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\begin{document}

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\section*{circle}

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

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{as_a_topological_space}{As a topological space}\dotfill \pageref*{as_a_topological_space} \linebreak
\noindent\hyperlink{as_a_homotopy_type}{As a homotopy type}\dotfill \pageref*{as_a_homotopy_type} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \textbf{circle} is a fantastic thing with lots and lots of [[stuff, structure, property|properties and extra structures]]. It is a:

\begin{itemize}%
\item [[topological space]]
\item [[smooth manifold]]
\item [[Lie group]] (the [[circle group]])
\item [[co-H-group]]

\end{itemize}
and it is one of the basic building blocks for lots of areas of mathematics, including:

\begin{itemize}%
\item [[homotopy theory]]
\item [[loop spaces]]
\item [[knots]] and [[links]]

\end{itemize}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

We consider the circle first as a [[topological space]], then as the [[homotopy type]] represented by that space.

\hypertarget{as_a_topological_space}{}\subsubsection*{{As a topological space}}\label{as_a_topological_space}

\begin{defn}
\label{circle}\hypertarget{circle}{}
The two most common definitions of the circle are:

\begin{enumerate}%
\item It is the subspace of $\mathbb{C}$ consisting of those numbers of length $1$:

\begin{displaymath}
U(1) \coloneqq \{z \in \mathbb{C} : {|z|} = 1\}
\end{displaymath}
(of course, $\mathbb{C}$ can be identified with $\mathbb{R}^2$ and an equivalent formulation of this definition given; also, to emphasise the reason for the notation $U(1)$, $\mathbb{C}$ can be replaced by $\mathbb{C}^* = GL_1(\mathbb{C})$)


\item It is the quotient of $\mathbb{R}$ by the integers:

\begin{displaymath}
S^1 \coloneqq \mathbb{R}/\mathbb{Z}
\end{displaymath}


\end{enumerate}
\end{defn}
The standard equivalence of the two definitions is given by the map $\mathbb{R} \to \mathbb{C}$, $t \mapsto e^{2 \pi i t}$.

\hypertarget{as_a_homotopy_type}{}\subsubsection*{{As a homotopy type}}\label{as_a_homotopy_type}

As a [[homotopy type]] the circle is for instance the [[homotopy pushout]]

\begin{displaymath}
S^1 \simeq * \coprod_{* \coprod * } * 
  \,.
\end{displaymath}
In [[homotopy type theory]], this can be formalized as a [[higher inductive type]] generated by a point {\colorbox[rgb]{1.00,0.93,1.00}{\tt base}} and a path from {\colorbox[rgb]{1.00,0.93,1.00}{\tt base}} to itself; see the references.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The topological circle is a [[compact space|compact]], [[connected space|connected]] [[topological space]]. It is a $1$-[[dimension|dimensional]] [[smooth manifold]] (indeed, it is the only $1$-dimensional compact, connected smooth manifold). It is \textbf{not} [[simply connected space|simply connected]].

The circle is a model for the [[classifying space]] for the [[abelian group]] $\mathbb{Z}$, the [[integer]]s. Equivalently, the circle is the [[Eilenberg-Mac Lane space]] $K({\mathbb{Z}},1)$. Explicitly, the first [[homotopy group]] $\pi_1(S^1)$ is the [[integer]]s $\mathbb{Z}$. But the higher [[homotopy groups]] $\pi_n(S^1) \simeq *$, $n \gt 1$ all vanish (and so is a [[homotopy type|homotopy 1-type]]). This can be deduced from the result that the loop space $\Omega S^1$ of the circle is the group ${\mathbb{Z}}$ of integers and that $S^1$ is path-connected. A proof of this in [[homotopy type theory]] is in \hyperlink{ShulmanP1S1}{Shulman P1S1}.

The result that $S^1\simeq K({\mathbb{Z}},1)$ holds in a general [[Grothendieck (∞,1)-topos]]. In fact, more generally, for $X$ a pointed object of a [[Grothendieck (∞,1)-topos]] ${\mathcal{H}}$, there is a natural equivalence between the [[suspension object]] $\Sigma X$ and the classifying space $B{\mathbb{Z}}\wedge X$. In particular, when $X$ is specifically the 0-truncated two-point space $S^0$, we get $S^1\simeq K({\mathbb{Z}},1)$.

The [[product]] of the circle with itself is the ($2$)-[[torus]]

\begin{displaymath}
T = S^1 \times S^1
  \,.
\end{displaymath}
Generally, the $n$-torus $T^n$ is $(S^1)^n$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[unit circle]]


\item [[circle action]]


\item [[sphere]]


\item [[pseudocircle]]


\item [[conic section]]

\begin{itemize}%
\item [[ellipse]], [[parabola]], [[hyperbola]]

\end{itemize}


\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

A formalization in [[homotopy type theory]], along with a proof that $\Omega S^1\simeq {\mathbb{Z}}$ (and hence $\pi_1(S^1) \simeq \mathbb{Z}$), can be found in

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

\end{itemize}
The proof is formalized therein using the [[Agda]] [[proof assistant]]. See also

\begin{itemize}%
\item The [[HoTT Book]], section 8.1


\item The HoTT Coq library: \href{https://github.com/HoTT/HoTT/blob/master/theories/hit/Circle.v}{theories/hit/Circle.v}


\item The HoTT Agda library: \href{https://github.com/HoTT/HoTT-Agda/blob/2.0/homotopy/LoopSpaceCircle.agda}{homotopy/LoopSpaceCircle.agda}



\end{itemize}
[[!redirects circle]] [[!redirects circles]] [[!redirects Circle]] [[!redirects Circles]]



\end{document}