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\begin{document}

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\section*{fundamental group}

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

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

[[!include homotopy - contents]]

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{naturality}{Naturality}\dotfill \pageref*{naturality} \linebreak
\noindent\hyperlink{RelationToSingularHomology}{Relation to singular homology}\dotfill \pageref*{RelationToSingularHomology} \linebreak
\noindent\hyperlink{relation_to_universal_covers_and_galois_groups}{Relation to universal covers and Galois groups}\dotfill \pageref*{relation_to_universal_covers_and_galois_groups} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{nonlocally_nice_spaces_and_generalised_spaces}{Non-locally `nice' spaces and `generalised' spaces}\dotfill \pageref*{nonlocally_nice_spaces_and_generalised_spaces} \linebreak
\noindent\hyperlink{proper_fundamental_groups}{Proper fundamental groups}\dotfill \pageref*{proper_fundamental_groups} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{related_concept}{Related concept}\dotfill \pageref*{related_concept} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{fundamental group} $\pi_1(X,x)$ of a [[pointed object|pointed]] [[topological space]] $(X,x)$ is the group of based [[homotopy classes]] of [[loops]] at $x$, with multiplication defined by concatenation (following one path by another).

This is also called \emph{the first [[homotopy group]]} of $X$.

The notion of \emph{fundamental group} $\pi_1(X,x)$ generalises in one direction to the [[fundamental groupoid]] $\Pi_1(X)$, or in another direction to the [[homotopy groups]] $\pi_n(X,a)$ for $n \in \mathbb{N}$. Both of this is contained within the [[fundamental ∞-groupoid]] $\Pi(X)$.

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

\begin{defn}
\label{}\hypertarget{}{}
For $X$ a [[topological space]] and $x : * \to X$ a [[point]]. A [[loop space|loop]] in $X$ based at $x$ is a [[continuous function]]

\begin{displaymath}
\gamma : \Delta^1  \to X
\end{displaymath}
from the topological 1-[[simplex]], such that $\gamma(0) = \gamma(1) = x$.

A \emph{based [[homotopy]]} between two loops is a [[homotopy]]

\begin{displaymath}
\itexarray{
    \Delta^1
    \\
    \downarrow^{\mathrlap{(id,\delta_0)}} & \searrow^{\mathrlap{f}}
    \\
    \Delta^1 \times \Delta^1 &\stackrel{\eta}{\to}& X
    \\
    \uparrow^{\mathrlap{(id,\delta_1)}} & \nearrow_{\mathrlap{g}}
    \\
    \Delta^1
  }
\end{displaymath}
such that $\eta(0,-) = \eta(1,-) = x$.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
This notion of based homotopy is an [[equivalence relation]].

\end{prop}
\begin{proof}
This is directly checked. It is also a special case of the general discussion at \emph{[[homotopy]]}.

\end{proof}
\begin{definition}
\label{Concatenation}\hypertarget{Concatenation}{}
Given two loops $\gamma_1, \gamma_2 : \Delta^1 \to X$, define their \textbf{concatenation} to be the loop

\begin{displaymath}
\gamma_2 \cdot \gamma_1 : t \mapsto
  \left\{
    \itexarray{
      \gamma_1(2 t) & ( 0 \leq t \leq 1/2 )
      \\
      \gamma_2(2 (t-1/2)) & (1/2 \leq t \leq 1) 
    }
  \right.
  \,.
\end{displaymath}
\end{definition}
\begin{prop}
\label{GroupStructure}\hypertarget{GroupStructure}{}
Concatenation of loops respects based homotopy classes where it becomes an [[associativity|associative]], [[unital]] binary pairing with [[inverses]], hence the product in a [[group]].

\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
See also at \emph{[[path groupoid]]} for similar constructions.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
For $X$ a topological space and $x \in X$ a point, the set of based homotopy equivalence classes of based loops in $X$ equipped with the group structure from prop. \ref{GroupStructure} is the \textbf{fundamental group} or \textbf{first [[homotopy group]]} of $(X,x)$, denoted

\begin{displaymath}
\pi_1(X,x) \in Grp
  \,.
\end{displaymath}
\end{defn}
Hence if we write $[\gamma] \in p_1(X,x)$ for the based homotopy class of a loop $p$, then then group operation is

\begin{displaymath}
[\gamma_1] \cdot [\gamma_2] \coloneqq [\gamma_1 \cdot \gamma_2]
  \,.
\end{displaymath}
\begin{defn}
\label{SimplyConnectedSpace}\hypertarget{SimplyConnectedSpace}{}
A [[topological space]] whose fundamental group is trivial is called a \emph{[[simply connected topological space]]}.

