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In
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 ...
, specifically
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 ...
, there is a distinguished class of spectra called Eilenberg–Maclane spectra HA for any
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 commut ...
Apg 134. Note, this construction can be generalized to
commutative rings In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
R as well from its underlying Abelian group. These are an important class of spectra because they model ordinary integral cohomology and cohomology with coefficients in an abelian group. In addition, they are a lift of the homological structure in the
derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
D(\mathbb) of abelian groups in the homotopy category of spectra. In addition, these spectra can be used to construct resolutions of spectra, called Adams resolutions, which are used in the construction of the
Adams spectral sequence In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now ca ...
.


Definition

For a fixed abelian group A let HA denote the set of
Eilenberg–MacLane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. ...
s
\
with the adjunction map coming from the property of loop spaces of Eilenberg–Maclane spaces: namely, because there is a homotopy equivalence
K(A,n-1)\simeq \Omega K(A,n)
we can construct maps \Sigma K(A,n-1) \to K(A,n) from the adjunction Sigma(X),Ysimeq ,\Omega(Y)/math> giving the desired structure maps of the set to get a spectrum. This collection is called the Eilenberg–Maclane spectrum of Apg 134.


Properties

Using the Eilenberg–Maclane spectrum H\mathbb we can define the notion of cohomology of a spectrum X and the homology of a spectrum Xpg 42. Using the functor
,H\mathbb\textbf^ \to \text
we can define cohomology simply as
H^*(E) = ,H\mathbb/math>
Note that for a CW complex X, the cohomology of the suspension spectrum \Sigma^\infty X recovers the cohomology of the original space X. Note that we can define the dual notion of homology as
H_*(X) = \pi_*(E\wedge X) = mathbb,E\wedge X/math>
which can be interpreted as a "dual" to the usual hom-tensor adjunction in spectra. Note that instead of H\mathbb, we take HA for some Abelian group A, we recover the usual (co)homology with coefficients in the abelian group A and denote it by H^*(X;A).


Mod-''p'' spectra and the Steenrod algebra

For the Eilenberg–Maclane spectrum H\mathbb/p there is an isomorphism
H^*(H\mathbb/p, \mathbb/p) \cong \mathbb/p,H\mathbb/p\cong \mathcal_p
for the p- Steenrod algebra \mathcal_p.


Tools for computing Adams resolutions

One of the quintessential tools for computing stable homotopy groups is the Adams spectral sequence. In order to make this construction, the use of Adams resolutions are employed. These depend on the following properties of Eilenberg–Maclane spectra. We define a generalized Eilenberg–Maclane spectrum K as a finite wedge of suspensions of Eilenberg–Maclane spectra HA_i, so
K := \Sigma^HA_1\wedge\cdots\wedge\Sigma^HA_n
Note that for \Sigma^kHA and a spectrum X
,\Sigma^kHA\cong H^(X;A)
so it shifts the degree of cohomology classes. For the rest of the article HA_i = HA for some fixed abelian group A


Equivalence of maps to ''K''

Note that a homotopy class f \in ,K/math> represents a finite collection of elements in H^*(X;A). Conversely, any finite collection of elements in H^*(X;A) is represented by some homotopy class f \in ,K/math>.


Constructing a surjection

For a locally finite collection of elements in H^*(X;A) generating it as an abelian group, the associated map f: X \to K induces a surjection on cohomology, meaning if we evaluate these spectra on some topological space S, there is always a surjection
f^*:K(S) \to X(S)
of Abelian groups.


Steenrod-module structure on cohomology of spectra

For a spectrum X taking the wedge X\wedge H\mathbb/p constructs a spectrum which is homotopy equivalent to a generalized Eilenberg–Maclane space with one wedge summand for each \mathbb/p generator or H^*(X;\mathbb/p). In particular, it gives the structure of a module over the Steenrod algebra \mathcal_p for H^*(X). This is because the equivalence stated before can be read as
H^*(X\wedge H\mathbb/p) \cong \mathcal_p\otimes H^*(X)
and the map f: X \to X \wedge H\mathbb/p induces the \mathcal_p-structure.


See also

*
Adams spectral sequence In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now ca ...
*
Spectrum (topology) In algebraic topology, a branch of mathematics, a spectrum is an object representable functor, representing a Cohomology#Generalized cohomology theories, generalized cohomology theory. Every such cohomology theory is representable, as follows from B ...
*
Homotopy groups of spheres In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure o ...


References


External links


Complex cobordism and stable homotopy groups of spheres

The Adams Spectral Sequence
{{DEFAULTSORT:Eilenberg-Maclane spectrum Algebraic topology Homological algebra Topology