Eilenberg–Maclane spectrum

In mathematics, specifically algebraic topology, there is a distinguished class of spectra called Eilenberg–Maclane spectra for any Abelian group [1]pg 134. 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.

Definition

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For a fixed abelian group   let   denote the set of Eilenberg–MacLane spaces

 

with the adjunction map coming from the property of loop spaces of Eilenberg–Maclane spaces: namely, because there is a homotopy equivalence

 

we can construct maps   from the adjunction   giving the desired structure maps of the set to get a spectrum. This collection is called the Eilenberg–Maclane spectrum of  [1]pg 134.

Properties

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Using the Eilenberg–Maclane spectrum   we can define the notion of cohomology of a spectrum   and the homology of a spectrum  [2]pg 42. Using the functor

 

we can define cohomology simply as

 

Note that for a CW complex  , the cohomology of the suspension spectrum   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. Note that instead of  , we take   for some Abelian group  , we recover the usual (co)homology with coefficients in the abelian group   and denote it by  .

Mod-p spectra and the Steenrod algebra

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For the Eilenberg–Maclane spectrum   there is an isomorphism

 

for the p-Steenrod algebra  .

Tools for computing Adams resolutions

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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. These depend on the following properties of Eilenberg–Maclane spectra. We define a generalized Eilenberg–Maclane spectrum   as a finite wedge of suspensions of Eilenberg–Maclane spectra  , so

 

Note that for   and a spectrum  

 

so it shifts the degree of cohomology classes. For the rest of the article   for some fixed abelian group  

Equivalence of maps to K

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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  .

Constructing a surjection

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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  , there is always a surjection

 

of Abelian groups.

Steenrod-module structure on cohomology of spectra

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For a spectrum   taking the wedge   constructs a spectrum which is homotopy equivalent to a generalized Eilenberg–Maclane space with one wedge summand for each   generator or  . In particular, it gives the structure of a module over the Steenrod algebra   for  . This is because the equivalence stated before can be read as

 

and the map   induces the  -structure.

See also

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References

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  1. ^ a b Adams, J. Frank (John Frank) (1974). Stable homotopy and generalised homology. Chicago: University of Chicago Press. ISBN 0-226-00523-2. OCLC 1083550.
  2. ^ a b Ravenel, Douglas C. (1986). Complex cobordism and stable homotopy groups of spheres. Orlando: Academic Press. ISBN 978-0-08-087440-1. OCLC 316566772.
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