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\begin{document}

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\section*{fundamental group of the circle is the integers}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{proofs}{Proofs}\dotfill \pageref*{proofs} \linebreak
\noindent\hyperlink{ProofInTopologicalHomotopyTheory}{In topological point-set homotopy theory}\dotfill \pageref*{ProofInTopologicalHomotopyTheory} \linebreak
\noindent\hyperlink{ProofInHomotopyTypeTheory}{In homotopy type theory}\dotfill \pageref*{ProofInHomotopyTypeTheory} \linebreak
\noindent\hyperlink{Consequences}{Consequences}\dotfill \pageref*{Consequences} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{in_topological_homotopy_theory}{In topological homotopy theory}\dotfill \pageref*{in_topological_homotopy_theory} \linebreak
\noindent\hyperlink{in_homotopy_type_theory_2}{In homotopy type theory}\dotfill \pageref*{in_homotopy_type_theory_2} \linebreak


\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

A basic statement in [[homotopy theory]]:

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(fundamental group of the circle is the integers)}

The [[fundamental group]] $\pi_1$ of the [[circle]] $S^1$ is the additive group of [[integers]]:

\begin{displaymath}
\pi_1(S^1) \overset{\simeq}{\longrightarrow} \mathbb{Z}
\end{displaymath}
and the isomorphism is given by assigning [[winding number]].

\end{prop}
Here in the context of [[topological homotopy theory]] the [[circle]] $S^1$ is the [[topological subspace]] $S^1 = \{x \in \mathbb{R}^2 \,\vert\, x_1^2 + x_2^2 = 1 \} \subset \mathbb{R}^2$ of the [[Euclidean plane]] with its [[metric topology]], or any [[topological space]] of the same [[homotopy type]]. More generally, the circle in question is, as a [[homotopy type]], the [[homotopy pushout]]

\begin{displaymath}
S^1 \simeq \ast \underset{\ast \sqcup \ast}{\coprod} \ast
  \,,
\end{displaymath}
hence the [[homotopy type]] with the [[universal property]] that it makes a homotopy commuting diagram of the form

\begin{displaymath}
\itexarray{
    \ast \sqcup \ast &\longrightarrow& \ast
    \\
    \downarrow &\swArrow& \downarrow
    \\
    \ast &\longrightarrow& S^1
  }
  \,.
\end{displaymath}
\hypertarget{proofs}{}\subsection*{{Proofs}}\label{proofs}

\hypertarget{ProofInTopologicalHomotopyTheory}{}\subsubsection*{{In topological point-set homotopy theory}}\label{ProofInTopologicalHomotopyTheory}

We discuss the classical proof in [[point-set topology|point-set]] [[topological homotopy theory]].

\begin{proof}
The [[universal covering space]] $\widehat{S^1}$ of $S^1$ is the [[real line]] (by \href{universal+covering+space#UniversalCoveringOfCircleRealLine}{this example}):

\begin{displaymath}
p \coloneqq (cos(2 \pi(-)), \sin(2 \pi(-))) 
   \;\colon\; \mathbb{R}^1 \longrightarrow S^1
  \,.
\end{displaymath}
Since the [[circle]] is [[locally path-connected topological space|locally path-connected]] (\href{locally+path-connected+space#LocallyPathConnectedCircle}{this example}) and [[semi-locally simply connected topological space|semi-locally simply connected]] (\href{semi-locally+simply+connected+topological+space#LocallySimplyConnectedCircle}{this example}) the [[fundamental theorem of covering spaces]] applies and gives that the [[automorphism group]] of $\mathbb{R}^1$ over $S^1$ equals the automorphism group of its [[monodromy]] [[permutation representation]]:

\begin{displaymath}
Aut_{Cov(S^1)}(\mathbb{R}^1)
    \;\simeq\;
  Aut_{\pi_1(S^1) Set}(Fib_{S^1}) 
  \,.
\end{displaymath}
Moreover, as a corollary of the [[fundamental theorem of covering spaces]] we have that the [[monodromy]] representation of a [[universal covering space]] is given by the [[action]] of the [[fundamental group]] $\pi_1(S)$ on itself (\href{universal+covering+space#ReconstructCoveringForFreeAndTransitiveMonodromyRepresentation}{this prop.}).

