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.
We discuss the model of Eilenberg-MacLane spectra as symmetric spectra and orthogonal spectra. To that end, notice the following model for Eilenberg-MacLane spaces.
For $A$ an abelian group and $n \in \mathbb{N}$, the reduced $A$-linearization $A[S^n]_\ast$ of the n-sphere $S^n$ is the topological space, whose underlying set is the quotient
of the tensor product with $A$ of the free abelian group on the underlying set of $S^n$, by the relation that identifies every formal linear combination of the (any fixed) basepoint of $S^n$ with 0. The topology is the induced quotient topology (of the disjoint union of product topological spaces, where $A$ is equipped with the discrete topology).
(Aguilar-Gitler-Prieto 02, def. 6.4.20)
For $A$ a countable abelian group, then the reduced $A$-linearization $A[S^n]_\ast$ (def. 1) is an Eilenberg-MacLane space, in that its homotopy groups are
(in particular for $n \geq 1$ then there is a unique connected component and hence we need not specify a basepoint for the homotopy group).
(Aguilar-Gitler-Prieto 02, corollary 6.4.23)
For $A$ a countable abelian group, then the orthogonal spectrum incarnation of the Eilenberg-MacLane spectrum $H A$ is the orthogonal spectrum with
components spaces
being the reduced $A$-linearization (def. 1) of the representation sphere $S^V$;
hence for $V = \mathbb{R}^n$ then
$O(V)$-action on $A[S^V]_\ast$ induced from the canonical $O(V)$-action on $S^V$ (representation sphere);
structure maps
hence
given by
The incarnation of $H A$ as a symmetric spectrum is the same, with the group action of $O(n)$ replaced by the subgroup action of the symmetric group $\Sigma(n) \hookrightarrow O(n)$.
If $R$ is a commutative ring, then the Eilenberg-MacLane spectrum $H R$ becomes a commutative orthogonal ring spectrum (or symmetric ring spectrum, respectively) by
taking the multiplication
to be given by
taking the unit maps
to be given by the canonical inclusion of generators
(Schwede 12, example I.1.14, Schwede 15, V, costruction 3.21)
Under the identification of connective spectra with “abelian infinity-groups” the Eilenberg-MacLane spectrum $H A$ simply is the group $A$.
Here the set $A$ 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 $\mathbf{B}^n A$ 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 $\mathbf{B}^n A$ 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 $A$ maps to the sequence of Eilenberg–Mac Lane spaces
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.1.14 in Symmetric spectra, 2012 (pdf)
John Greenlees, example 3.6, example 4.16 in Spectra for commutative algebraists (pdf)