In
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
are understood to mean
.
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
, 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
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 of
finite type, meaning
for
and
is a finitely generated Abelian group in each degree. Then, there is a spectral sequence
such that
#
for
the mod
Steenrod algebra
# For
of finite type,
is a bigraded group associated with a filtration of
(the
p-adic integers)
Note that this implies for
, this computes the
-torsion of the homotopy groups of the
sphere spectrum, i.e. the stable homotopy groups of the spheres. Also, because for any CW-complex
we can consider the suspension spectrum
, this gives the statement of the previous formulation as well.
This statement generalizes a little bit further by replacing the
-module
with the cohomology groups
for some connective spectrum
(or topological space
). This is because the construction of the spectral sequence uses a "free" resolution of
as an
-module, hence we can compute the Ext groups with
as the second entry. We therefore get a spectral sequence with
-page given by
which has the convergence property of being isomorphic to the graded pieces of a filtration of the
-torsion of the stable homotopy group of homotopy classes of maps between
and
, that is
Spectral sequence for the stable homotopy groups of spheres
For example, if we let both spectra be the sphere spectrum, so
, then the Adams spectral sequence has the convergence property
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 information
pp 23–25. Also note that we can rewrite
, so the
-page is
We include this calculation information below for
.
Ext terms from the resolution
Given the
Adams resolutionwe have the
-terms as
for the graded Hom-groups. Then the
-page can be written as
so the degree of
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
th Adams differential always goes to the left 1, and up
. That is,
.
Examples with Eilenberg–Maclane spectra
Some of the simplest calculations are with
Eilenberg–Maclane spectra such as
and
.
[ For the first case, we have the pagegiving a collapsed spectral sequence, hence . This can be rewritten asgiving the -page. For the other case, note there is a cofiber sequencewhich ends up giving a splitting in cohomology, so as -modules. Then, the -page of can be read asThe expected -page is].
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 .
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: 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 we can compute several terms explicitly, giving some of the first stable homotopy groups of spheres.[ For this amounts to looking at the -page withThis can be done by first looking at the Adams resolution of . Since is in degree , we have a surjectionwhere has a generator in degree denoted . The kernel consists of all elements for admissible monomials generating , hence we have a mapand we denote each of the generators mapping to in the direct sum as , and the rest of the generators as for some . For example,Notice that the last two elements of map to the same element, which follows from the Adem relations. Also, there are elements in the kernel, such as sincebecause of the Adem relation. We call the generator of this element in , . We can apply the same process and get a kernel , resolve it, and so on. When we do, we get an -page which looks like
which can be expanded by computer up to degree with relative ease. Using the found generators and relations, we can calculate the -page with relative ease. Sometimes homotopy theorists like to rearrange these elements by having the horizontal index denote and the vertical index denote giving a different type of diagram for the -page][pg 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