In mathematics, specifically algebraic topology, there is a distinguished class of spectra called Eilenberg–Maclane spectra
Note, this construction can be generalized to commutative rings
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
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.
For a fixed abelian group
with the adjunction map coming from the property of loop spaces of Eilenberg–Maclane spaces: namely, because there is a homotopy equivalence
giving the desired structure maps of the set to get a spectrum.
we can define the notion of cohomology of a spectrum
{\displaystyle [-,H\mathbb {Z} ]:{\textbf {Spectra}}^{op}\to {\text{GrAb}}}
recovers the cohomology of the original space
Note that we can define the dual notion of homology as
which can be interpreted as a "dual" to the usual hom-tensor adjunction in spectra.
, we recover the usual (co)homology with coefficients in the abelian group
One of the quintessential tools for computing stable homotopy groups is the Adams spectral sequence.
[2] In order to make this construction, the use of Adams resolutions are employed.
as a finite wedge of suspensions of Eilenberg–Maclane spectra
so it shifts the degree of cohomology classes.
for some fixed abelian group
Note that a homotopy class
represents a finite collection of elements in
Conversely, any finite collection of elements in
is represented by some homotopy class
For a locally finite collection of elements in
generating it as an abelian group, the associated map
induces a surjection on cohomology, meaning if we evaluate these spectra on some topological space
constructs a spectrum which is homotopy equivalent to a generalized Eilenberg–Maclane space with one wedge summand for each
In particular, it gives the structure of a module over the Steenrod algebra