= Fundamental group
{title2=$\pi_1(X,x_0)$}
{wiki}
The <fundamental group> $\pi_1(X,x_0)$ consists of <based loops> modulo <based homotopy>. Multiplication follows one <based loop> and then the other; the constant <based loop> represents the identity, and <path reversal> gives the inverse. Passing to <homotopy classes> makes this operation associative, as explained by <well-defined loop concatenation>.
Identify the <circle> with $S^1=\{z\in\mathbb C:|z|=1\}$ and choose basepoint $1$. Its <based loops>
$$
\gamma_n(s)=e^{2\pi i n s},\qquad 0\leq s\leq1,\quad n\in\mathbb Z,
$$
have <winding number> $n$: positive $n$ counts counterclockwise turns, negative $n$ counts clockwise turns, and $n=0$ gives the constant <based loop>. Every <based loop> on the <circle> is joined by a <based homotopy> to exactly one $\gamma_n$. Consequently
$$
\pi_1(S^1,1)\cong(\mathbb Z,+),\qquad
[\gamma_m*\gamma_n]=[\gamma_{m+n}].
$$
This is Hatcher's Theorem 1.7 in https://pi.math.cornell.edu/~hatcher/AT/ATch1.pdf#page=9[Algebraic Topology, §1.1, pp. 29–30]. For example, two counterclockwise turns followed by one clockwise turn represent $2+(-1)=1$, rather than three turns. The winding graph in the figure above makes the signed arithmetic visible.
A proof outline uses the <covering map> $q:\mathbb R\to S^1$, $q(u)=e^{2\pi i u}$. By the <path lifting theorem>, a <based loop> has a unique lift starting at zero and ending at an <integer> $n$. Interpolating its lift to $s\mapsto ns$ produces a <based homotopy> to $\gamma_n$. The <homotopy lifting theorem for a covering map> keeps that endpoint unchanged during a <based homotopy>, proving uniqueness.
In the filled <disk>, every <based loop> $\gamma$ at $x_0$ has a <based homotopy> to the constant <based loop>:
$$
K(s,t)=(1-t)\gamma(s)+t x_0.
$$
Both endpoints remain $x_0$, and the <convex set> property of the <disk> keeps every intermediate point inside. Thus $\pi_1(D^2,x_0)$ is the <trivial group>. Together with the <deformation retraction> above, this computes three concrete answers: the filled <disk> has trivial <fundamental group>, while the <circle> and the <annulus> have <infinite cyclic groups>.
Maps have computable algebraic shadows too. The <continuous map> $p_n:S^1\to S^1$, $p_n(z)=z^n$, fixes $1$ and sends $\gamma_k$ to $\gamma_{nk}$. Its induced <group homomorphism> is therefore $k\mapsto nk$, and its <mapping degree> is $n$. For $n=2$ the image consists of even <integers>, so this map cannot be a <homotopy equivalence>. Composing $p_2$ with $p_{-3}$ gives $p_{-6}$: map composition multiplies degrees, whereas concatenating <based loops> adds winding numbers. Distinct degrees distinguish these maps up to <homotopy>; this calculation does not classify arbitrary <topological spaces>.
Historically, Poincaré's 1895 Analysis Situs introduced the <fundamental group>; its original §12 explicitly names it. The https://analysis-situs.pages.math.cnrs.fr/site/p-12-Groupe-fondamental.html[CNRS transcription of Poincaré's §12, Groupe fondamental] provides a direct view of this early formulation.
Back to article page
