Adams filtration

In mathematics, especially in the area of algebraic topology known as stable homotopy theory, the Adams filtration and the Adams-Novikov filtration allow a stable homotopy group to be understood as built from layers, the nth layer containing just those maps which require at most n auxiliary spaces in order to be a composition of homologically trivial maps. These filtrations are of particular interest because the Adams (-Novikov) spectral sequence converges to them.

Definition

The group of stable homotopy classes [X,Y] between two spectra X and Y can be given a filtration by saying that a map f: X → Y has filtration n if it can be written as a composite of maps X = X₀ → X₁ → ... → X_n = Y such that each individual map X_i → X_i+1 induces the zero map in some fixed homology theory E. If E is ordinary mod-p homology, this filtration is called the Adams filtration, otherwise the Adams-Novikov filtration.