But the [[automorphism group]] of any group regarded as an [[action]] on itself by left multiplication is canonically isomorphic to that group itself (by \href{permutation+representation#AutomorphismsOfGAsGTorsor}{this example}), hence we have

\begin{displaymath}
Aut_{{\pi_1(S^1)} Set}(Fib_{S^1}) 
   \;\simeq\;
  Aut_{{\pi_1(S^1)} Set}( \pi_1(S^1) )
   \;\simeq\;
  \pi_1(S^1)
  \,.
\end{displaymath}
Therefore to conclude the proof it is now sufficient to show that

\begin{displaymath}
\mathrm{Aut}_{Cov(S^1)}(\mathbb{R}^1) \simeq \mathbb{Z}
  \,.
\end{displaymath}
\# To that end, consider a [[homeomorphism]] of the form

\begin{displaymath}
\itexarray{
    \mathbb{R}^1 && \underoverset{\simeq}{f}{\longrightarrow} && \mathbb{R}^1
    \\
    & {}_{\mathllap{p}}\searrow && \swarrow_{\mathrlap{p}}
    \\
    && S^1
  }
  \,.
\end{displaymath}
Let $s \in S^1$ be any point, and consider the restriction of $f$ to the fibers over the [[complement]]:

\begin{displaymath}
\itexarray{
    p^{-1}(S^1 \setminus \{s\}) && \underoverset{\simeq}{f}{\longrightarrow} && p^{-1}(S^1 \setminus \{s\})
    \\
    & {}_{\mathllap{p}}\searrow && \swarrow_{\mathrlap{p}}
    \\
    && S^1 \setminus \{s\}
  }
  \,.
\end{displaymath}
By the [[covering space]] property we have (via \href{universal+covering+space#UniversalCoveringOfCircleRealLine}{this example}) a [[homeomorphism]]

\begin{displaymath}
p^{-1}(S^1 \setminus \{s\}) 
   \simeq
  (0,1) \times Disc(\mathbb{Z})
  \,.
\end{displaymath}
Therefore, up to homeomorphism, the restricted function is of the form

\begin{displaymath}
\itexarray{
    (0,1)\times Disc(\mathbb{Z}) 
      && 
      \underoverset{\simeq}{f}{\longrightarrow} 
      && 
    (0,1) \times Disc(\mathbb{Z})
    \\
    & {}_{pr_1}\searrow && \swarrow_{pr_1}
    \\
    && (0,1)
  }
  \,.
\end{displaymath}
By the [[universal property]] of the [[product topological space]] this means that $f$ is equivalently given by its two components

\begin{displaymath}
(0,1) \times Disc(\mathbb{Z})
    \overset{pr_1 \circ f}{\longrightarrow}
  (0,1)
  \phantom{AAAA}
  (0,1) \times Disc(\mathbb{Z})
    \overset{pr_2 \circ f}{\longrightarrow}
  Disc(\mathbb{Z})
  \,.
\end{displaymath}
By the [[commuting diagram|commutativity]] of the above [[diagram]], the first component is fixed to be $pr_1$. Moreover, by the fact that $Disc(\mathbb{Z})$ is a [[discrete space]] it follows that the second component is a [[locally constant function]] (by \href{locally+constant+function#LocallyConstantFunctionIntoDiscreteSpace}{this example}). Therefore, since the [[product space]] with a [[discrete space]] is a [[disjoint union space]] (via \href{product+topological+space#ProductWithDiscreteFiniteTopologicalSpace}{this example})

\begin{displaymath}
(0,1) \times Disc(\mathbb{Z}) \simeq \underset{n \in \mathbb{Z}}{\sqcup}(0,1)
\end{displaymath}
and since the disjoint summands $(0,1)$ are [[connected topological spaces]] (\href{connected+space#ConnectedSubspacesOfRealLineAreTheIntervals}{this example}), it follows that the second component is a [[constant function]] on each of these summands (by \href{connected+space#LocallyConstantFunctionsOnConnectedSpaces}{this example}).

