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, named after Frank Adams and Sergei Novikov, are of particular interest because the Adams (–Novikov) spectral sequence converges to them.^[1][2]

Definition

The group of stable homotopy classes ${\displaystyle [X,Y]}$ between two spectra X and Y can be given a filtration by saying that a map ${\displaystyle f\colon X\to Y}$ has filtration n if it can be written as a composite of maps

${\displaystyle X=X_{0}\to X_{1}\to \cdots \to X_{n}=Y}$

such that each individual map ${\displaystyle X_{i}\to X_{i+1}}$ 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.