group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $A$ an abelian group, the Eilenberg-Mac Lane spectrum $H A$ is the spectrum that represents the ordinary cohomology theory/ordinary homology with coefficients in $A$.
By default $A$ is taken to be the integers and hence “the Eilenberg-MacLane spectrum” is $H \mathbb{Z}$, representing integral cohomology.
As a symmetric spectrum: (Schwede 12, example I.2.6)
As a symmetric monoidal ∞-groupoid the Eilenberg–Mac Lane spectrum is the abelian group of integers (under addition)
Here the set $\mathbb{Z}$ is regarded as a discrete groupoid (one object per integer, no nontrivial morphisms) whose symmetric monoidal structure is that given by the additive group structure on the integers.
Accordingly, the infinite tower of suspensions induced by this is the sequence of ∞-groupoids
that in this case happen to be strict omega-groupoids. The strict omega-groupoid $\mathbf{B}^n \mathbb{Z}$ has only identity $k$-morphisms for all $k$, except for $k = n$, where $\mathrm{Mor}_n(\mathbf{B}^n \mathbb{Z}) = \mathbb{Z}$ are the endomorphisms of the unique identity $(n-1)$-morphism.
The strict ∞-groupoid $\mathbf{B}^n \mathbb{Z}$ is the one given under the Dold-Kan correspondence by the crossed complex of groupoids that is trivial everywhere and has the group $\mathbb{Z}$ in degree $n$.
Under the Quillen equivalence
between infinity-groupoids and topological spaces (see homotopy hypothesis) this sequence of suspensions of $\mathbb{Z}$ maps to the sequence of Eilenberg–Mac Lane spaces
that give the Eilenberg–Mac Lane spectrum
its name.
chromatic level | complex oriented cohomology theory | E-∞ ring/A-∞ ring | real oriented cohomology theory |
---|---|---|---|
0 | ordinary cohomology | Eilenberg-MacLane spectrum $H \mathbb{Z}$ | HZR-theory |
0th Morava K-theory | $K(0)$ | ||
1 | complex K-theory | complex K-theory spectrum $KU$ | KR-theory |
first Morava K-theory | $K(1)$ | ||
first Morava E-theory | $E(1)$ | ||
2 | elliptic cohomology | elliptic spectrum $Ell_E$ | |
second Morava K-theory | $K(2)$ | ||
second Morava E-theory | $E(2)$ | ||
algebraic K-theory of KU | $K(KU)$ | ||
3 …10 | K3 cohomology | K3 spectrum | |
$n$ | $n$th Morava K-theory | $K(n)$ | |
$n$th Morava E-theory | $E(n)$ | BPR-theory | |
$n+1$ | algebraic K-theory applied to chrom. level $n$ | $K(E_n)$ (red-shift conjecture) | |
$\infty$ | complex cobordism cohomology | MU | MR-theory |
ordinary homology spectra split: For $S$ any spectrum and $H A$ an Eilenberg-MacLane spectrum, then the smash product $S\wedge H A$ (the $A$-ordinary homology spectrum) is non-canonically equivalent to a product of EM-spectra (hence a wedge sum of EM-spectra in the finite case).
Textbook accounts include
Frank Adams, part III, section 2 of Stable homotopy and generalised homology, 1974
Stanley Kochmann, section 3.5 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Lecture notes include
Peter May, chapter 22 of A concise course in algebraic topology (pdf)
John Rognes, section 3.2 3.4 of The Adams spectral sequence, 2012 (pdf)
Stefan Schwede, Example I.2.7 in Symmetric spectra, 2012 (pdf)