For questions about or involving fundamental groups of topological spaces, as well as related topics such as fundamental groupoids and étale fundamental groups.
For a topological space $X$, we can form the set of all loops in $X$ based at a given point $x_0 \in X$, i.e. $\mathcal{F}(X, x_0) = \{\alpha : [0, 1] \to X \mid \alpha\ \text{continuous}, \alpha(0) = \alpha(1) = x_0\}$.
Let $\alpha, \beta$ be two loops in $X$ based at $x_0$; we say $\alpha$ is homotopic to $\beta$ if there exists a continuous map $H : [0, 1]\times [0, 1] \to X$ such that $H(s, 0) = \alpha(s)$ for all $s \in [0, 1]$, $H(s, 1) = \beta(s)$ for all $s \in [0, 1]$, and $H(0, t) = H(1, t) = x_0$ for all $t \in [0, 1]$. We call $H$ a homotopy of paths (or a homotopy relative to $\{0, 1\}$).
Setting $\alpha \sim \beta$ when $\alpha$ is homotopic to $\beta$, we find that $\sim$ is an equivalence relation on $\mathcal{F}(X, x_0)$. We define $\pi_1(X, x_0)$ to be the quotient space, i.e. $\pi_1(X, x_0) = \mathcal{F}(X, x_0)/\sim$. The set $\pi_1(X, x_0)$ obtains a group structure via concatenation of paths. We call $\pi_1(X, x_0)$ the fundamental group of $X$ at $x_0$.
If $x_0, x_1 \in X$ are in the same path-connected component of $X$, then $\pi(X, x_0)$ and $\pi(X, x_1)$ are (non-canonically) isomorphic. In particular, if $X$ is connected, we often supress the base point and just write $\pi_1(X)$ for the fundamental group of $X$.
The fundamental groups at different basepoints $X$ can be assembled into a single object called the fundamental groupoid of $X$ (often written as $\pi_{\leq 1}(X)$ or $\Pi_1(X)$). This is the groupoid whose objects are points of $X$ and in which a morphism $x\to y$ is a homotopy class of paths from $x$ to $y$, with composition given by concatenation of paths. For each $x\in X$, the automorphism group of $x$ as an object of the fundamental groupoid is the fundamental group $\pi_1(X,x)$.
