nullhomotopic

Nullhomotopic is a term in algebraic topology describing a property of a continuous map. A map f: X -> Y is nullhomotopic if it is homotopic to a constant map. In other words, there exists a continuous function H: X × I -> Y with H(x,0) = f(x) for all x in X and H(x,1) = y0 for some fixed point y0 in Y. If X and Y are based spaces and f is basepoint-preserving, the homotopy can be chosen so that the basepoint remains fixed throughout.

Several equivalent viewpoints appear in practice. If Y is contractible, every map X -> Y is nullhomotopic. If X is contractible, every map f: X -> Y is nullhomotopic as well, since f factors through the contraction of X to a point. Conversely, f is nullhomotopic precisely when it represents the trivial element in the homotopy class [X, Y]. For X = S^n, a map f: S^n -> Y is nullhomotopic exactly when the corresponding element of the n-th homotopy group π_n(Y) is trivial.

Examples help illustrate the concept. Constant maps are always nullhomotopic. Any map into a contractible target space is nullhomotopic. A map f: S^1 -> S^1 has a trivial degree if and only if it is nullhomotopic. In the context of basepointed spaces, nullhomotopic maps determine the zero element in the set of based homotopy classes [X, Y].