Adams filtration explained

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

[X,Y] between two spectra X and Y can be given a filtration by saying that a map f\colon X\to Y has filtration n if it can be written as a composite of maps X = X _{0} \to X _{1} \to … \to X _{n} = Y such that each individual map 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.

Notes and References

1. Web site: mypapers. people.math.rochester.edu.
2. https://scholar.harvard.edu/files/rastern/files/adamsspectralsequence.pdf&ved=2ahUKEwjI-JDstJOEAxU4WEEAHThuAOI4ChAWegQIGxAB&usg=AOvVaw3_Ga18v-jBF18FS-U4KgPV