In 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 invariants that classify topological spaces up to homeomorphism, though usually most classif ...

known as

stable homotopy theory In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the ...

, 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 research ...

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 Spectra may refer to: * The plural of spectrum, conditions or values that vary over a continuum, especially the colours of visible light * ''Spectra'' (journal), of the Museum Computer Network (MCN) * The plural of spectrum (topology), an object ...

''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 filte ...

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 In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topol ...

''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 chromo ...

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