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, named after Frank Adams and Sergei Novikov, are of particular interest because the Adams (–Novikov) spectral sequence converges to them.
Definition.
The group of stable homotopy classes formula_1 between two spectra "X" and "Y" can be given a filtration by saying that a map formula_2 has filtration "n" if it can be written as a composite of maps 
such that each individual map formula_4 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.