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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Adams spectral sequence is a
spectral sequence In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they h ...
introduced by which computes the stable homotopy groups of
topological spaces In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called
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 ...
. It is a reformulation using
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, and an extension, of a technique called 'killing homotopy groups' applied by the French school of
Henri Cartan Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of c ...
and
Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inau ...
.


Motivation

For everything below, once and for all, we fix a prime ''p''. All spaces are assumed to be
CW complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
es. The ordinary
cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
s H^*(X) are understood to mean H^*(X; \Z/p\Z). The primary goal of algebraic topology is to try to understand the collection of all maps, up to homotopy, between arbitrary spaces ''X'' and ''Y''. This is extraordinarily ambitious: in particular, when ''X'' is S^n, these maps form the ''n''th
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
of ''Y''. A more reasonable (but still very difficult!) goal is to understand the set
, Y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> of maps (up to homotopy) that remain after we apply the suspension functor a large number of times. We call this the collection of stable maps from ''X'' to ''Y''. (This is the starting point of
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 ...
; more modern treatments of this topic begin with the concept of a
spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
. Adams' original work did not use spectra, and we avoid further mention of them in this section to keep the content here as elementary as possible.) The set
, Y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> turns out to be an abelian group, and if ''X'' and ''Y'' are reasonable spaces this group is finitely generated. To figure out what this group is, we first isolate a prime ''p''. In an attempt to compute the ''p''-torsion of
, Y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math>, we look at cohomology: send
, Y The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
/math> to Hom(''H''*(''Y''), ''H''*(''X'')). This is a good idea because cohomology groups are usually tractable to compute. The key idea is that H^*(X) is more than just a graded
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
, and more still than a graded
ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...
(via the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p+q. This defines an associative (and distributive) graded commutative product opera ...
). The representability of the cohomology functor makes ''H''*(''X'') a module over the algebra of its stable
cohomology operation In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if ''F'' is a functor defining a cohomology theory, then a cohomo ...
s, the
Steenrod algebra Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology. Life He was born in Dayton, Ohio, and educated at Miami University and University of ...
''A''. Thinking about ''H''*(''X'') as an ''A''-module forgets some cup product structure, but the gain is enormous: Hom(''H''*(''Y''), ''H''*(''X'')) can now be taken to be ''A''-linear! A priori, the ''A''-module sees no more of 'X'', ''Y''than it did when we considered it to be a map of vector spaces over F''p''. But we can now consider the derived functors of Hom in the category of ''A''-modules,
Ext Ext, ext or EXT may refer to: * Ext functor, used in the mathematical field of homological algebra * Ext (JavaScript library), a programming library used to build interactive web applications * Exeter Airport Exeter Airport , formerly ''Ex ...
''A''''r''(''H''*(''Y''), ''H''*(''X'')). These acquire a second grading from the grading on ''H''*(''Y''), and so we obtain a two-dimensional "page" of algebraic data. The Ext groups are designed to measure the failure of Hom's preservation of algebraic structure, so this is a reasonable step. The point of all this is that ''A'' is so large that the above sheet of cohomological data contains all the information we need to recover the ''p''-primary part of 'X'', ''Y'' which is homotopy data. This is a major accomplishment because cohomology was designed to be computable, while homotopy was designed to be powerful. This is the content of the Adams spectral sequence.


Classical formulation


Formulation for computing homotopy groups of spectra

The classical Adams spectral sequence can be stated for any connective spectrum X of finite type, meaning \pi_i(X)=0 for i < 0 and \pi_i(X) is a finitely generated Abelian group in each degree. Then, there is a spectral sequence E_*^(X) such that # E_2^ = \text_^(H^*(X), \Z/p) for A_p the mod p Steenrod algebra # For X of finite type, E_\infty^ is a bigraded group associated with a filtration of \pi_*(X)\otimes \Z_p (the p-adic integers) Note that this implies for X = \mathbb, this computes the p-torsion of the homotopy groups of the sphere spectrum, i.e. the stable homotopy groups of the spheres. Also, because for any CW-complex Y we can consider the suspension spectrum \Sigma^\infty Y, this gives the statement of the previous formulation as well. This statement generalizes a little bit further by replacing the \mathcal_p-module \mathbb/p with the cohomology groups H^*(Y) for some connective spectrum Y (or topological space Y). This is because the construction of the spectral sequence uses a "free" resolution of H^*(X) as an \mathcal_p-module, hence we can compute the Ext groups with H^*(Y) as the second entry. We therefore get a spectral sequence with E_2-page given by
E_2^ = \text_^(H^*(X),H^*(Y))
which has the convergence property of being isomorphic to the graded pieces of a filtration of the p-torsion of the stable homotopy group of homotopy classes of maps between X and Y, that is
E_2^ \Rightarrow \pi_^( ,Y\otimes \mathbb_


