# 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 *n*th 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* = *X*_{0} → *X*_{1} → ... → *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**.