homotopy of maps

Imagine two roads drawn between the same two towns. Sometimes you can slide one road continuously across the open countryside until it lies exactly on top of the other, never lifting your pen and never leaving the map. When that sliding is possible we say the two roads are 'the same up to deformation.' Homotopy of maps is the precise version of this idea: two continuous maps are homotopic if one can be continuously deformed into the other.

Precisely, let f, g: X -> Y be continuous maps. A homotopy from f to g is a continuous map H: X x [0,1] -> Y with H(x, 0) = f(x) and H(x, 1) = g(x) for all x. Think of the second coordinate t in [0,1] as time: at t = 0 the picture is f, at t = 1 it is g, and in between H gives a continuous movie deforming one into the other. The single map H being continuous on the whole product X x [0,1] is exactly what makes the deformation 'continuous in both space and time.' We write f ≃ g. Often one fixes a subspace A and demands H(a, t) be constant in t for a in A; this is homotopy rel A, used for instance to pin down basepoints when building pi_1.

Homotopy is an equivalence relation on the set of maps X -> Y: it is reflexive (the constant-in-time homotopy), symmetric (run the movie backwards, t -> 1 - t), and transitive (play one movie then the other, reparametrising the two halves of [0,1] and using the pasting lemma for continuity). The resulting equivalence classes, written [X, Y], are the real objects of homotopy theory. The whole subject is the study of spaces and maps remembered only up to this relation — a deliberate blurring that throws away rigid geometry but keeps the features that survive bending and stretching.