Spectral sequence for the stable homotopy groups of spheres

For example, if we let both spectra be the sphere spectrum, so X = Y = \mathbb, then the Adams spectral sequence has the convergence property
E_2^ = \text_^(H^*(\mathbb),H^*(\mathbb)) \Rightarrow \pi_(\mathbb)\otimes \mathbb_p
giving a technical tool for approaching a computation of the stable homotopy groups of spheres. It turns out that many of the first terms can be computed explicitly from purely algebraic informationpp 23–25. Also note that we can rewrite H^*(\mathbb) = \mathbb/p, so the E_2-page is
E_2^ = \operatorname_^(\mathbb/p,\mathbb/p) \Rightarrow \pi_(\mathbb)\otimes \mathbb_p
We include this calculation information below for p=2.


Ext terms from the resolution

Given the Adams resolution
\cdots \to H^*(F_2) \to H^*(F_1) \to H^*(F_0) \to H^*(X)
we have the E_1-terms as
E_1^ = \operatorname^t_(H^*(F_s),H^*(Y))
for the graded Hom-groups. Then the E_1-page can be written as
E_1 = \begin 3 & \vdots & \vdots & \vdots\\ 2 & \text^2(H^*(F_0), H^*(Y)) & \text^2(H^*(F_1), H^*(Y)) & \text^2(H^*(F_2), H^*(Y)) & \cdots \\ 1 & \text^1(H^*(F_0), H^*(Y)) & \text^1(H^*(F_1), H^*(Y)) & \text^1(H^*(F_2), H^*(Y)) & \cdots \\ 0 & \text^0(H^*(F_0), H^*(Y)) & \text^0(H^*(F_1), H^*(Y)) & \text^0(H^*(F_2), H^*(Y)) & \cdots \\ \hline & 0 & 1 & 2 \end
so the degree of s can be thought of how "deep" in the Adams resolution we go before we can find the generators.


Calculations

The sequence itself is not an algorithmic device, but lends itself to problem solving in particular cases.


Grading of the Differential

The rth Adams differential always goes to the left 1, and up r. That is,
d_r \colon E_r^ \to E_r^.


Examples with Eilenberg–Maclane spectra

Some of the simplest calculations are with Eilenberg–Maclane spectra such as X = H\Z and X = H\Z/(p^k). For the first case, we have the E_1 page
E_1^ = \begin \Z/p & \text t = s \\ 0 & \text \end
giving a collapsed spectral sequence, hence E_1 = E_\infty. This can be rewritten as
\text^_(H^*(H\Z), \Z/p) = \begin \Z/p & \text t = s \\ 0 & \text t \neq s \end
giving the E_2-page. For the other case, note there is a cofiber sequence
H\Z\xrightarrow H\Z \to H\Z/p^k \to \Sigma H\Z
which ends up giving a splitting in cohomology, so H^*(H\Z/p^k) = H^*(H\Z)\oplus H^*(\Sigma H\Z) as \mathcal_p-modules. Then, the E_2-page of H^*(H\Z/p) can be read as
E_2^ = \begin \Z/p & \text t-s = 0,1 \\ 0 & \text \end
The expected E_\infty-page is
E_\infty^ = \begin \mathbb/p^k & \text t = s \\ 0 & \text \end.
The only way for this spectral sequence to converge to this page is if is there are non-trivial differentials supported on every element with Adams grading (s, s+1).


Other applications

Adams' original use for his spectral sequence was the first proof of the
Hopf invariant In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between ''n''-spheres. __TOC__ Motivation In 1931 Heinz Hopf used Clifford parallels to construct the '' Hopf map'' :\eta\colon S^ ...
1 problem: \R^n admits a division algebra structure only for ''n'' = 1, 2, 4, or 8. He subsequently found a much shorter proof using cohomology operations in
K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
. The Thom isomorphism theorem relates differential topology to stable homotopy theory, and this is where the Adams spectral sequence found its first major use: in 1960,
John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
and Sergei Novikov used the Adams spectral sequence to compute the coefficient ring of complex cobordism. Further, Milnor and C. T. C. Wall used the spectral sequence to prove Thom's conjecture on the structure of the oriented
cobordism In mathematics, cobordism is a fundamental equivalence relation on the class of compact space, compact manifolds of the same dimension, set up using the concept of the boundary (topology), boundary (French ''wikt:bord#French, bord'', giving ''cob ...
ring: two oriented manifolds are cobordant if and only if their Pontryagin and Stiefel–Whitney numbers agree.


