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Eilenberg-Mac Lane spectrum

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Contents

Idea

For AA an abelian group, the Eilenberg-Mac Lane spectrum HAH A is the spectrum that represents the ordinary cohomology theory/ordinary homology with coefficients in AA.

By default AA is taken to be the integers and hence “the Eilenberg-MacLane spectrum” is HH \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 AA an abelian group and nn \in \mathbb{N}, the reduced AA-linearization A[S n] *A[S^n]_\ast of the n-sphere S nS^n is the topological space, whose underlying set is the quotient

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

of the tensor product with AA of the free abelian group on the underlying set of S nS^n, by the relation that identifies every formal linear combination of the (any fixed) basepoint of S nS^n with 0. The topology is the induced quotient topology (of the disjoint union of product topological spaces, where AA is equipped with the discrete topology).

(Aguilar-Gitler-Prieto 02, def. 6.4.20)

Proposition

For AA a countable abelian group, then the reduced AA-linearization A[S n] *A[S^n]_\ast (def. 1) is an Eilenberg-MacLane space, in that its homotopy groups are

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

(in particular for n1n \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 AA a countable abelian group, then the orthogonal spectrum incarnation of the Eilenberg-MacLane spectrum HAH A is the orthogonal spectrum with

  • components spaces

    (HA) VA[S V] * (H A)_V \coloneqq A[S^V]_\ast

    being the reduced AA-linearization (def. 1) of the representation sphere S VS^V;

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

    (HA) n=A[S n] * (H A)_n = A[S^n]_\ast
  • O(V)O(V)-action on A[S V] *A[S^V]_\ast induced from the canonical O(V)O(V)-action on S VS^V (representation sphere);

  • structure maps

    σ V,W:(HA) VS W(HA) VW \sigma_{V,W} \;\colon\; (H A)_V \wedge S^W \longrightarrow (H A)_{V\oplus W}

    hence

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

    given by

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

The incarnation of HAH A as a symmetric spectrum is the same, with the group action of O(n)O(n) replaced by the subgroup action of the symmetric group Σ(n)O(n)\Sigma(n) \hookrightarrow O(n).

If RR is a commutative ring, then the Eilenberg-MacLane spectrum HRH R becomes a commutative orthogonal ring spectrum (or symmetric ring spectrum, respectively) by

  1. taking the multiplication

    (HR) V 1(HR) V 2=R[S V 1] *R[S V 2] *R[S V 1V 2]=(HR) V 1V 2 (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

    ((ia ix i),(jb jy j))i,j(a ib j)(x i,y j) \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 VA[S V] *=(HR) V S^V \longrightarrow A[S^V]_\ast = (H R)_V

    to be given by the canonical inclusion of generators

    x1x. 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 HAH A simply is the group AA.

Here the set AA 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,BA,B 2A,B 3A, 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 B nA\mathbf{B}^n A has only identity kk-morphisms for all kk, except for k=nk = n, where Mor n(B n)=\mathrm{Mor}_n(\mathbf{B}^n \mathbb{Z}) = \mathbb{Z} are the endomorphisms of the unique identity (n1)(n-1)-morphism.

The strict ∞-groupoid B nA\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 nn.

[B nA] =([B nA] n+1[B nA] n[B nA] n1[B nA] 1[B nA] 0) =(*A***). \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

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

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

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

Properties

Chromatic filtration

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum HH \mathbb{Z}HZR-theory
0th Morava K-theoryK(0)K(0)
1complex K-theorycomplex K-theory spectrum KUKUKR-theory
first Morava K-theoryK(1)K(1)
first Morava E-theoryE(1)E(1)
2elliptic cohomologyelliptic spectrum Ell EEll_E
second Morava K-theoryK(2)K(2)
second Morava E-theoryE(2)E(2)
algebraic K-theory of KUK(KU)K(KU)
3 …10K3 cohomologyK3 spectrum
nnnnth Morava K-theoryK(n)K(n)
nnth Morava E-theoryE(n)E(n)BPR-theory
n+1n+1algebraic K-theory applied to chrom. level nnK(E n)K(E_n) (red-shift conjecture)
\inftycomplex cobordism cohomologyMUMR-theory

Ordinary homology spectra split

ordinary homology spectra split: For SS any spectrum and HAH A an Eilenberg-MacLane spectrum, then the smash product SHAS\wedge H A (the AA-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

References

Textbook accounts include

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

as orthogonal spectra:

Revised on July 1, 2016 15:13:24 by Urs Schreiber (131.220.184.222)