mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, especially in the area of

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

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 John Frank Adams (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory. Life He was born in Woolwich, a suburb in south-east London, and attended Bedford School. He began researc ...

and Sergei Novikov, are of particular interest because the Adams (–Novikov) spectral sequence converges to them.

Definition

The

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

of stable homotopy classes

,Y /math> between two spectra''X'' and ''Y'' can be given a

filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...

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 \cdots \to X_n = Y

such that each individual map

X_i\to X_

induces the zero map in some fixed homology theory ''E''. If ''E'' is ordinary mod-''p''

homology Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor * Sequence homology, biological homology between DNA, RNA, or protein sequences *Homologous chrom ...

, this filtration is called the Adams filtration, otherwise the Adams–Novikov filtration. Homotopy theory {{topology-stub