Eilenberg–MacLane spaces

E634842

homotopy-theoretic construction mathematical object topological space

Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.

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Observed surface forms (2)

Surface form	Occurrences
Eilenberg–MacLane space	1
Postnikov systems	1

Statements (47)

Predicate	Object
instanceOf	homotopy-theoretic construction ⓘ mathematical object ⓘ topological space ⓘ
appearsIn	homological algebra ⓘ spectra representing ordinary cohomology ⓘ stable homotopy theory ⓘ
characterizedBy	having exactly one nontrivial homotopy group ⓘ
construction	can be constructed as CW complexes with prescribed homotopy groups ⓘ can be obtained via bar constructions in some cases ⓘ
definedBy	π_n(X) ≅ G and π_k(X) = 0 for k ≠ n ⓘ
field	algebraic topology ⓘ homotopy theory ⓘ
generalizationOf	circle as K(ℤ,1) ⓘ infinite projective spaces as K(G,n) for suitable G and n ⓘ
hasHomotopyGroup	π_k(K(G,n)) = 0 for k ≠ n ⓘ π_n(K(G,n)) ≅ G ⓘ
hasProperty	can be taken as CW complexes ⓘ corepresent cohomology functors in the homotopy category ⓘ homotopy type determined by group G and integer n ⓘ path-connected when n ≥ 1 ⓘ serves as a classifying space for cohomology with coefficients in G ⓘ serves as building block for decomposing spaces via Postnikov towers ⓘ unique up to homotopy equivalence for fixed G and n ⓘ
introducedIn	mid 20th century ⓘ
namedAfter	Samuel Eilenberg NERFINISHED ⓘ Saunders Mac Lane NERFINISHED ⓘ
notation	K(G,n) NERFINISHED ⓘ
parameter	group G ⓘ integer n ≥ 0 ⓘ
relatedConcept	Postnikov system NERFINISHED ⓘ classifying space ⓘ cohomology group ⓘ homotopy group ⓘ
relatedTo	Brown representability theorem NERFINISHED ⓘ ordinary cohomology theories ⓘ
representsFunctor	[X,K(G,n)] ≅ H^n(X;G) ⓘ
specialCase	K(0,n) is contractible for all n ⓘ K(ℤ,1) is homotopy equivalent to the circle S^1 ⓘ K(ℤ,2) is homotopy equivalent to the infinite complex projective space ℂP^∞ ⓘ K(ℤ/2ℤ,1) is homotopy equivalent to ℝP^∞ ⓘ
usedFor	classifying cohomology classes ⓘ constructing Postnikov towers ⓘ constructing spectral sequences in algebraic topology ⓘ defining cohomology operations ⓘ defining singular cohomology ⓘ representing cohomology theories ⓘ studying homotopy types ⓘ

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Samuel Eilenberg → notableWork → Eilenberg–MacLane spaces ⓘ

Samuel Eilenberg → knownFor → Eilenberg–MacLane spaces ⓘ

Samuel Eilenberg → notableConcept → Eilenberg–MacLane spaces ⓘ

this entity surface form: Eilenberg–MacLane space

Characteristic Classes → hasSubject → Eilenberg–MacLane spaces ⓘ

this entity surface form: Postnikov systems