Path (topology) - Homotopy of Paths

Homotopy of Paths

Paths and loops are central subjects of study in the branch of algebraic topology called homotopy theory. A homotopy of paths makes precise the notion of continuously deforming a path while keeping its endpoints fixed.

Specifically, a homotopy of paths, or path-homotopy, in X is a family of paths f_t : I → X indexed by I such that

• f_t(0) = x₀ and f_t(1) = x₁ are fixed.
• the map F : I × I → X given by F(s, t) = f_t(s) is continuous.

The paths f₀ and f₁ connected by a homotopy are said to homotopic (or more precisely path-homotopic, to distinguish between the relation defined on all continuous functions between fixed spaces). One can likewise define a homotopy of loops keeping the base point fixed.

The relation of being homotopic is an equivalence relation on paths in a topological space. The equivalence class of a path f under this relation is called the homotopy class of f, often denoted .