Math 2.0

Algebraic topology Stable homotopy theory and generalized cohomology Object

Eilenberg–MacLane spectrum

Builds on SpectrumEilenberg–MacLane spaceAbelian groupSingular cohomologyStable homotopy category

Description

Let GG be an abelian group. The Eilenberg–MacLane spectrum HGHG is the spectrum with (HG)n=K(G,n)(HG)_n = K(G, n), an Eilenberg–MacLane space, for n≥0n \ge 0. Its structure maps ΣK(G,n)→K(G,n+1)\Sigma K(G, n) \to K(G, n+1) are adjoint to weak homotopy equivalences K(G,n)→ΩK(G,n+1)K(G, n) \to \Omega K(G, n+1), so HGHG is an Ω\Omega-spectrum. Its homotopy groups are π0(HG)=G\pi_0(HG) = G and πk(HG)=0\pi_k(HG) = 0 for k≠0k \neq 0, and these properties determine HGHG up to isomorphism in the stable homotopy category.

Remarks

The spectrum HGHG represents ordinary singular cohomology with coefficients in GG: for a CW complex XX and n≥0n \ge 0, H~n(X;G)≅[Σ∞X,ΣnHG]\tilde H^n(X; G) \cong [\Sigma^\infty X, \Sigma^n HG]. It also represents homology: H~n(X;G)≅πn(HG∧X)\tilde H_n(X; G) \cong \pi_n(HG \wedge X). For pp prime, the graded ring [H𝔽p,Σ∗H𝔽p][H\F_p, \Sigma^* H\F_p] of stable self-maps is the mod pp Steenrod algebra. If RR is a commutative ring, the product of RR makes HRHR a commutative ring spectrum.

Reading path

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Binary operation, definition, no evidence levelBinary operationdefinitionEilenberg–MacLane space, concept, no evidence levelEilenberg–MacLane spa…conceptGroup, concept, no evidence levelGroupconceptSingular cohomology, concept, no evidence levelSingular cohomologyconceptSpectrum, concept, no evidence levelSpectrumconceptStable homotopy category, concept, no evidence levelStable homotopy categ…conceptCommutativity, definition, no evidence levelCommutativitydefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionEilenberg–MacLane spectrum, object, no evidence levelEilenberg–MacLane spe…object
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Binary operation definition
  2. Eilenberg–MacLane space concept
  3. Group concept
  4. Singular cohomology concept
  5. Spectrum concept
  6. Stable homotopy category concept
  7. Commutativity definition
  8. Abelian group definition
  9. Eilenberg–MacLane spectrum object

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Eilenberg%E2%80%93MacLane_spectrum
About this page
TagXJPYJT9Z1P
URIm2://XJPYJT9Z1P
Kindobject
WritingW0 stub
MSC55P42, 55P20
Instance ofSpectrum
UsesSpectrum, Eilenberg–MacLane space, Abelian group, Singular cohomology, Stable homotopy category
Same as
Changed2 November 2026

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