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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

As a symmetric spectrum: (Schwede 12, example I.2.6)

Properties

As symmetric monoidal \infty-groupoids

As a symmetric monoidal ∞-groupoid the Eilenberg–Mac Lane spectrum is the abelian group of integers (under addition)

E EM:=. E_{EM} := \mathbb{Z} \,.

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

,B,B 2,B 3, \mathbb{Z}, \mathbf{B} \mathbb{Z}, \mathbf{B}^2 \mathbb{Z}, \mathbf{B}^3 \mathbb{Z}, \cdots

that in this case happen to be strict omega-groupoids. The strict omega-groupoid B n\mathbf{B}^n \mathbb{Z} 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 n\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 nn.

[B n] =([B n] n+1[B n] n[B n] n1[B n] 1[B n] 0) =(****). \begin{aligned} [\mathbf{B}^n \mathbb{Z}] &= ( \cdots \to [\mathbf{B}^n\mathbb{Z}]_{n+1} \to [\mathbf{B}^n\mathbb{Z}]_{n} \to [\mathbf{B}^n\mathbb{Z}]_{n-1} \cdots \to [\mathbf{B}^n\mathbb{Z}]_{1} \stackrel{\to}{\to} [\mathbf{B}^n\mathbb{Z}]_{0}) \\ &= ( \cdots \to {*} \to \mathbb{Z} \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 \mathbb{Z} maps to the sequence of Eilenberg–Mac Lane spaces

|B n|K(,n) |\mathbf{B}^n \mathbb{Z}| \simeq K(\mathbb{Z}, n)

that give the Eilenberg–Mac Lane spectrum

(E n):=(K(,n)) (E_n) := (K(\mathbb{Z},n))

its name.

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

Revised on April 15, 2016 09:58:32 by Urs Schreiber (131.220.184.222)