linewidth]{../../Projects/Tilings/SS/LaTortilleriaFloor.jpg}
\hfill\includegraphics[width=0.49\linewidth]{../../Projects/Tilings/So/IndianaUniversityCarpet.jpg}

\end{document}