Finally, since every function out of a [[discrete topological space]] is continuous, it follows in conclusion that the restriction of $f$ to the fibers over $S^1 \setminus \{s\}$ is entirely encoded in an [[endofunction]] of the set of [[integers]]

\begin{displaymath}
\phi \;\colon\; \mathbb{Z} \to \mathbb{Z}
\end{displaymath}
by

\begin{displaymath}
\itexarray{
    S^1 \setminus \{s\} \times Disc(\mathbb{Z})
      &\overset{f}{\longrightarrow}&
    S^1 \setminus \{s\} \times Disc(\mathbb{Z})
    \\
    (t,k) &\mapsto& (t, \phi(k))
  }
  \,.
\end{displaymath}
Now let $s' \in S^1$ be another point, distinct from $s$. The same analysis as above applies now to the restriction of $f$ to $S^1 \setminus \{s'\}$ and yields a function

\begin{displaymath}
\phi' \;\colon\; \mathbb{Z} \longrightarrow \mathbb{Z}  
  \,.
\end{displaymath}
Since

\begin{displaymath}
\left\{
    p^{-1}(S^1 \setminus \{s\})
    \subset \mathbb{R}^1
     \,,\,
    p^{-1}(S^1 \setminus \{s'\})
    \subset \mathbb{R}^1
  \right\}
\end{displaymath}
is an [[open cover]] of $\mathbb{R}^1$, it follows that $f$ is unqiuely fixed by its restrictions to these two subsets.

Now unwinding the definition of $p$ shows that the condition that the two restrictions coincide on the intersection $S^1 \setminus \{s,s'\}$ implies that there is $n \in \mathbb{Z}$ such that $\phi(k) = k+ n$ and $\phi'(k) = k+n$.

This shows that $Aut_{Cov(S^1)}(\mathbb{R}^1) \simeq \mathbb{Z}$.

\end{proof}
\hypertarget{ProofInHomotopyTypeTheory}{}\subsubsection*{{In homotopy type theory}}\label{ProofInHomotopyTypeTheory}

There is also a purely synthetic a proof in [[homotopy type theory]] (\hyperlink{LicataShulman13}{Licata-Shulman 13}, \hyperlink{UF}{UF, corollary 8.1.10}).

\hypertarget{Consequences}{}\subsection*{{Consequences}}\label{Consequences}

\begin{example}
\label{CoveringOfCircleAndConjugacyClassesOfSymmetricGroup}\hypertarget{CoveringOfCircleAndConjugacyClassesOfSymmetricGroup}{}
\textbf{([[isomorphism classes]] of [[covering spaces|coverings]] of the circle are [[conjugacy classes]] in the [[symmetric group]])}

The [[monodromy]] construction assigns to an [[isomorphism class]] of covering spaces over the [[circle]] $S^1$ with [[fibers]] consisting of $n$ elements [[conjugacy classes]] of elements the [[symmetric group]] $\Sigma(n)$:

\begin{displaymath}
\left \{
    \itexarray{
      \text{isomorphism classes of}
      \\
      \text{finite covering spaces }
      \\
      \text{over the circle}
    }
  \right\}
   \;\simeq\;
  \left\{
    \itexarray{
      \text{conjugacy classes of}
      \\
      \text{elements of a symmetric group}
    }
  \right\}
\end{displaymath}
To see this, we may without restriction (via \href{groupoid#representation#GroupoidRepresentationsAreProductsOfGroupRepresentations}{this prop.}) choose a basepoint $x \in S^1$ so that a monodromy representation is equivalently a groupoid morphism of the form

\begin{displaymath}
\rho
    \;\colon\;
  B \mathbb{Z}
    \overset{\simeq}{\longrightarrow}
  B \pi_1(S^1,x)
    \overset{\rho}{\longrightarrow}
  Core(Set)
  \,.
\end{displaymath}
Since $\mathbb{Z}$ is the [[free abelian group]] on a single generator, such as morphism is uniquely determined by the image of $1 \in \mathbb{Z}$. This is taken to some isomorphism of the set $p^{-1}(x)$. If we choose any identification $\phi \colon p^{-1}(x) \overset{\simeq}{\to} \{1, \cdots, n\}$, then this defines an element $\sigma \in \Sigma(n)$ in the [[symmetric group]]:

\begin{displaymath}
\itexarray{
    x &\mapsto&  p^{-1}(x) &\underoverset{\simeq}{\phi}{\longrightarrow}& \{1, \cdots, n\}
    \\
    {}^{\mathllap{1}}\downarrow && {}^{\mathllap{\rho(1)}}\downarrow && \downarrow^{\sigma}
    \\
    x &\mapsto& p^{-1}(x) &\underoverset{\phi}{\simeq}{\longrightarrow}& \{1, \cdots, n\}
  }
  \,.
\end{displaymath}
Now if

\begin{displaymath}
f \;\colon\; E_1 \overset{\simeq}{\longrightarrow} E_2
\end{displaymath}
is an isomorphism of covering spaces, then by the [[fundamental theorem of covering spaces]] this corresponds bijectively to a homomorphism of representations

\begin{displaymath}
Fib(f) \;\colon\; Fib_{E_1} \overset{\simeq}{\longrightarrow} Fib_{E_2}
\end{displaymath}
which in turn is by definition a homotopy (natural isomorphism) between the monodromy functors $Fib_{E_i} \;\colon\; B \mathbb{Z} \to Core(Set)$.

