Adams spectral sequence
Encyclopedia
In mathematics
, the Adams spectral sequence is a spectral sequence
introduced by . Like all spectral sequences, it is a computational tool; it relates homology
theory to what is now called stable homotopy theory
. It is a reformulation using homological algebra
, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan
and Jean-Pierre Serre
.
es. The ordinary cohomology groups H*(X) are understood to mean H*(X; Z/pZ).
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 Sn, these maps form the nth homotopy group
of Y. A more reasonable (but still very difficult!) goal is to understand [X, Y], the 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
; more modern treatments of this topic begin with the concept of a spectrum
. 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.)
[X, Y] 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 [X, Y], we look at cohomology: send [X, Y] 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
, and more still than a graded ring
(via the cup product
). The representability of the cohomology functor makes H*(X) a module
over the algebra of its stable cohomology operation
s, the Steenrod algebra
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 Fp. But we can now consider the derived functors of Hom in the category of A-modules, Ext
Ar(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.
and differentials of bidegree (r, r-1).
Adams' original use for his spectral sequence was the first proof of the Hopf invariant
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
.
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, Milnor and Novikov used the Adams spectral sequence to compute the coefficient ring of complex cobordism
. Further, Milnor and Wall used the spectral sequence to prove Thom's conjecture on the structure of the oriented cobordism
ring: two oriented manifolds are cobordant if and only if their Pontryagin and Stiefel-Whitney numbers agree.
. 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Adams spectral sequence is a spectral sequence
Spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations...
introduced by . Like all spectral sequences, it is a computational tool; it relates homology
Homology
Homology may refer to:* Homology , analogy between human beliefs, practices or artifacts owing to genetic or historical connections* Homology , any characteristic of biological organisms that is derived from a common ancestor....
theory to what is now called stable homotopy theory
Stable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
. It is a reformulation using homological algebra
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan
Henri Cartan
Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the son of the French mathematician Élie Cartan.-Life:...
and Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
.
Motivation
For everything below, we need to once and for all fix a prime p. All spaces are assumed to be CW complexCW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
es. The ordinary cohomology groups H*(X) are understood to mean H*(X; Z/pZ).
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 Sn, these maps form the nth homotopy group
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, which records information about loops in a space...
of Y. A more reasonable (but still very difficult!) goal is to understand [X, Y], the 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
Stable homotopy theory
In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor...
; more modern treatments of this topic begin with the concept of a spectrum
Spectrum (homotopy theory)
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, any of which gives a context for the same stable homotopy theory....
. 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.)
[X, Y] 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 [X, Y], we look at cohomology: send [X, Y] 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
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
, and more still than a graded ring
Ring
Ring may refer to:*Ring , a decorative ornament worn on fingers, toes, or around the arm or neck-Computing:* Ring , a layer of protection in computer systems...
(via the cup product
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 graded commutative product operation in cohomology, turning the cohomology of a space X into a...
). The representability of the cohomology functor makes H*(X) a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over the algebra of its stable cohomology operation
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 cohomology operation should be a natural transformation from...
s, the Steenrod algebra
Steenrod algebra
In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.For a given prime number p, the Steenrod algebra Ap is the graded Hopf algebra over the field Fp of order p, consisting of all stable cohomology operations for mod p...
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 Fp. But we can now consider the derived functors of Hom in the category of A-modules, Ext
Ext functor
In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics.- Definition and computation :...
Ar(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
For X and Y spaces of finite type, with Y a finite dimensional CW-complex, there is a spectral sequence, called the classical Adams spectral sequence, converging to the p-torsion in [X, Y], with E2-term given by- E2r,s = ExtAr,s(H*(Y), H*(X)),
and differentials of bidegree (r, r-1).
Calculations
The sequence itself is not an algorithmic device, but lends itself to problem solving in particular cases.Adams' original use for his spectral sequence was the first proof of the Hopf invariant
Hopf invariant
In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between spheres.- Motivation :In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map\eta\colon S^3 \to S^2,...
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-theoryK-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
.
The Thom isomorphism theorem
Thom space
In mathematics, the Thom space, Thom complex, or Pontryagin-Thom construction of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space....
relates differential topology to stable homotopy theory, and this is where the Adams spectral sequence found its first major use: in 1960, Milnor and Novikov used the Adams spectral sequence to compute the coefficient ring of complex cobordism
Complex cobordism
In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories...
. Further, Milnor and Wall used the spectral sequence to prove Thom's conjecture on the structure of the oriented cobordism
Cobordism
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary of a manifold. Two manifolds are cobordant if their disjoint union is the boundary of a manifold one dimension higher. The name comes...
ring: two oriented manifolds are cobordant if and only if their Pontryagin and Stiefel-Whitney numbers agree.
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, often complex bordism or Brown–Peterson cohomologyBrown–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 idempotent:...
. 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.