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.

Examples

References