Order of a homotopy invariant in the stable case

TL;DR

This paper investigates the order of homotopy invariants in the stable case, establishing that under certain conditions, the order equals the degree with respect to the Curtis filtration.

Contribution

It proves that for finite CW-complexes in the stable case, the order of a homotopy invariant matches its degree in the Curtis filtration.

Findings

Order of homotopy invariants equals their Curtis degree in the stable case.

Provides a characterization of the order for invariants on finite CW-complexes.

Connects the order of invariants with algebraic filtration degrees.

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].