\end{defn}
\begin{defn}
\label{EMSpace}\hypertarget{EMSpace}{}
Conversely, a topological space whose only non-trivial [[homotopy group]] is the fundamental group is called an [[Eilenberg-MacLane space]] denoted $K(\pi_1(X), 1)$.

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

\hypertarget{naturality}{}\subsubsection*{{Naturality}}\label{naturality}

\begin{prop}
\label{}\hypertarget{}{}
If a topological space $X$ is path-[[connected space]], then all of the fundamental groups $\pi_1(X,x)$ are [[isomorphism|isomorphic]], for all choices of base points $x \in X$.

\end{prop}
\begin{proof}
For $x_0, x_1 \in X$ any two basepoints, there is by assumption a path connecting them, hence a [[continuous function]]

\begin{displaymath}
p : \Delta^1 \to X
\end{displaymath}
such that $p(0) = x_0$ and $p(1) = x_1$.

Write $\bar p \coloneqq (p(1-(-)))$ for the same path with the orientation reversed. Then for $\gamma_0$ any loop based at $x_0$, the concatenation $[ p \cdot (\gamma_0 \cdot \bar p) ]$, def. \ref{Concatenation} yield a loop based at $x_1$ (obtained from $[\gamma_0]$ by [[conjugation]] with $[p]$).

It is immediate to check that this induces an [[isomorphism]]

\begin{displaymath}
Ad_{[p]} : \pi_1(X,x_0) \to \pi_1(X,x_1)
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Therefore one sometimes loosely speaks of `the' fundamental group of a connected space. But beware that the isomorphism in the above construction is not unique. Therefore forming fundamental groups is not a [[functor]] on connected spaces.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
It is, however, a functor on [[pointed object|pointed topological spaces]]: $\pi_1(-) :$ [[Top]] ${}^{*/} \to$ [[Grp]].

\end{prop}
\hypertarget{RelationToSingularHomology}{}\subsubsection*{{Relation to singular homology}}\label{RelationToSingularHomology}

The fundamental group is in general non-abelian (i.e. is not an [[abelian group]]), the \hyperlink{Examples}{Examples} below. For a connected topological space $X$, its \emph{[[abelianization]]} is equivalent to the first [[singular homology]] group

\begin{displaymath}
\pi_1(X,x)^{ab} \simeq H_1(X)
  \,.
\end{displaymath}
See at \emph{\href{singular%20homology#RelationToHomotopyGroups}{singular homology -- Relation to homotopy groups}} for more on this.

\hypertarget{relation_to_universal_covers_and_galois_groups}{}\subsubsection*{{Relation to universal covers and Galois groups}}\label{relation_to_universal_covers_and_galois_groups}

There is a relation to [[universal cover|universal covers]]: Under suitable conditions the group of cover automorphisms of a universal cover is isomorphic to the fundamental group of the covered space. This is the topic of the \emph{[[étale fundamental group]]}, also referred to at \emph{[[Chevalley fundamental group]]}, see there for more.

In particular in [[algebraic geometry]] and [[arithmetic geometry]] this essentially identifies the concept of fundamental group with that of \emph{[[Galois groups]]}. For this reason one also speaks of the \emph{[[algebraic fundamental group]]} in this context. See at \emph{[[Galois theory]]} for more on this.

See also at \emph{[[link between Galois theory and fundamental groups]]}.

In [[Grothendieck's Galois theory]], the role of the basepoint is replaced by considering a `fibre functor' $F:\mathcal{C}\to Sets$ or to $FinSets$, where $\mathcal{C}$ is the category of coverings of the given space. This theory extends to other situations and the term [[algebraic fundamental group]] is used in particular for the case of [[scheme]]s (of a suitable type); see (SGA1).

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

\hypertarget{nonlocally_nice_spaces_and_generalised_spaces}{}\subsubsection*{{Non-locally `nice' spaces and `generalised' spaces}}\label{nonlocally_nice_spaces_and_generalised_spaces}

The definition of fundamental group in terms of homotopy classes of loops at a base point does not work well for the spaces that occur in [[algebraic geometry]], nor for many spaces considered in analysis as there may be very few loops. For instance, for a [[scheme]] there are in general very few paths, and [[Grothendieck]] gave a definition of a fundamental group in SGA1 which is closely related to the [[Galois group|Galois groups]] of number theory, but in cases where both the path-based group and this [[algebraic fundamental group]] make sense, the algebraic form tends to be related to the [[profinite completion of a group|profinite completion]] of the topological fundamental group; see the example in that entry.

A similar type of construction gives the [[fundamental group of a topos]]. Other related forms include a ech version of the fundamental group used in shape theory, and linked to ech homology groups of a compact space.

