Definition (fundamental group):

Let ${\displaystyle X}$ be a topological space and let ${\displaystyle x_{0}\in X}$. The fundamental group of ${\displaystyle X}$ based at ${\displaystyle x_{0}}$ is the group of homotopy equivalence classes of loops at ${\displaystyle x_{0}}$ leaving the endpoints fixed

${\displaystyle \pi _{1}(X,x_{0}):=\{[\gamma ]|\gamma :[0,1]\to X,\gamma (0)=\gamma (1)=x_{0}\}}$

where the group law is given by concatenation of paths.