The combination of the naturality square of this natural isomorphism with the above identification yields the following diagram

\begin{displaymath}
\itexarray{
    \{1,\cdots, n\}
      &\overset{\phi_1^{-1}}{\longrightarrow}&
    p_1^{-1}(x)
      &\overset{f\vert_{\{x\}}}{\longrightarrow}&
    p_2^{-1}(x)
      &\overset{\phi_2}{\longrightarrow}&
    \{1, \cdots, n\}
    \\
    {}^{\mathllap{\sigma_1}}
    \downarrow
      &&
    {}^{\mathllap{ Fib_{E_1}(1) }}\downarrow
      &&
    \downarrow^{\mathrlap{ Fib_{E_2}(1) }}
      &&
    \downarrow^{\mathrlap{ \sigma_2 }}
    \\
    \{1,\cdots, n\}
      &\underset{\phi_1^{-1}}{\longrightarrow}&
    p_1^{-1}(x)
      &\underset{f\vert_{\{x\}}}{\longrightarrow}&
    p_2^{-1}(x)
      &\underset{\phi_2}{\longrightarrow}&
    \{1, \cdots, n\}
  }
  \,.
\end{displaymath}
The commutativity of the total rectangle says that the permutations $\sigma_1$ and $\sigma_2$ are related by conjugation with the element $\phi_2 \circ f\vert_{\{x\}}\circ \phi_1^{-1}$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
\textbf{(three-sheeted covers of the circle)}

Consider the three-sheeted [[covering spaces]] of the [[circle]].

By example \ref{CoveringOfCircleAndConjugacyClassesOfSymmetricGroup} these are, up to isomorphism, given by the [[conjugacy classes]] of the elements of the [[symmetric group]] $\Sigma(3)$ on three elements. These in turn are labeled by the [[cycle]] structure of the elements (\href{symmetric+group#ConjugacycClassesOfSymmetricGroupCorrespondToCycleSet}{this prop.}).

For the symmetric group on three elements there are three such classes

\begin{displaymath}
\itexarray{
    (1\; 2\; 3)
    \\
    (1 \; 2) (3)
    \\
    (1) (2) (3)
    
  }
\end{displaymath}
The corresponding covering spaces of the circle are shown in the graphics.

\begin{quote}%
graphics grabbed from \href{homotopy+equivalence#Hatcher}{Hatcher}


\end{quote}
\end{example}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{in_topological_homotopy_theory}{}\subsubsection*{{In topological homotopy theory}}\label{in_topological_homotopy_theory}

Lecture notes on the classical discussion include

\begin{itemize}%
\item [[Friedhelm Waldhausen]], p. 63-77 of \emph{Topologie} (\href{https://www.math.uni-bielefeld.de/~fw/ein.pdf}{pdf})


\item [[Jesper Møller]], theorem 3.1 in \emph{The fundamental group and covering spaces} (2011) (\href{http://www.math.ku.dk/~moller/f03/algtop/notes/covering.pdf}{pdf})



\end{itemize}
\hypertarget{in_homotopy_type_theory_2}{}\subsubsection*{{In homotopy type theory}}\label{in_homotopy_type_theory_2}

The proof in [[homotopy type theory]] is discussed in

\begin{itemize}%
\item [[Daniel Licata]], [[Michael Shulman]], \emph{Calculating the Fundamental Group of the Circle in Homotopy Type Theory}, (\href{https://arxiv.org/abs/1301.3443}{arXiv:1301.3443})


\item [[UF-IAS-2012|Univalent Foundations Project]], section 8.1 of \emph{[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]}



\end{itemize}
the HoTT-[[Coq]]-code is at

\begin{itemize}%
\item [[Mike Shulman]], \emph{\href{https://github.com/HoTT/HoTT/blob/master/Coq/HIT/Pi1S1.v}{P1S1.v}}

\end{itemize}


\end{document}