nLab Eilenberg-Mac Lane spectrum

Idea

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.

Definition

Incarnations

As symmetric/orthogonal spectra

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.

Definition

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

$$\underset{k \in \mathbb{N}}{\sqcup} A^k \times (S^n)^k \longrightarrow A[S^n]_\ast$$

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)

Proposition

For $A$ a countable abelian group, then the reduced $A$-linearization $A[S^n]_\ast$ (def. ) is an Eilenberg-MacLane space, in that its homotopy groups are

$$\pi_q(A[S^n]_\ast) \simeq \left\{ \array{ A & if \; q = n \\ \ast & otherwise } \right.$$

(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)

Definition

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

  $$(H A)_V \coloneqq A[S^V]_\ast$$

  being the reduced $A$-linearization (def. ) of the representation sphere $S^V$;

  hence for $V = \mathbb{R}^n$ then

  $$(H A)_n = A[S^n]_\ast$$

• $O(V)$-action on $A[S^V]_\ast$ induced from the canonical $O(V)$-action on $S^V$ (representation sphere);

• structure maps

  $$\sigma_{V,W} \;\colon\; (H A)_V \wedge S^W \longrightarrow (H A)_{V\oplus W}$$

  hence

  $$A[S^V] \wedge S^W \longrightarrow A[S^{V \oplus W}]$$

  given by

  $$\left( \left( \underset{i}{\sum} a_i x_i \right), y \right) \mapsto \underset{i}{\sum} a_i (x_i, y) \,.$$

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

1. taking the multiplication

   $$(H R)_{V_1} \wedge (H R)_{V_2} = R[S^{V_1}]_\ast \wedge R[S^{V_2}]_\ast \longrightarrow R[S^{V_1 \oplus V_2}] = (H R)_{V_1 \oplus V_2}$$

   to be given by

   $$\left( \left( \underset{i}{\sum} a_i x_i \right) , \left( \underset{j}{\sum} b_j y_j \right) \right) \;\mapsto\; \underset{i,j}{\sum} (a_i \cdot b_j)(x_i, y_j)$$

2. taking the unit maps

   $$S^V \longrightarrow A[S^V]_\ast = (H R)_V$$

   to be given by the canonical inclusion of generators

   $$x \mapsto 1 x \,.$$

(Schwede 12, example I.1.14, Schwede 15, V, costruction 3.21)

As symmetric monoidal $\infty$-groupoids

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

$$A, \mathbf{B} A, \mathbf{B}^2 A, \mathbf{B}^3 A, \cdots$$

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 $\mathrm{Mor}_n(\mathbf{B}^n A) = A$ 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 $n$.

$$\begin{aligned} [\mathbf{B}^n A] &= ( \cdots \to [\mathbf{B}^n A]_{n+1} \to [\mathbf{B}^n A]_{n} \to [\mathbf{B}^n A]_{n-1} \cdots \to [\mathbf{B}^n A]_{1} \stackrel{\to}{\to} [\mathbf{B}^n A]_{0}) \\ &= ( \cdots \to {*} \to A \to {*} \cdots \to {*} \stackrel{\to}{\to} {*}) \end{aligned} \,.$$

Under the Quillen equivalence

$$|-| : \infty Grpds \to Top$$

between infinity-groupoids and topological spaces (see homotopy hypothesis) this sequence of suspensions of $A$ maps to the sequence of Eilenberg?Mac Lane spaces?

$$|\mathbf{B}^n A| \simeq K(A, n) \,.$$

Properties

Chromatic filtration

chromatic homotopy theory

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

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).

Fibrant-cofibrant models

MO comment

Related concepts

• generalized Eilenberg-MacLane spectrum

• Moore spectrum

• cohomology theory, ordinary cohomology

References

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)

As symmetric spectra

• 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)

as orthogonal spectra:

• Stefan Schwede, construction V 3.21 in Global homotopy theory, 2015