homotopy

homotopy, 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 endpoints are called homotopic if one can be continuously deformed into the other leaving the end points fixed and remaining within its defined region. In part A of the figure, the shaded region has a hole in it; f and g are homotopic paths, but g′ is not homotopic to f or g since g′ cannot be deformed into f or g without passing through the hole and leaving the region.

More formally, homotopy involves defining a path by mapping points in the interval from 0 to 1 to points in the region in a continuous manner—that is, so that neighbouring points on the interval correspond to neighbouring points on the path. A homotopy map h(x, t) is a continuous map 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. The map 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 part A of the figure; the points f(x) and g(x) are joined by a straight line segment, and for each fixed value of t, h(x, t) defines a path joining the same two endpoints.

Of particular interest are the homotopic paths starting and ending at a single point (see part B of the figure). 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 can be given an algebraic structure called a group, the fundamental group of the region, whose structure varies according to the type of region. In a region with no holes, all closed paths are homotopic and the fundamental group consists of a single element. In a region with a single hole, all paths are homotopic that wind around the hole the same number of times. In the figure, paths a and b are homotopic, as are paths c and d, but path e is not homotopic to any of the other paths.

One defines in the same way homotopic paths and the fundamental group of regions in three or more dimensions, as well as on general manifolds. In higher dimensions one can also define higher-dimensional homotopy groups.