Põhigruppideks

Põhigruppideks, also known as fundamental groups, are a concept in algebraic topology that provide a way to study the topological properties of spaces. They are defined for path-connected topological spaces and are homotopy invariants, meaning that homeomorphic spaces have isomorphic fundamental groups. The fundamental group of a space X, denoted as π1(X), is the set of homotopy classes of loops based at a point x0 in X, with the operation of concatenation of loops. This group captures information about the loops in the space and their homotopy classes, which can be used to distinguish between different topological spaces. For example, the fundamental group of the circle S1 is isomorphic to the integers Z, while the fundamental group of the torus T2 is isomorphic to Z x Z. The concept of fundamental groups was introduced by Poincaré in the late 19th century and has since become a fundamental tool in topology and geometry.