The fundamental group counts loops up to stretching
Fix a base point and gather every path that leaves and returns. Treat two loops as the same when a continuous deformation turns one into the other without tearing. Concatenate them, and the resulting group—Poincaré's 1895 invention—records the space's one-dimensional holes.
In algebraic topology this is the first and simplest homotopy group. It is a homotopy invariant: homotopy-equivalent spaces, and hence homeomorphic ones, have isomorphic fundamental groups. Intuitively it measures whether loops wrap around holes. Combining loops means travel the first, then the second; the constant loop is the identity, and reversing a loop gives its inverse. Working with homotopy classes rather than the huge raw loop space keeps the object computable. Two loops count as equivalent precisely when a continuous deformation carries one into the other without tearing the path.
Henri Poincaré defined the notion in his 1895 paper "Analysis situs." The idea grew from Riemann surfaces and work by Riemann, Poincaré, and Klein on monodromy of complex functions, and it completely classifies closed surfaces topologically. A loop based at the chosen point is a continuous map from the circle (or unit interval with ends identified) into the space. A homotopy is a continuous interpolation between two such loops. Concatenation is well-defined on classes even though literal path products need reparametrization to look associative.
Changing the base point in a path-connected space yields an isomorphic group, so the choice is largely bookkeeping. Nonabelian fundamental groups are common—think of figure-eights—and that noncommutativity is why homology, which abelianizes, is often easier to compute. Still, for many spaces the fundamental group is the sharpest elementary invariant: it detects holes that no amount of local Euclidean looking will reveal. From Riemann surfaces to modern homotopy theory, tracking deformable loops remains the first algebraic snapshot of shape.
Source: Fundamental group
