HOMOTOPY

Figure 1: (A) Homotopic and nonhomotopic paths; (B) closed paths in mathematics, a way of classifying geometric regions by studying the different types of paths that can be drawn in the region. Two paths with common end points are called homotopic if one can be continuously deformed into the other without leaving its end points or passing outside its defined region. In Figure 1A, f and g are homotopic paths; since g cannot be deformed into f or g without leaving the region defined by the single hole, g is not homotopic to f or g. Figure 1: (A) Homotopic and nonhomotopic paths; (B) closed paths More formally, homotopy involves defining a function that corresponds to points in the interval from 0 to 1 with points on the path in a continuous mannerthat is, so that neighbouring points on the interval correspond to neighbouring points on the path. A homotopy function h(x,t) is a function that associates with two suitable paths, f (x) and g(x), a function of two variables x and t that is equal to f (x) when t = 0 and equal to g(x) when t = 1, and corresponds to the intuitive idea of a gradual deformation without leaving the region as t changes from 0 to 1. For example, h(x,t) = (1 - t)f(x) + tg (x) is a homotopic function for paths f and g in Figure 1A. Figure 1: (A) Homotopic and nonhomotopic paths; (B) closed paths Figure 2: Homotopy classes Of particular interest are the homotopic paths starting and ending at a single point (see Figure 1B). The class of all such paths homotopic to each other in a given geometric region is called a homotopy class. The set of all such classes forms an algebraic entity called a group, the structure of which varies according to the type of region. In a region with no holes, all closed paths are homotopic. In a region with a single hole, all paths are homotopic that wind around the hole the same number of times. In Figure 2, paths a and b are homotopic, as are paths c and d, but path e is not homotopic to any of the other paths.