Stable homotopy groups of spheres

Using the spectral sequence above for X = Y = \mathbb we can compute several terms explicitly, giving some of the first stable homotopy groups of spheres. For p=2 this amounts to looking at the E_2-page with
E_2^ = \text_^(\mathbb/2,\mathbb/2)
This can be done by first looking at the Adams resolution of \mathbb/2. Since \mathbb/2 is in degree 0, we have a surjection
\mathcal_2\cdot \iota \to \mathbb/2
where \mathcal_2 has a generator in degree 0 denoted \iota. The kernel K_0 consists of all elements Sq^I\iota for admissible monomials Sq^I generating \mathcal_2, hence we have a map
\bigoplus_ \mathcal_2\cdot Sq^I\iota \to K_0
and we denote each of the generators mapping to Sq^i\iota in the direct sum as \alpha_i, and the rest of the generators as Sq^I\alpha_j for some j. For example,
\begin \alpha_1 \mapsto Sq^1\iota & &Sq^2\alpha_1 \mapsto Sq^\iota \\ \alpha_2 \mapsto Sq^2\iota & & Sq^1\alpha_2 \mapsto Sq^3\iota \\ \alpha_4 \mapsto Sq^4\iota & & Sq^3\alpha_1 \mapsto Sq^\iota \\ \alpha_8 \mapsto Sq^8 & & Sq^2\alpha_2 \mapsto Sq^\iota \end
Notice that the last two elements of \alpha_i map to the same element, which follows from the Adem relations. Also, there are elements in the kernel, such as Sq^1\alpha_1 since
Sq^1\alpha_1 \mapsto Sq^1Sq^1\iota = 0
because of the Adem relation. We call the generator of this element in F_2, \beta_2. We can apply the same process and get a kernel K_1, resolve it, and so on. When we do, we get an E_1-page which looks like
E_1^ = \begin \vdots & \vdots & \vdots & \vdots\\ 4 & Sq^4\iota, Sq^\iota & Sq^\alpha_1, Sq^\alpha_1, Sq^2\alpha_2, \alpha_4 & Sq^2\beta_2 & \cdots \\ 3 & Sq^3\iota, Sq^\iota &Sq^2\alpha_1, Sq^1\alpha_2 & Sq^1\beta_2& \cdots \\ 2 & Sq^2\iota & \alpha_2, Sq^1\alpha_1 & \beta_2 & \cdots \\ 1 & Sq^1\iota & \alpha_1 & 0 & \cdots \\ 0 & \iota & 0 & 0 & \cdots \\ \hline & 0 & 1 & 2 \end
which can be expanded by computer up to degree 100 with relative ease. Using the found generators and relations, we can calculate the E_2-page with relative ease. Sometimes homotopy theorists like to rearrange these elements by having the horizontal index denote s and the vertical index denote t - s giving a different type of diagram for the E_2-pagepg 21. See the diagram above for more information.


Generalizations

The Adams–Novikov spectral sequence is a generalization of the Adams spectral sequence introduced by where ordinary cohomology is replaced by a
generalized cohomology theory In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
, often complex bordism or
Brown–Peterson cohomology In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by , depending on a choice of prime ''p''. It is described in detail by . Its representing spectrum is denoted by BP. Complex cobordism and Quillen's idempot ...
. This requires knowledge of the algebra of stable cohomology operations for the cohomology theory in question, but enables calculations which are completely intractable with the classical Adams spectral sequence.


See also

* Postnikov system *
Steenrod algebra Norman Earl Steenrod (April 22, 1910October 14, 1971) was an American mathematician most widely known for his contributions to the field of algebraic topology. Life He was born in Dayton, Ohio, and educated at Miami University and University of ...
*
Spectrum (topology) In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomo ...
* Adams resolution * Ravenel's conjectures


References

* * * * * * * .


Overviews of computations

* – computes all Adams spectral sequences for the stable homotopy groups of spheres up to degree 90


Higher-order terms

* * *


External links

* *


Notes

{{Reflist Homotopy theory Spectral sequences