The notion of fundamental group generalizes to that of [[fundamental groupoid]] in both the loop based theory and in [[Grothendieck's Galois theory]] as described in [[SGA1]]. In this form it has been used to give generalisations for simplicial profinite spaces in work by Quick and to [[pro-spaces]] in work of Isaksen.

\hypertarget{proper_fundamental_groups}{}\subsubsection*{{Proper fundamental groups}}\label{proper_fundamental_groups}

In the context of [[proper homotopy theory]] there are two related fundamental groups for single ended spaces.

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\begin{example}
\label{EuclideanSpaceFundamentalGroup}\hypertarget{EuclideanSpaceFundamentalGroup}{}
\textbf{(Euclidean space is [[simply connected topological space|simply connected]])}

For $n \in \mathbb{N}$, let $\mathbb{R}^n$ be the $n$-dimensional [[Euclidean space]] with its [[metric topology]]. Then for every point $x \in \mathbb{R}^n$ the fundamental group is trivial:

\begin{displaymath}
\pi_1(\mathbb{R}^n, x) = 1
  \,.
\end{displaymath}
\end{example}
\begin{proof}
Let

\begin{displaymath}
\gamma \;\colon\; [0,1] \longrightarrow \mathbb{R}^n
\end{displaymath}
be [[loop]] at $x$, hence a [[continuous function]] with $\gamma(0) = x$ and $\gamma(1) = x$. Using the [[real vector space]] structure on $\mathbb{R}^n$, we may define the function

\begin{displaymath}
\itexarray{
    [0,1] \times [0,1] &\overset{\eta}{\longrightarrow}& \mathbb{R}^n
    \\
    (t,s) &\mapsto& x + s (\gamma(t) - x)
  }
  \,.
\end{displaymath}
This is a [[continuous function]], since it is the composite of the continuous function $(id_{[0,1]} \times \gamma) \;\colon\; (t,s) \mapsto (\gamma(t),s)$ (which is continuous as the [[product topological space|product]] of two continuous functions) and the function $(v,s) \mapsto x + s ( v - x )$ (which is continuous since [[polynomials are continuous]]).

Moreover, by construction we have

\begin{displaymath}
\eta(-,1) = \gamma(-)
   \phantom{AAAA}
  \eta(-,0) = const_x
  \,.
\end{displaymath}
Therefore this is a [[homotopy]] from $\gamma$ to the constant loop at $x$.

\end{proof}
\begin{example}
\label{}\hypertarget{}{}
By definition \ref{SimplyConnectedSpace}, the fundamental group of every [[simply connected topological space]] is trivial.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
The [[fundamental group of the circle is the integers]]:

\begin{displaymath}
\pi_1(S^1) \simeq \mathbb{Z}
  \,.
\end{displaymath}
\end{example}
\begin{remark}
\label{}\hypertarget{}{}
An instructive formalization of this basic statement in [[homotopy type theory]] is in (\hyperlink{Shulman}{Shulman}).

\end{remark}
\begin{example}
\label{}\hypertarget{}{}
By definition \ref{EMSpace}, the fundamental group of any [[Eilenberg-MacLane space]] $K(G,1)$ is $G$: $\pi_1(K(G,1)) = G$.

\end{example}
\hypertarget{related_concept}{}\subsection*{{Related concept}}\label{related_concept}

\begin{itemize}%
\item [[fundamental theorem of covering spaces]]


\item [[winding number]]


\item [[algebraic fundamental group]]


\item [[anabelian geometry]]


\item [[Chevalley fundamental group]]


\item [[etale homotopy]]


\item [[Grothendieck's Galois theory]]



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

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


\item Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 2.5 of \emph{Algebraic topology from a homotopical viewpoint}, Springer (2002) (\href{http://tocs.ulb.tu-darmstadt.de/106999419.pdf}{toc pdf})



\end{itemize}
Discussion from the point of view of [[Galois theory]] is in

\begin{itemize}%
\item [[Luis Javier Hernández-Paricio]], \emph{Fundamental pro-groupoids and covering projections}(\href{http://matwbn.icm.edu.pl/ksiazki/fm/fm156/fm15611.pdf}{pdf})

\end{itemize}
Isaksen's work is

\begin{itemize}%
\item [[D. C. Isaksen]], \emph{A model structure on the category of pro-simplicial sets}, Trans. Amer. Math. Soc., 353, (2001), 2805--2841

\end{itemize}
whilst Quick's is in

\begin{itemize}%
\item [[G. Quick]], \emph{Profinite homotopy theory}, Documenta Mathematica, 13, (2008), 585--612.

\end{itemize}
Proof that the [[fundamental group of the circle is the integers]] in [[homotopy type theory]] is 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}
[[!redirects fundamental groups]]



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