Order of a homotopy invariant in the stable case

Abstract

Let X and Y be CW-complexes, U be an abelian group, and f:[X,Y]->U be a map (a homotopy invariant). We say that f has order at most r if the characteristic function of the r'th Cartesian power of the graph of a continuous map a:X->Y Z-linearly determines f([a]). Suppose that the CW-complex X is finite and we are in the stable case: dim X<2n-1 and Y is (n-1)-connected. We prove that then the order of f equals its degree with respect to the Curtis filtration of the group [X,Y].